Frobenius Algebras I: Basic Representation Theory

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EMS Textbooks in Mathematics EMS Textbooks in Mathematics is a series of books aimed at students or professional mathematicians seeking an introduction into a particular field. The individual volumes are intended not only to provide relevant techniques, results, and applications, but also to afford insight into the motivations and ideas behind the theory. Suitably designed exercises help to master the subject and prepare the reader for the study of more advanced and specialized literature. Jørn Justesen and Tom Høholdt, A Course In Error-Correcting Codes Markus Stroppel, Locally Compact Groups Peter Kunkel and Volker Mehrmann, Differential-Algebraic Equations Dorothee D. Haroske and Hans Triebel, Distributions, Sobolev Spaces, Elliptic Equations Thomas Timmermann, An Invitation to Quantum Groups and Duality Oleg Bogopolski, Introduction to Group Theory Marek Jarnicki und Peter Pflug, First Steps in Several Complex Variables: Reinhardt Domains Tammo tom Dieck, Algebraic Topology Mauro C. Beltrametti et al., Lectures on Curves, Surfaces and Projective Varieties Wolfgang Woess, Denumerable Markov Chains Eduard Zehnder, Lectures on Dynamical Systems. Hamiltonian Vector Fields and Symplectic Capacities

Andrzej Skowroński Kunio Yamagata

Frobenius Algebras I Basic Representation Theory

Authors: Andrzej Skowron´ ski Faculty of Mathematics and Computer Science Nicolaus Copernicus University Chopina 12/18 87-100 Torun´ Poland

Kunio Yamagata Department of Mathematics Tokyo University of Agriculture and Technology Nakacho 2-24-16, Koganei Tokyo 184-8588 Japan

E-mail: [email protected]

E-mail: [email protected]

2010 Mathematical Subject Classification (primary; secondary): 16-01; 13E10, 15A63, 15A69, 16Dxx, 16E30, 16G10, 16G20, 16G70, 16K20, 16W30, 51F15 Key words: Algebra, module, representation, quiver, ideal, radical, simple module, semisimple module, uniserial module, projective module, injective module, simple algebra, semisimple algebra, separable algebra, hereditary algebra, Nakayama algebra, Frobenius algebra, symmetric algebra, selfinjective algebra, Brauer tree algebra, enveloping algebra, Coxeter group, Coxeter graph, Hecke algebra, coalgebra, comodule, Hopf algebra, Hopf module, syzygy module, periodic module, periodic algebra, irreducible homomorphism, almost split sequence, Auslander–Reiten translation, Auslander–Reiten quiver, extension spaces, projective dimension, injective dimension, category, functor, Nakayama functor, Nakayama automorphism, Morita equivalence, Morita–Azumaya duality

ISBN 978-3-03719-102-6 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © European Mathematical Society 2012 Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum FLI C4 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: [email protected] Homepage: www.ems-ph.org Typeset using the authors’ TEX files: I. Zimmermann, Freiburg Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 987654321

To our wives Mira and Taeko and children Magda, Akiko, Ikuo and Taketo

Contents

Introduction I

ix

Algebras and modules 1 Algebras . . . . . . . . . . . . . . . . . . . . . 2 Representations of algebras and modules . . . 3 The Jacobson radical . . . . . . . . . . . . . . 4 The Krull–Schmidt theorem . . . . . . . . . . 5 Semisimple modules . . . . . . . . . . . . . . 6 Semisimple algebras . . . . . . . . . . . . . . 7 The Jordan–Hölder theorem . . . . . . . . . . 8 Projective and injective modules . . . . . . . . 9 Hereditary algebras . . . . . . . . . . . . . . . 10 Nakayama algebras . . . . . . . . . . . . . . . 11 The Grothendieck group and the Cartan matrix 12 Exercises . . . . . . . . . . . . . . . . . . . .

II Morita theory 1 Categories and functors . . . . . . . . 2 Bimodules . . . . . . . . . . . . . . . 3 Tensor products of modules . . . . . 4 Adjunctions and natural isomorphisms 5 Progenerators . . . . . . . . . . . . . 6 Morita equivalence . . . . . . . . . . 7 Morita–Azumaya duality . . . . . . . 8 Exercises . . . . . . . . . . . . . . .

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123 123 125 133 140 151 157 175 188

III Auslander–Reiten theory 1 The radical of a module category . . . . . . . 2 The Harada–Sai lemma . . . . . . . . . . . . 3 The space of extensions . . . . . . . . . . . . 4 The Auslander–Reiten translations . . . . . . 5 The Nakayama functors . . . . . . . . . . . . 6 The Auslander–Reiten formulas . . . . . . . 7 Irreducible and almost split homomorphisms 8 Almost split sequences . . . . . . . . . . . . 9 The Auslander–Reiten quiver . . . . . . . . . 10 The Auslander theorem . . . . . . . . . . . .

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203 203 207 209 232 247 252 257 269 282 301

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viii

Contents

11 The Bautista–Smalø theorem . . . . . . . . . . . . . . . . . . . . . 312 12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 IV Selfinjective algebras 1 The Frobenius theorem . . . . . . . . . . 2 The Brauer–Nesbitt–Nakayama theorems 3 Frobenius algebras . . . . . . . . . . . . 4 Symmetric algebras . . . . . . . . . . . . 5 Simple algebras . . . . . . . . . . . . . . 6 The Nakayama theorems . . . . . . . . . 7 Non-Frobenius selfinjective algebras . . . 8 The syzygy functors . . . . . . . . . . . 9 The higher extension spaces . . . . . . . 10 Periodic modules . . . . . . . . . . . . . 11 Periodic algebras . . . . . . . . . . . . . 12 The Green–Snashall–Solberg theorems . 13 Dynkin and Euclidean graphs . . . . . . 14 Canonical mesh algebras of Dynkin type . 15 The Riedtmann–Todorov theorem . . . . 16 Exercises . . . . . . . . . . . . . . . . .

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332 332 336 345 355 368 375 386 392 402 414 427 442 447 452 455 470

V Hecke algebras 1 Finite reflection groups 2 Coxeter graphs . . . . 3 The Coxeter theorems 4 The Iwahori theorem . 5 Hecke algebras . . . . 6 Exercises . . . . . . .

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489 489 499 507 515 528 533

VI Hopf algebras 1 Coalgebras . . . . . . . . . . . . . . . . . . . . 2 Hopf algebras . . . . . . . . . . . . . . . . . . . 3 The Larson–Sweedler theorems . . . . . . . . . 4 The Radford theorem . . . . . . . . . . . . . . . 5 The Fischman–Montgomery–Schneider formula . 6 The module category . . . . . . . . . . . . . . . 7 Exercises . . . . . . . . . . . . . . . . . . . . .

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539 539 552 584 595 609 616 630

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Bibliography

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Index

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Introduction The major concern of this book is the representation theory of finite dimensional associative algebras with an identity over a field. In simplest terms, it is an approach to the problem of describing how a finite number of linear transformations can act simultaneously on a finite dimensional vector space over a field. The representation theory of finite dimensional algebras traces its origin to the middle part of the nineteenth century with Hamilton’s discovery of the quaternions, the first noncommutative field, and investigations of finite groups via their representations in matrix algebras over the field of complex numbers. The main achievements of the representation theory of algebras of the latter part of the nineteenth and the beginning of the twentieth century concerned the structure of semisimple finite dimensional algebras over fields and their representations. A new and fundamental view on the representation theory of finite dimensional algebras over fields came in the 1930s from the papers by Noether who gave the theory its modern setting by interpreting representations as modules. The module theoretical approach allowed one to apply in the representation theory of finite dimensional algebras the language and techniques of category theory and homological algebra. The modern representation theory of finite dimensional algebras over fields can be regarded as the study of the categories of their finite dimensional modules and the associated combinatorial and homological invariants. A prominent role in the representation theory of finite dimensional algebras over fields is played by the Frobenius algebras. This is a wide class of algebras containing the semisimple algebras, blocks of group algebras of finite groups, the Hecke algebras of finite Coxeter groups, the finite dimensional Hopf algebras, and the orbit algebras of the repetitive categories of algebras. The Frobenius algebras have their origin in the 1903 papers by Frobenius who discovered that the left and right regular representations of a finite dimensional algebra over a field, defined in terms of structure constants with respect to a fixed linear basis, are equivalent if and only if there is a very special invertible matrix intertwining both representations. Later Brauer, Nesbitt and Nakayama realized that the study of finite dimensional algebras with the property that the left and right regular representations are equivalent is crucial for a better understanding of the structure of nonsemisimple algebras and their modules, and called them Frobenius algebras. In a series of papers from 1937–1941, Brauer, Nesbitt and Nakayama established characterizations of Frobenius algebras which were independent of the choice of a linear basis of the algebra. In particular, we may say that a finite dimensional algebra A over a field K is a Frobenius algebra if there exists a nondegenerate K-bilinear form .; / W A A ! K which is associative, in the sense that .ab; c/ D .a; bc/ for all elements a, b, c of A. Moreover, if such a nondegenerate, associative, K-bilinear form is symmetric, A is called a symmetric algebra. We also mention that the Frobenius algebras are selfinjective

x

Introduction

algebras (projective and injective modules coincide), and the module category of every finite dimensional selfinjective algebra over a field is equivalent to the module category of a Frobenius algebra. The main aim of the book is to provide a comprehensive introduction to the representation theory of finite dimensional algebras over fields, via the representation theory of Frobenius algebras. The book is primarily addressed to graduate students starting research in the representation theory of algebras as well as mathematicians working in other fields. It is hoped that the book will provide a friendly access to the representation theory of finite dimensional algebras as the only prerequisite is a basic knowledge of linear algebra. We present complete proofs of all results exhibited in the book. Moreover, a rich supply of examples and exercises will help the reader to understand the theory presented in the book. We divide the book into two volumes. The aim of the first volume of the book is two fold. Firstly, it serves as a general introduction to basic results and techniques of the modern representation theory of finite dimensional algebras over fields, with special attention to the representation theory of Frobenius algebras. The second aim is to exhibit prominent classes of Frobenius algebras, or more generally selfinjective algebras. The first volume of the book is divided into six chapters, each of which is subdivided into sections. We start with Chapter I presenting background on the finite dimensional algebras over a field and their finite dimensional modules. In Chapter II we present the Morita equivalences and the Morita–Azumaya dualities for the module categories of finite dimensional algebras over fields. Chapter III is devoted to presenting background on the Auslander–Reiten theory of irreducible homomorphisms and almost split sequences, and the associated combinatorial and homological invariants. Chapter IV forms the heart of the first volume of the book and contains fundamental classical and recent results concerning the selfinjective algebras and their module categories. In Chapter V we present the classification of finite reflection groups of real Euclidean spaces via the associated Coxeter graphs and show that they provide a wide class of symmetric algebras over an arbitrary field, called the Hecke algebras. In the final Chapter VI we describe the basic theory of finite dimensional Hopf algebras over fields and show that they form a distinguished class of Frobenius algebras for which the Nakayama automorphisms are of finite order. The main aim of the second volume of the book, “Frobenius Algebras II. Orbit Algebras” is to study the Frobenius algebras as the orbit algebras of repetitive categories of finite dimensional algebras with respect to actions of admissible automorphism groups. In particular, we will introduce covering techniques which frequently allow us to reduce the representation theory of Frobenius algebras to the representation theory of algebras of small homological dimension. A prominent role in these investigations will be played by tilting theory and the authors theory of selfinjective algebras with deforming ideals.

Introduction

xi

We thank our universities for their continuous support as well as for financial support from Research Grant N N201 269135 of the Polish Ministry of Science and Higher Education and the Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (B) 21340003, allowing the realization of this book project during the authors’ visits in Toruń and Tokyo. We would like to express our deep gratitude to Jerzy Białkowski for typing and proper computer edition of all chapters of the book. We take also pleasure in thanking our younger colleagues Marta Błaszkiewicz, Alicja Jaworska, Maciej Karpicz, Marta Kwiecień and Adam Skowyrski for carefully reading parts of the book and their helpful corrections and suggestions. We also thank the European Mathematical Society Publishing House, in particular Manfred Karbe and Irene Zimmermann, for their very friendly cooperation.

Chapter I

Algebras and modules

This chapter is devoted to presenting background on finite dimensional algebras (associative, with an identity) over a field and their finite dimensional modules (representations). We present complete proofs of all results exhibited in this chapter. We start with a discussion of examples of finite dimensional algebras over a field, including the bound quiver algebras of the path algebras of finite quivers modulo admissible ideals, playing a prominent role in the modern representation theory of algebras. Moreover, we give a very useful interpretation of modules over bound quiver algebras as linear representations of quivers bound by the relations generating the related admissible ideals. Then we introduce and describe the basic properties of the (Jacobson) radical of an algebra and of a module, semisimple modules and algebras, the socle and the top of a module, projective modules and injective modules, hereditary algebras, uniserial modules and Nakayama algebras, the Grothendieck group and the Cartan matrix of an algebra. In particular, we prove the following classical results: the Krull–Schmidt decomposition theorem for modules, the Wedderburn structure theorem for semisimple algebras, Maschke’s theorem for group algebras, the Jordan–Hölder theorem on composition series of modules, the structure theorems on Nakayama algebras and their modules, as well as on projective modules and injective modules.

1 Algebras By a ring we mean a system .A; C; ; 0A ; 1A / consisting of a set A, two binary operations, the addition C W A A ! A, .a; b/ 7! a C b, the multiplication W A A ! A, .a; b/ 7! a b (simply ab), and two different elements 0A and 1A of A, such that the following conditions are satisfied: (i) .A; C; 0A / is an abelian group, with zero element 0A , (ii) .ab/c D a.bc/, (iii) a.b C c/ D ab C ac and .b C c/a D ba C ca, (iv) 1A a D a D a1A , for all elements a; b; c 2 A. Therefore, the multiplication is associative and both left and right distributive with respect to the addition, and 1A is the identity of A with respect to the multiplication. A ring A is commutative if ab D ba for all

2

Chapter I. Algebras and modules

elements a; b 2 A. We will frequently abbreviate 0 D 0A and 1 D 1A , if it will not lead to confusion. Throughout, we identify the ring .A; C; ; 0A ; 1A / with its underlying set A. A ring F is said to be a skew field (or division ring) if every nonzero element a in F is invertible with respect to the multiplication, that is, there exists b 2 F such that ab D 1F and ba D 1F . A skew field F is said to be a field if F is commutative. A field K is said to be algebraically closed if every nonconstant polynomial f .x/ in one variable x with coefficients in K has a root in K. (We refer to [Coh1] for a general theory of skew fields.) If A and B are rings, then a map f W A ! B is called a ring homomorphism if f .a C b/ D f .a/ C f .b/ and f .ab/ D f .a/f .b/, for all elements a; b 2 A, and f .0A / D 0B , f .1A / D 1B . Let K be a field. A K-algebra is a ring A with an additional structure of a (left) K-vector space, compatible with the multiplication of the ring, that is, such that .ab/ D .a/b D a.b/ for all 2 K and all a; b 2 A, and 1K 1A D 1A , where 1K is the identity of K and 1A is the identity of A (see [Noe1]). Observe that then 1K a D 1K .1A a/ D .1K 1A /a D 1A a D a for all a 2 A. Moreover, A has also a right K-vector space structure given by a D a.1A / D .a1A / D a for all 2 K and a 2 A. We will identify the field K with the K-subspace K1A D 1A K of A. A K-algebra A is said to be finite dimensional if the dimension dimK A of the K-vector space A is finite. If A and B are K-algebras, then a ring homomorphism f W A ! B is called a K-algebra homomorphism if f is a K-linear map. Two K-algebras A and B are called isomorphic if there is a K-algebra isomorphism f W A ! B, that is, a bijective K-algebra homomorphism. In this case, we write A Š B. A K-vector subspace B of a K-algebra A is called a K-subalgebra of A if the identity of A belongs to B and bb 0 2 B for all elements b; b 0 2 B. Observe that then B is a K-algebra with 1B D 1A . A K-vector subspace I of a K-algebra A is a left ideal of A (respectively, right ideal of A) if ax 2 I (respectively, xa 2 I ) for all elements x 2 I and a 2 A. A two-sided ideal of A (or simply an ideal of A) is a K-vector subspace I of A which is both a left ideal and a right ideal of A. Observe that if I is a two-sided ideal of A, then the quotient K-vector space A=I has a unique K-algebra structure such that the canonical surjective K-linear map W A ! A=I , a 7! aN D a C I , becomes a K-algebra homomorphism. A left, right, or two-sided ideal I of A is said to be proper if I ¤ A, or equivalently, 1A … I . Examples 1.1. (a) The field C D R ˚ Ri of complex numbers is a 2-dimensional R-algebra over the field R of real numbers, and an infinite dimensional Q-algebra over the field Q of rational numbers.

1. Algebras

3

(b) Let H D R ˚ Ri ˚ Rj ˚ Rk be the division R-algebra of quaternions (discovered in 1843 by W. R. Hamilton [Ham]) with the multiplication of basis elements: ij D j i D k;

j k D kj D i;

ki D i k D j;

i 2 D j 2 D k 2 D 1:

Then H is a 4-dimensional (noncommutative) R-algebra. Observe that H is not a C-algebra, because R is the set of all elements of H commuting with every element of H, and obviously R is not a C-algebra. On the other hand, H is a left C-vector space and a right C-vector space. We would like to mention that by a theorem of F. G. Frobenius [Fro0] from 1877, the only finite dimensional division R-algebras are R, C and H (for a proof see [Coh3], Corollary 5.4.2, or [DK]), Theorem 4.6.1. We also note that C is an algebraically closed field, and hence every finite dimensional division C-algebra is isomorphic to C (see Exercise 12.38). (c) Let K be a field and n a positive natural number. Then the set Mn .K/ of all square n n matrices over K is a K-algebra of dimension n2 . A K-basis of Mn .K/ is formed by the set of elementary matrices Eij , 1 i; j n, where Eij has coefficient 1 in the position .i; j / and the coefficient 0 elsewhere. We will denote by 0n the zero matrix and by In the identity matrix in Mn .K/. (d) For a positive natural number n, the subset 2 3 K 0 0 6K K 0 7 6 7 Tn .K/ D 6 : :: : : :: 7 : 4: : :5 : K K K of Mn .K/ consisting of all triangular matrices Œaij in Mn .K/ with zeros above the main diagonal is a K-subalgebra of Mn .K/. (e) Let G D .G; ; e/ be a finite group with identity element e and K a field. The group algebra ofP G with coefficients in K is the K-vector space KG consisting of all formal sums g2G g g, where g 2 K for all g 2 G, with the multiplication defined by the formula X X X g g h h D g h gh: g2G

h2G

g;h2G

Then KG is a K-algebra of dimension jGj (the order of G) and the identity 1KG D e. Moreover, the K-algebra KG is commutative if and only if the group G is abelian. For K D C, the group algebra CG has been introduced by F. G. Frobenius in the paper [Fro2] from 1897. (f) Assume that A1 and A2 are K-algebras. The product of the K-algebras A1 and A2 is the product A D A1 A2 of the K-vector spaces A1 and A2 ,

4


with the multiplication given by .a1 ; a2 /.b1 ; b2 / D .a1 b1 ; a2 b2 /, for all elements a1 ; b1 2 A1 , a2 ; b2 2 A2 . Then A D A1 A2 is a K-algebra with the identity element .1A1 ; 1A2 /. Clearly, if A1 and A2 are finite dimensional K-algebras, then A is a finite dimensional K-algebra, and dimK A D dimK A1 C dimK A2 . (g) Let A be a K-algebra. Then the opposite algebra Aop of A is the K-algebra whose underlying K-vector space is A, but the multiplication in Aop is defined by the formula a b D ba, for all elements a; b 2 A. The classical nineteenths century way of defining finite dimensional associative algebras over a field K (K. Weierstrass [Wei], R. Dedekind [Ded], T. Molien [Mol], F. G. Frobenius [Fro1]) involved some set of constants from the field K assumed to satisfy certain conditions. Let A be a finite dimensional K-algebra over a field K. Choose a basis a1 ; a2 ; : : : ; an of the K-vector space A. Then there exist elements ˛ij k 2 K, i; j; k 2 f1; : : : ; ng such that aj ak D

n X

˛ij k ai

iD1

for all j; k 2 f1; : : : ; ng, called the structure constants of A, with respect to the basis a1 ; a2 ; : : : ; an . Moreover, there exist 1 ; : : : ; n 2 K such that the identity 1A of A has the expression n X j aj : 1A D j D1

Lemma 1.2. (i) For all elements j; k; h; l 2 f1; : : : ; ng, the following equality holds: n n X X ˛ij k ˛hil D ˛ikl ˛hj i : iD1

iD1

(ii) For all elements i; k 2 f1; : : : ; ng, the following equalities hold: n X

j ˛ij k D ıik and

j D1

n X

j ˛ikj D ıik ;

j D1

where ıik is the Kronecker delta. Proof. (i) For j; k; l 2 f1; : : : ; ng, we have the equalities .aj ak /al D D

n X iD1 n X

n X ˛ij k ai al D ˛ij k .ai al /

˛ij k

iD1

iD1 n X hD1

n X n X ˛hil ah D ˛ij k ˛hil ah ; hD1

iD1

1. Algebras

5

and aj .ak al / D aj

n X

n X ˛ikl ai D ˛ikl aj ai

iD1

D

n X

˛ikl

iD1

iD1 n X

n X n X ˛hj i ah D ˛ikl ˛hj i ah :

hD1

hD1

iD1

Since a1 ; a2 ; : : : ; an form a K-basis of A, the required equalities are equivalent to the associativity conditions .aj ak /al D aj .ak al /, for all elements j; k; l 2 f1; : : : ; ng. (ii) For k 2 f1; : : : ; ng, we have the equalities 1A ak D

n X

n X j aj ak D j aj ak

j D1

D

n X

j

j D1 n X

j D1

n X n X ˛ij k ai D j ˛ij k ai ;

iD1

iD1

j D1

and ak 1A D ak

n X

j aj D

j D1

D

n X j D1

j

n X

j ak aj

j D1

n X iD1

n X n X ˛ikj ai D j ˛ikj ai : iD1

j D1

Again, the required equalities are equivalent to the identity conditions 1A ak D ak D ak 1A , for all elements k 2 f1; : : : ; ng. Observe also that the associativity .ab/c D a.bc/, for all a; b; c 2 A, is equivalent to the associativity conditions .aj ak /al D aj .ak al /, for all j; k; l 2 f1; : : : ; ng, for the basis elements. Similarly, the identity condition 1A a D a D a1A , for all a 2 A, is equivalent to the identity conditions 1A ak D ak D ak 1A , for all k 2 f1; : : : ; ng. Therefore, given a finite dimensional K-vector space A and a K-basis a1 ; a2 ; : : : ; an of A, the choice of n3 C n elements ˛ij k 2 K and j 2 K, i; j; k 2 f1; : : : ; ng, satisfying the equalities described in Lemma 1.2, determines a K-algebra structure on A. Conversely, every finite dimensional K-algebra A can be constructed in that way. In fact, if we choose the first basis element a1 to be the identity of A, then j D ı1j , ˛i1k D ıik , ˛ij1 D ıij , for all i; j; k 2 f1; : : : ; ng, and the condition (ii) of Lemma 1.2 is trivially satisfied. Observe that it is the case for the R-algebras C and H (see Examples 1.1 (a) and (b)), where additionally the structure constants are from f1; 1g.

6


We will present now an important construction of finite dimensional K-algebras proposed by P. Gabriel [Ga1], [Ga2] in the early 1970s, playing a fundamental role in the modern representation theory of associative algebras. By a quiver we mean a quadruple Q D .Q0 ; Q1 ; s; t / consisting of two sets: Q0 (whose elements are called vertices) and Q1 (whose elements are called arrows), and two maps s; t W Q1 ! Q0 which associate to each arrow ˛ 2 Q1 its source s.˛/ 2 Q0 and its target t .˛/ 2 Q0 , respectively. An arrow ˛ 2 Q1 with source ˛ b. A a D s.˛/ and target b D t .˛/ is usually denoted by ˛ W a ! b, or a ! quiver Q D .Q0 ; Q1 ; s; t / is usually denoted briefly by Q D .Q0 ; Q1 /, or simply by Q. A quiver Q D .Q0 ; Q1 / is said to be finite if Q0 and Q1 are finite sets. The x of a quiver Q is obtained from Q by forgetting the orientation underlying graph Q x is a connected graph. of the arrows. The quiver Q is said to be connected if Q Let Q D .Q0 ; Q1 ; s; t / be a quiver and a; b 2 Q0 . A path of length l 1 with source a and target b (or a path from a to b) is a sequence .aj˛1 ; ˛2 ; : : : ; ˛l jb/, where ˛k 2 Q1 for all k 2 f1; : : : ; lg, and we have s.˛1 / D a, t .˛k / D s.˛kC1 / for each k 2 f1; : : : ; l 1g, and t .˛l / D b. Such a path will be denoted briefly by ˛1 ˛2 : : : ˛l , and may be visualized as ˛1

˛2

˛l

a D a0 ! a1 ! a2 ! !al1 ! al D b: We denote by Ql the set of all paths of Q of length l. We may also associate with each vertex a 2 Q0 a path of length l D 0, called the trivial path at a, denoted by "a D .aka/. Thus the paths of lengths 0 and 1 are in bijective correspondence with the elements of Q0 and Q1 , respectively. A path of Q of length l 1 is called a cycle whenever its source and target coincide. A cycle of length 1 in Q is called a loop. The quiver Q without cycles is said to be acyclic. Let Q D .Q0 ; Q1 / be a finite quiver and K be a field. The path algebra KQ of Q over K is the K-algebra whose underlying K-vector space has as its basis the set of all paths .aj˛1 ; : : : ; ˛l jb/ of length l 0 in Q and the product of two basis vectors .aj˛1 ; : : : ; ˛l jb/ and .cjˇ1 ; : : : ; ˇk jd / of KQ is defined by .aj˛1 ; : : : ; ˛l jb/.cjˇ1 ; : : : ; ˇk jd / D ıbc .aj˛1 ; : : : ; ˛l ; ˇ1 ; : : : ; ˇk jd /; where ıbc is the Kronecker delta. Therefore, the product of two paths ˛1 : : : ˛l and ˇ1 : : : ˇk is equal to zero if t .˛l / 6D s.ˇ1 / and is equal to the composed path ˛1 : : : ˛l ˇ1 : : : ˇk if t .˛l / D s.ˇ1 /. The product of basis elements is extended uniquely to arbitrary elements of KQ by the distributivity and the K-algebra requirement .xy/ D .x/y D .x/y D x.y/ D x.y/ D .xy/, for x; y 2 KQ, 2 K. Observe also that KQ has the K-vector space decomposition KQ D KQ0 ˚ KQ1 ˚ KQ2 ˚ ˚ KQl ˚ where, for each l 0, KQl is the subspace of KQ spanned by the set Ql of all paths in Q of length l. Moreover, we have .KQl /.KQm / KQlCm for all

1. Algebras

7

l; m 0, because the product of a path of length l by a path of length m is either zero or a path of length l C m. Hence KQ is an N-graded K-algebra. Lemma 1.3. Let Q be a finite quiver and K a field. Then P (i) KQ is a K-algebra with the identity 1KQ D a2Q0 "a . (ii) KQ is finite dimensional if and only if Q is an acyclic quiver. P Proof. (i) Since the quiver Q is finite, the sum a2Q0 "a of the trivial paths "a D .aka/, a 2 Q0 , is an element of KQ. Moreover, for a path ˛1 : : : ˛l in Q from b D s.˛1 / to c D t .˛l /, we have X X "a ˛1 : : : ˛l D "a ˛1 : : : ˛l D "b ˛1 : : : ˛l D ˛1 : : : ˛l ; a2Q0

.˛1 : : : ˛l /

X a2Q0

a2Q0

X

"a D

˛1 : : : ˛l "a D ˛1 : : : ˛l "c D ˛1 : : : ˛l :

a2Q0

P Therefore, a2Q0 "a is indeed the identity 1KQ of KQ. (ii) Assume that w D ˛1 ˛2 : : : ˛l is a cycle in Q. Then, for each r 1, we have a basis vector w r D .˛1 ˛2 : : : ˛l /r , and hence KQ is infinite dimensional. Therefore, if KQ is finite dimensional then the quiver Q is acyclic. Conversely, if Q is an acyclic quiver, then Q contains only finitely many paths, because Q is finite, and so KQ is finite dimensional. Examples 1.4. Let K be a field. (a) Let Q be the quiver

1 d

˛

consisting of a single vertex and a single loop. Then "1 ; ˛; ˛ 2 ; : : : ; ˛ l ; : : : is the defining basis of the path algebra KQ, and the multiplication of basis vectors is given by "21 D "1 , "1 ˛ l D ˛ l D ˛ l "1 and ˛ l ˛ k D ˛ lCk , for all l; k 1. Hence KQ is isomorphic to the polynomial algebra KŒx in one variable x over K. (b) Let Q be the Kronecker quiver o 1 o

˛ ˇ

2:

Then "1 , "2 , ˛, ˇ is the defining basis of the path algebra KQ and the multiplication table of the basis elements is "1 "2 ˛ ˇ

"1 "1 0 ˛ ˇ

"2 0 "2 0 0

˛ 0 ˛ 0 0

ˇ 0 ˇ 0 0 :

8


Then a simple checking shows that KQ is isomorphic to the matrix K-algebra ² ³ K 0 a 0 ˇˇ D a; b; c; d 2 K ; K2 K .b; c/ d where a K-algebra isomorphism f W by 1 f ."1 / D .0; 0/ 0 f .˛/ D .1; 0/

KQ !

K 0 K2 K

is given on the basis elements

0 0 ; f ."2 / D 0 .0; 0/ 0 0 ; f .ˇ/ D 0 .0; 1/

0 ; 1 0 : 0

In particular, it follows that for K D R the path algebra RQ is isomorphic to the R-subalgebra ² ³ ˇ a 0 R 0 ˇ D 2 M2 .C/ a; b 2 R; c 2 C C R c b of the matrix algebra M2 .C/ of the R-algebra C of complex numbers. On the other hand, for K D C, the path algebra CQ is not isomorphic to the C-subalgebra ² ³ ˇ C 0 a 0 D 2 M2 .H/ ˇ a; b 2 C; c 2 H H C c b of the matrix algebra M2 .H/ of the quaternions H, because the left action and the right action of C on H do not coincide. (c) Let Q be the quiver o 1

˛

o 2

ˇ

3

/ : 4

Then "1 , "2 , "3 , "4 , ˛, ˇ, , ˇ˛ is the defining basis of the path algebra KQ, and the multiplication basis elements table is "1 "2 "3 "4 ˛ ˇ ˇ˛

"1 "1 0 0 0 ˛ 0 0 ˇ˛

"2 0 "2 0 0 0 ˇ 0 0

"3 0 0 "3 0 0 0 0 0

"4 0 0 0 "4 0 0 0

˛ 0 ˛ 0 0 0 ˇ˛ 0 0

ˇ 0 0 ˇ 0 0 0 0 0

0 0 0 0 0 0 0

ˇ˛ 0 0 ˇ˛ 0 0 0 0 0 :

1. Algebras

9

Then a simple checking shows that KQ is isomorphic to the matrix K-subalgebra of M4 .K/ of the form 9 2 3 82 3 K 0 0 0 a 0 0 0 ˆ > ˆ > 6K K 0 0 7 ˆ > : ; 0 0 0 K 0 0 0 d where a K-algebra isomorphism f W KQ ! A is given on the basis elements by 2 3 2 3 2 3 1 0 0 0 0 0 0 0 0 0 0 0 60 0 0 07 60 1 0 0 7 60 0 0 07 7 6 7 6 7 f ."1 / D 6 40 0 0 05 ; f ."2 / D 40 0 0 05 ; f ."3 / D 40 0 1 05 ; 0 0 0 0 0 0 0 0 0 0 0 0 2 3 2 3 2 3 0 0 0 0 0 0 0 0 0 0 0 0 60 0 0 07 6 7 6 7 7 ; f .˛/ D 61 0 0 07 ; f .ˇ/ D 60 0 0 07 ; f ."4 / D 6 40 0 0 05 40 0 0 0 5 40 1 0 05 0 0 0 1 0 0 0 0 0 0 0 0 2 3 2 3 0 0 0 0 0 0 0 0 60 0 0 07 60 0 0 07 7 6 7 f ./ D 6 40 0 0 15 ; f .ˇ˛/ D 41 0 0 05 : 0 0 0 0 0 0 0 0 Observe that f .ˇ˛/ D f .ˇ/f .˛/. Let Q be a finite quiver and K a field. Then the K-subspace RQ D KQ1 ˚ KQ2 ˚ ˚ KQl ˚ of the path algebra KQ is a two-sided ideal, called the arrow ideal of KQ. Observe that, for each l 1, we have M l D KQm ; RQ ml l and hence RQ is the ideal of KQ generated by all paths in Q of length l. A two-sided ideal I of KQ is said to be admissible if there exists m 2 such that m 2 I RQ : RQ

If I is an admissible ideal of KQ, then the pair .Q; I / is said to be a bound quiver. The quotient algebra KQ=I is said to be the bound quiver algebra of the bound quiver .Q; I /. Lemma 1.5. Let Q be a finite quiver, K a field, and I an admissible ideal of KQ. Then the bound quiver algebra KQ=I is a finite dimensional K-algebra.

10


P Proof. Clearly, KQ=I is a K-algebra whose identity is the coset a2Q0 "a CI of m 2 the identity of KQ. Since I is an admissible ideal of KQ, we have RQ I RQ , for some m 2. Hence every path ˛1 ˛2 : : : ˛l of Q of length l m becomes the zero element ˛1 ˛2 : : : ˛l C I D 0 C I of KQ=I . Therefore, KQ=I is finite dimensional. Let Q be a finite quiver. By a relation in KQ we mean a K-linear combination %D

n X

i wi

iD1

of paths w1 ; : : : ; wn in Q of length at least 2 having a common source and a common target, with 1 ; : : : ; n 2 K. Lemma 1.6. Let Q be a finite quiver, K a field, and I an admissible ideal of KQ. Then I is generated by a finite number of relations %1 ; : : : ; %r in KQ. m 2 Proof. Since I is an admissible ideal, we have RQ I RQ , for some m 2. Observe that there are at most finitely many paths in Q of a given length l 2. m m Hence KQ=RQ is a finite dimensional K-algebra and I =RQ is a two-sided ideal m 2 m of KQ=RQ , contained in the ideal RQ =RQ . Let u1 ; : : : ; us 2 I be such that m m m ; : : : ; us C RQ form a basis of the K-vector space I =RQ . the cosets u1 C RQ Moreover, let v1 ; : : : ; vr be all paths in Q of length m, if such paths exist. Clearly, m m v1 ; : : : ; vr generate the ideal RQ , and belong to I , because RQ I . Observe that the elements u1 ; : : : ; us ; v1 ; : : : ; vr generate the ideal I of KQ. Multiplying all elements u1 ; : : : ; us by the trivial paths "a D .aka/, a 2 Q0 , from left and right, we obtain a finite set "a ui "b , a; b 2 Q0 , i 2 f1; : : : ; sg, v1 ; : : : ; vr , of relations in KQ which generates the ideal I .

Examples 1.7. Let K be a field. (a) Let Q be the quiver

1 d

˛

m and I D RQ for some m 2. Then clearly I is an admissible ideal of KQ generated by ˛ m , and the bound quiver algebra KQ=I of .Q; I / is isomorphic to the quotient polynomial algebra KŒx=.x m /. (b) Let Q be the quiver

? @1 ~~ @@@˛ ~ @@ ~~ @ ~~ 2 3 o

ˇ

and I the ideal of KQ generated by ˛ˇ. Then I is an admissible ideal of KQ 4 , and the associated bound quiver algebra KQ=I is a 9-dimensional containing RQ

1. Algebras

11

K-algebra with a K-basis given by the cosets "1 C I , "2 C I , "3 C I , ˛ C I , ˇ C I , C I , ˇ C I , ˛ C I , ˇ ˛ C I . (c) Let Q be the quiver 2 t ˛ tt t ytt 1 eKK KK K K 3

eKK ˇ KK KK 4 tt tt t y t

and I be the ideal of KQ generated by the element ˇ˛ . Then I is an admissible 3 ideal of KQ (since RQ D 0), and the bound quiver algebra KQ=I of .Q; I / is a 9dimensional K-algebra with a K-basis given by the cosets "N1 D "1 CI , "N2 D "2 CI , "N3 D "3 C I , "N4 D "4 C I , ˛N D ˛ C I , ˇN D ˇ C I , N D C I , N D C I , and ˇN ˛N D ˇ˛ C I D C I D N N . Consider the matrix K-subalgebra 9 2 3 82 3 K 0 0 0 a 0 0 0 ˆ > ˆ > 6K K 0 0 7 ˆ > : ; K K K K z t u d of M4 .K/. Then there is a K-algebra isomorphism f W KQ=I ! A given on the basis elements of KQ=I by 2 3 2 3 2 3 1 0 0 0 0 0 0 0 0 0 0 0 60 0 0 07 6 7 6 7 7 ; f .N"2 / D 60 1 0 07 ; f .N"3 / D 60 0 0 07 ; f .N"1 / D 6 40 0 0 05 40 0 0 0 5 40 0 1 05 0 0 0 0 0 0 0 0 0 0 0 0 2 3 2 3 2 3 0 0 0 0 0 0 0 0 0 0 0 0 60 0 0 07 61 0 0 07 60 0 0 07 N 7 7 6 7 N D6 f .N"4 / D 6 40 0 0 05 ; f .˛/ 40 0 0 05 ; f .ˇ/ D 40 0 0 05 ; 0 0 0 1 0 0 0 0 0 1 0 0 2 3 2 3 0 0 0 0 0 0 0 0 60 0 0 07 60 0 0 07 7 6 7 f .N / D 6 41 0 0 05 ; f .N / D 40 0 0 05 ; 0 0 0 0 0 0 1 0 2 3 0 0 0 0 60 0 0 0 7 N 7 f .ˇN ˛/ N D6 N D f .N /f .N /: 40 0 0 05 D f .ˇ/f .˛/ 1 0 0 0

12


(d) Let Q be the quiver $

ˇ

o 1 2 3 and I the ideal of KQ generated by ˛ . Then I is an admissible ideal of KQ 4 containing RQ , and the bound quiver algebra KQ=I is a 7-dimensional K-algebra with a K-basis given by the cosets "N1 D "1 C I , "N2 D "2 C I , ˛N D ˛ C I , ˛N 2 D ˛ 2 C I , ˇN D ˇ C I , ˇN ˛N D ˇ˛ C I , ˇN ˛N 2 D ˇ˛ 2 C I . Then KQ=I is isomorphic to the matrix K-algebra of the form KŒx=.x 3 / 0 ƒD : KŒx=.x 3 / K ˛

Indeed, there is a K-algebra isomorphism f W KQ=I ! ƒ given on the basis elements of KQ=I by 2 0 0 0 xN 0 xN 1N 0 2 f .N"1 / D ; ; f .N"2 / D ; f .˛/ N D ; f .˛N / D 0 0 0 1 0 0 0 0 0 0 0 0 0 0 2 N N N f .ˇ/ D N ; f .ˇ ˛/ N D ; f .ˇ ˛N / D 2 ; xN 0 0 xN 1 0 where 1N D 1 C .x 3 /, xN D x C .x 3 /. (e) Let Q D .Q0 ; Q1 ; s; t / be a finite quiver, K a field and I an admissible ideal of KQ. Denote by Qop the opposite quiver Qop D .Q0 ; Q1 ; s 0 ; t 0 / of Q, where, for ˛ 2 Q1 , s 0 .˛/ D t .˛/ and t 0 .˛/ D s.˛/. Moreover, let %1 ; : : : ; %r be a finite set of relations in KQ generating the ideal I (see Lemma 1.6). Denote by op op %1 ; : : : ; %r the relations in KQop obtained by reversing the orientation of all paths occurring in the relations %1 ; : : : ; %r , respectively. Then the ideal I op of KQop op op generated by %1 ; : : : ; %r is an admissible ideal of KQop , and the bound quiver op op algebra KQ =I is isomorphic to the opposite algebra .KQ=I /op of the bound quiver algebra KQ=I .

2 Representations of algebras and modules Let K be a field. For a positive natural number n, we denote by GLn .K/ the subset of Mn .K/ consisting of all invertible matrices. Then GLn .K/ is a group with respect to multiplication of matrices and the identity n n matrix In as the identity element. Let G D .G; ; e/ be a finite group. Following F. G. Frobenius [Fro2], by a representation of G over K we mean a group homomorphism ' W G ! GLn .K/ for some n 1. The integer n is called the degree of the representation '. Hence, ' assigns to any element g 2 G an invertible matrix '.g/ 2 GLn .K/ and '.gh/ D '.g/'.h/, for all g; h 2 G, and '.e/ D In . Two representations ' W G ! GLn .K/

2. Representations of algebras and modules

13

and W G ! GLn .K/ are said to be equivalent if there exists a matrix C 2 GLn .K/ such that .g/ D C '.g/C 1 for all g 2 G. Let A D .A; C; ; 0A ; 1A / be a finite dimensional K-algebra. By a representation of A over K we mean (see [Fro3]) a K-algebra homomorphism ˆ W A ! Mn .K/, for some n 1. The integer n is called the dimension of the representation ˆ. Hence, ˆ assigns to any element a 2 A a matrix ˆ.a/ 2 Mn .K/, and ˆ.a C b/ D ˆ.a/ C ˆ.b/, ˆ.ab/ D ˆ.a/ˆ.b/, for all a; b 2 A, and ˆ.0A / D 0n , ˆ.1A / D In . Two representations ˆ W A ! Mn .K/ and ‰ W A ! Mn .K/ of A over K (of the same dimension) are said to be equivalent if there exists a matrix C 2 GLn .K/ such that ‰.a/ D C ˆ.a/C 1 for all a 2 A. The following lemma describes the relationship between the representations of groups and the representations of finite dimensional algebras. Lemma 2.1. Let G be ˇ a finite group and n a positive natural number. The restriction map ˆ 7! ' D ˆˇG induces a bijection between the equivalence classes of the representations of the group algebra KG of G over K of dimension n and the equivalence classes of the representations of G over K of degree n. Proof. Let ˆ W KG ! Mn .K/ be a representation of KG over K of dimension n. Then the identity e of G is the identity element of KG, and hence ˆ.e/ D In . Moreover, for g; h 2 G, we have ˆ.gh/ D ˆ.g/ˆ.h/. Thus ˆ.g/ˆ.g 1 / D ˆ.gg 1ˇ / D ˆ.e/ D In , and so ˆ.g/ 2 GLn .K/. Therefore, the restriction ' D ˆˇG W G ! GLn .K/ is a representation of G of degree n. Conversely, assume that ' W G ! GLn .K/ is a representation of G over K of degree n. Consider the map ˆ W KG ! Mn .K/ defined by X X ˆ g g D g '.g/ g2G

g2G

P

for an element g2G g g of KG. It follows immediately from definition of the K-algebra structure on KG that ˆ is a representation of KG of dimension n such ˇ that ' D ˆˇG . Observe that trivially two representations ˆ and ‰ of KGˇ of dimension ˇn are equivalent if and only if the associated representations ' D ˆˇG and D ‰ ˇG of G of degree n are equivalent. Example 2.2. Let m 1 and be an element of K with m D 1. Consider the cyclic group G D fe; g; g 2 ; : : : ; g m1 g of order m. Then, for any n 1, the map 'n W G ! GLn .K/ given by 'n .g k / D k In , for k 2 f0; 1; : : : ; m 1g, defines a representation of G of degree n. Observe also that the group algebra KG is isomorphic to the K-algebra KŒx=.x m 1/. Let A be a finite dimensional K-algebra, a1 ; a2 ; : : : ; an a basis of the K-vector space A, and ˛ij k 2 K, i; j; k 2 f1; : : : ; ng, the associated structure constants, that

14


is, aj ak D

n X

˛ij k ai

iD1

for all j; k 2 f1; : : : ; ng. Moreover, let j 2 K, j 2 f1; : : : ; ng, be such that n X

1A D

j aj :

j D1

Following F. G. Frobenius [Fro4], [Fro5], consider the matrices L.aj / D ŒL.aj /ik D Œ˛ij k ik 2 Mn .K/; R.al / D ŒR.al /ik D Œ˛ikl ik 2 Mn .K/;

j 2 f1; : : : ; ng; l 2 f1; : : : ; ng:

Therefore, L.aj / is the matrix of the K-linear map aj ./ W A ! A, aj .x/ D aj x, for x 2 A, in the basis a1 ; : : : ; an . Similarly, R.al / is the matrix of the K-linear map ./al W A ! A, .x/al D xal , for x 2 A, in the basis a1 ; : : : ; an . The matrices L.aj / and R.al /, j; l 2 f1; : : : ; ng, determine K-linear maps L W A ! Mn .K/ and R W A ! Mn .K/, respectively. Denote by Rt W A ! Mn .K/ the linear map such that Rt .a/ D .R.a//t is the transpose of the matrix R.a/, for any a 2 A. Lemma 2.3. The maps L; Rt W A ! Mn .K/ are representations of the algebra A over K. Proof. For j; k 2 f1; : : : ; ng, we have the equalities n X

L.aj ak / D L

n X ˛ij k ai D ˛ij k L.ai /

iD1

D

n X

˛ij k Œ˛hil hl D

iD1

iD1 n hX

i ˛ij k ˛hil

iD1

L.aj /L.ak / D Œ˛hj i hi Œ˛ikl il D

n hX

hl

i ˛hj i ˛ikl

iD1

hl

; n hX

D

i ˛ikl ˛hj i

iD1

hl

:

Then it follows from Lemma / D L.aj /L.ak / for all j; k 2 P 1.2 (i) that L.aj akP f1; : : : ; ng. Hence, for a D jnD1 j aj and b D nkD1 k ak from A, we obtain n n X X

L.ab/ D L

j D1 kD1

D

n n X X

n n X X j k aj ak D j k L aj ak j D1 kD1

j k L.aj /L.ak / D

j D1 kD1

D L.a/L.b/:

n X j D1

n X

j L.aj /

kD1

k L.ak /

15


Moreover, by Lemma 1.2 (ii), we also obtain that L.1A / D L

n X

n n X X j L.aj / D j Œ˛ij k ik j aj D

j D1

D

n hX

j D1

j D1

i j ˛ij k

j D1

ik

D Œıik ik D In :

Therefore, L W A ! Mn .K/ is a representation of A. For k; l 2 f1; : : : ; ng, we have the equalities R.ak al / D R

n X

n X ˛ikl ai D ˛ikl R.ai /

iD1

D

n X

˛ikl Œ˛hj i hj D

iD1

iD1 n hX

iD1 n hX

R.al /R.ak / D Œ˛hil hi Œ˛ij k ij D

i ˛ikl ˛hj i

˛hil ˛ij k

iD1

hj

i hj

; D

n hX

i ˛ij k ˛hil

iD1

hj

:

Applying again Lemma 1.2 (i), we obtain that R.ak al / D R.al /R.ak / for all k; l 2 f1; : : : ; ng. Hence, as above, we conclude that R.ab/ D R.b/R.a/ for all a; b 2 A. This implies that Rt .ab/ D R.ab/t D .R.b/R.a//t D R.a/t R.b/t D Rt .a/Rt .b/. Moreover, by Lemma 1.2 (ii), we also obtain R.1A / D R

n X

j aj D

j D1

D

n hX j D1

j R.aj / D

j D1

i j ˛ikj

n X

ik

n X

j Œ˛ikj ik

j D1

D Œıik ik D In ;

and hence Rt .1A / D R.1A /t D Int D In . Therefore, Rt W A ! Mn .K/ is a representation of A.

Following F. G. Frobenius, the representation L W A ! Mn .K/ is called the first (left) regular representation of A over K, and the representation Rt W A ! Mn .K/ is called the second (right) regular representation of A over K (see [Fro4], [Fro5]). This depends on the choice of a K-basis a1 ; : : : ; an of the K-algebra A. But any other choice of the K-basis of A leads to representations of A equivalent to L and Rt , respectively (Exercise 12.1). A new and fundamental view on the representation theory of algebras came from two papers [Noe1] and [Noe2] by E. Noether who gave to the theory its modern setting by interpreting representations as modules.

16


Let A be a K-algebra. Following E. Noether, a left A-module (or a left module over A) is a pair .M; /, where M is a left K-vector space and W A M ! M , .a; m/ 7! a m (simply am), is a binary operation satisfying the conditions (i) a.x C y/ D ax C ay, (ii) .a C b/x D ax C bx, (iii) .ab/x D a.bx/, (iv) 1A x D x, (v) a.x/ D .a/x D .ax/, for all x; y 2 M , a; b 2 A, and 2 K. Dually, a right A-module (or a right module over A) is a pair .M; /, where M is a right K-vector space and W M A ! M , .m; a/ 7! ma, is a binary operation satisfying the conditions (i) .x C y/a D xa C ya, (ii) x.a C b/ D xa C xb, (iii) x.ab/ D .xa/b, (iv) x1A D x, (v) .x/a D x.a/ D .xa/, for all x; y 2 M , a; b 2 A, and 2 K. We will denote by 0M the zero element of a (left or right) A-module M . A (left or right) A-module M is said to be finite dimensional if the dimension dimK M of the underlying K-vector space of M is finite. Moreover, we denote by 0 the zero (left or right) A-module on the zero K-vector space. We note that there is a bijection between the left (respectively, right) modules over a K-algebra A and the right (respectively, left) modules over the opposite algebra Aop . Indeed, for a left (respectively, right) A-module M D .M; / the corresponding right (respectively, left) Aop -module M op D .M; / is such that x a D ax (respectively, a x D xa) for x 2 M , a 2 A. Throughout, we identify a module .M; / with its underlying set M . Let M and N be left (respectively, right) modules over a K-algebra A. Then a K-linear map f W M ! N of left (respectively, right) K-vector spaces is said to be an A-module homomorphism if f .ax/ D af .x/ (respectively, f .xa/ D f .x/a) for all x 2 M and a 2 A. Moreover, if f is bijective, then f is called an A-module isomorphism. Two A-modules M and N are said to be isomorphic if there exists an A-module isomorphism f W M ! N (equivalently, g W N ! M ). In this case we write M Š N . The following proposition was the key observation of E. Noether.


17

Proposition 2.4. Let A be a K-algebra. There is a bijection between the equivalence classes of representations of A over K of the dimension n and the isomorphism classes of left A-modules of the dimension n. Proof. Let ˆ W A ! Mn .K/ be a representation of A over K. Consider the left K-vector space M D Mn1 .K/ consisting of all n 1 matrices of K. We define the binary operation W A M ! M by ax D ˆ.a/x for a 2 A and x 2 M , where the right side is the multiplication of matrices. Then for all a; b 2 A, x; y 2 M and 2 K, the following equalities hold: a.x C y/ D ˆ.a/.x C y/ D ˆ.a/x C ˆ.a/y D ax C ay; .a C b/x D ˆ.a C b/x D .ˆ.a/ C ˆ.b// x D ˆ.a/x C ˆ.b/x D ax C bx; .ab/x D ˆ.ab/x D .ˆ.a/ˆ.b// x D ˆ.a/ .ˆ.b/x/ D a.bx/; 1A x D ˆ.1A /x D In x D x; a.x/ D ˆ.a/.x/ D .ˆ.a// x D .ˆ.a// x D ˆ.a/x D .a/x; a.x/ D .ˆ.a// x D .ˆ.a/x/ D .ax/: This defines on M a left A-module structure, denoted by Mˆ . Let ˆ W A ! Mn .K/ and ‰ W A ! Mn .K/ be two representations of A over K of the dimension n. Assume that ˆ and ‰ are equivalent. Then there exists a matrix C 2 GLn .K/ such that ‰.a/ D C ˆ.a/C 1 for all a 2 A. Let, as above, M D Mn1 .K/ and f W M ! M be the K-linear isomorphism given by f .x/ D C x for all x 2 M . Then af .x/ D ‰.a/.C x/ D .‰.a/C / x D .C ˆ.a// x D C .ˆ.a/x/ D f .ax/ for all a 2 A, x 2 M . Therefore, f defines an isomorphism f W Mˆ ! M‰ of left A-modules. Conversely, assume that there exists an isomorphism f W Mˆ ! M‰ of left A-modules. Since Mˆ D Mn1 .K/ D M‰ as K-vector spaces, there exists a matrix C 2 GLn .K/ such that f .x/ D C x for all x 2 Mn1 .K/. Moreover, for any a 2 A and x 2 M D Mn1 .K/, we have .C ˆ.a// x D C .ˆ.a/x/ D f .ax/ D af .x/ D ‰.a/.C x/ D .‰.a/C / x; and hence C ˆ.a/ D ‰.a/C , or equivalently, ‰.a/ D C ˆ.a/C 1 . Therefore, the representations ˆ and ‰ are equivalent. Let N be a left A-module of dimension n. We show that N Š Mˆ for a representation ˆ W A ! Mn .K/ of A over K of dimension n. Since dimK N D n, there exists an isomorphism g W N ! M D Mn1 .K/ of K-vectorspaces. Then we transport the left A-module structure from N to M by ax D g ag 1 .x/ for all a 2 A and x 2 M . A direct checking shows that this defines a left A-module structure on M , and clearly then g W N ! M is an isomorphism of left A-modules. Fix now an element a 2 A and consider the K-linear map 'a W M ! M given

18


by 'a .x/ D ax for x 2 M . Then there exists a matrix ˆ.a/ 2 Mn .K/ such that 'a .x/ D ˆ.a/x for any matrix x 2 M D Mn1 .K/. Therefore, we defined a map ˆ W A ! Mn .K/. Moreover, for a; b 2 A, 2 K, and x 2 M , we have the equalities ˆ.a C b/x ˆ.ab/x ˆ.0A /x ˆ.1A /x ˆ.a/x

D .a C b/x D ax C bx D ˆ.a/x C ˆ.b/x D .ˆ.a/ C ˆ.b// x; D .ab/x D a.bx/ D ˆ.a/ .ˆ.b/x/ D .ˆ.a/ˆ.b// x; D 0n x D 0 (D zero matrix in M ); D 1A x D x D In x; D .a/x D .ax/ D .ˆ.a/x/ D .ˆ.a// x;

and hence ˆ is a K-algebra homomorphism. Summing up, we have proved that N Š Mˆ as left A-modules. Although historically the left modules were more natural objects of study than the right modules, in the modern representation theory of algebras and from technical reasons, the right modules became more popular. We will follow this trend and concentrate on study of right modules. Let A be a K-algebra and M be a right A-module. A K-subspace M 0 of M is said to be a right A-submodule of M if xa 2 M 0 for all x 2 M 0 and a 2 A. Then the K-vector space M=M 0 has a natural structure of a right A-module, called a factor right A-module of M , given by .x C M 0 /a D xa C M 0 for x 2 M , a 2 A, and the canonical K-linear epimorphism W M ! M=M 0 is an A-module homomorphism. A submodule M 0 of M with M 0 ¤ M is called a proper submodule of M . For a right ideal I of A, the set MI consisting of all finite sums x1 a1 C C xs as , where x1 ; : : : ; xs 2 M and a1 ; : : : ; as 2 I , for some s 1, is an A-submodule of M . The module M is said to be finitely generated if there exist elements x1 ; : : : ; xs 2 M such that any element x of M has the form x D x1 a1 C C xs as for some a1 ; : : : ; as 2 A. In such a case, the elements x1 ; : : : ; xs are called generators of M . For A-submodules M1 ; : : : ; Ms of M we define M1 C C Ms to be the Asubmodule of M consisting of all sums x1 C Cxs , where x1 2 M1 ; : : : ; xs 2 Ms . We note the following simple fact. Proposition 2.5. Let A be a finite dimensional K-algebra and M a right A-module. Then M is a finitely generated A-module if and only if M is a finite dimensional A-module. Proof. Recall that the field K is identified with the K-subalgebra of A consisting of all elements of the form 1A D 1A , 2 K. Hence, if M is generated by elements x1 ; : : : ; xr as a right K-vector space, then clearly M is generated by x1 ; : : : ; xr as a right A-module. Conversely, assume that M is generated by some elements m1 ; : : : ; ms as a right A-module. Let a1 ; : : : ; an be a K-basis of A. Then the set of elements mi aj , i 2 f1; : : : ; sg, j 2 f1; : : : ; ng, generates M as a (right) K-vector space.


19

Let A be a K-algebra. For right A-modules M , N , we denote by HomA .M; N / the set of all A-module homomorphisms from M to N . We note that HomA .M; N / is a right K-vector space with respect to the addition .f; g/ 7! f C g given by .f C g/.x/ D f .x/ C g.x/ for f; g 2 HomA .M; N / and x 2 M , and the scalar multiplication .f; / 7! f given by .f /.x/ D f .x/ D f .x/ for f 2 HomA .M; N /, 2 K and x 2 M . Obviously the zero homomorphism 0M;N which assigns to any element m 2 M the zero element 0N of N is the zero element of HomA .M; N / with respect to C. If the modules M and N are finite dimensional then the K-vector space HomA .M; N / is finite dimensional, since it is a K-subspace of the K-vector space HomK .M; N / of all K-linear maps from M to N . It is easy to check that for any triple L; M; N of right A-modules the composition map W HomA .M; N / HomA .L; M / ! HomA .L; N /; .h; g/ 7! hg, for h 2 HomA .M; N / and g 2 HomA .L; M /, is K-bilinear. For a right A-module M the K-vector space EndA .M / D HomA .M; M / of all A-module endomorphisms of M has a K-algebra structure with respect to composition of maps, and whose identity is the identity map idM of M . For a homomorphism f 2 HomA .M; N /, the kernel Ker f D fx 2 M j f .x/ D 0N g, the image Im f D ff .x/ j x 2 M g, and the cokernel Coker f D N= Im f have natural structures of right A-modules, and moreover the K-linear isomorphism M= Ker f ! Im f , x C Ker f 7! f .x/, is an isomorphism of right A-modules. The direct sum of right A-modules M1 ; : : : ; Mr , r 1, is defined to be the Kvector space direct sum M1 ˚ ˚ Mr equipped with the right A-module structure defined by .x1 ; : : : ; xr /a D .x1 a; : : : ; xr a/ for x1 2 M1 ; : : : ; xr 2 Mr , and a 2 A. We denote the direct sum M ˚ ˚ M of r 1 copies of an A-module M by M r . Moreover, for r D 0, we define M 0 to be the zero A-module. A right A-module M is said to be the direct sum of right A-submodules M1 ; : : : ; Mr , r 1, denoted by M D M1 ˚ ˚ Mr , if every element m 2 M has a unique expression of the form m D m1 C Cmr with m1 2 M1 ; : : : ; mr 2 Mr . Moreover, a right A-module M is said to be indecomposable if M is nonzero and not a direct sum of two nonzero right A-submodules of M . The following simple lemma is very useful. Lemma 2.6. Let A be a K-algebra, M be a right A-module and M1 ; : : : ; Mr , r 1, right A-submodules of M . Then M D M1 ˚ ˚ Mr if and only if the following conditions are satisfied: (1) M D M1 C C Mr , P (2) Mi \ j 2f1;:::;rgnfig Mj D 0 for each i 2 f1; : : : ; rg.

20


Proof. Clearly, M D M1 C C Mr is just the fact that every element m 2 M has an expression of the form m D m1 C C mr with m1 2 M1 ; : : : ; mr 2 Mr . Moreover, if m1 C C mr D m D m01 C C m0r are two expressions of m 2 M with m1 ; m01 2 M1 ; : : : ; mr ; m0r 2 Mr , then, for each i 2 f1; : : : ; rg, we have X X 0 Mj : mj mj 2 Mi \ mi m0i D j 2f1;:::;rgnfig

j 2f1;:::;rgnfig

Hence the required equivalence follows.

We denote by Mod A the category of all right A-modules over a K-algebra A, that is, the category whose objects are right A-modules, the morphisms are A-module homomorphisms, and the composition of morphisms is the usual composition of maps. Moreover, we denote by mod A the full subcategory of Mod A whose objects are the finite dimensional (over K) right A-modules. We note that if A is a finite dimensional K-algebra then, by Proposition 2.5, mod A is the category of all finitely generated right A-modules. The following proposition summarizes the properties of the categories Mod A and mod A described above (we refer to [ML2] for background on abelian categories). Proposition 2.7. Let A be a K-algebra. Then Mod A and mod A are abelian K-categories. We also note that every left A-module over a K-algebra A can be viewed as a right Aop -module. Therefore, the category Mod Aop (respectively, mod Aop ) will be identified with the category A-Mod of all (respectively, A-mod of all finite dimensional) left A-modules. For the categories mod A and mod Aop of finite dimensional K-algebras A we have moreover the standard K-duality of module categories mod A o

D D

/

mod Aop

with 1mod A Š D B D and 1mod Aop Š D B D, where D D HomK .; K/. The functor D W mod A ! mod Aop assigns to a module M in mod A the dual K-vector space D.M / D HomK .M; K/ endowed with the left A-module structure given by .a'/.m/ D '.ma/, for ' 2 HomK .M; K/, m 2 M , a 2 A, and to each morphism f W M ! N in mod A the dual K-homomorphism D.f / W D.N / ! D.M / of left A-modules such that D.f /. / D f for any 2 D.N / D HomK .N; K/. The quasi-inverse functor D W mod Aop ! mod A assigns to a module X in mod Aop the dual K-vector space D.X / D HomK .X; K/ endowed with the right A-module structure given by .'a/.x/ D '.ax/, for ' 2 HomK .X; K/, x 2 X , a 2 A, and to each morphism g W X ! Y in mod Aop the dual K-homomorphism


21

D.g/ W D.Y / ! D.X / of right A-modules such that D.g/. / D g for any 2 D.Y / D HomK .Y; K/. Then the standard evaluation isomorphism eV W V ! DD.V / D HomK .HomK .V; K/; K/, for a finite dimensional K-vector space V , given by eV .x/.f / D f .x/, where x 2 V , f 2 D.V /, defines natural isomorphisms of functors 1mod A Š D B D and 1mod Aop Š D B D (in the sense of II.4). For bound quiver algebras A D KQ=I of bound quivers .Q; I /, the categories Mod A and mod A have a very useful description as the categories of K-linear representations of .Q; I /, which we define below. Let Q D .Q0 ; Q1 ; s; t / be a finite quiver. A K-linear representation of Q (or a representation of Q over K) is a system M D .Ma ; '˛ /a2Q0 ;˛2Q1 ; briefly M D .Ma ; '˛ /, consisting of K-vector spaces Ma , a 2 Q0 , and K-linear maps '˛ W Ms.˛/ ! M t.˛/ , ˛ 2 Q1 . The representation M is said to be finite dimensional if each K-vector space Ma is finite dimensional. Let M 0 D .Ma0 ; '˛0 /a2Q0 ;˛2Q1 and M D .Ma ; '˛ /a2Q0 ;˛2Q1 be K-linear representations of the quiver Q. Then M 0 is said to be a subrepresentation of M 0 0 ! M t.˛/ if Ma0 is a K-vector subspace of Ma , for any a 2 Q0 , and '˛0 W Ms.˛/ 0 0 is the restriction '˛ jMs.˛/ of '˛ W Ms.˛/ ! M t.˛/ to Ms.˛/ , for any ˛ 2 Q1 . Then the factor representation M 00 D M=M 0 of M by M 0 is defined as M 00 D 00 .Ma00 ; '˛00 /a2Q0 ;˛2Q1 , where Ma00 D Ma =Ma0 for any a 2 Q0 , and '˛00 W Ms.˛/ ! 0 0 00 00 M t.˛/ , for any ˛ 2 Q1 , is given by '˛ .x CMs.˛/ / D '˛ .x/CM t.˛/ for x 2 Ms.˛/ . Let M D .Ma ; '˛ / and N D .Na ; ˛ / be two representations of Q over K. A morphism (of representations) f W M ! N is a family f D .fa /a2Q0 of K-linear maps fa W Ma ! Na , a 2 Q0 , such that ˛ fs.˛/ D f t.˛/ '˛ for any arrow ˛ 2 Q1 , or equivalently the square of K-linear maps Ms.˛/

'˛

fs.˛/

Ns.˛/

˛

/ M t.˛/

f t .˛/

/ N t.˛/

is commutative. A morphism f D .fa /a2Q0 W M ! N of representations is called an isomorphism if all K-linear maps fa , a 2 Q0 , are isomorphisms. We denote by HomQ .M; N / the set of all morphisms of representations from M to N . Observe that HomQ .M; N / has a K-vector space structure given by f Cg D .fa Cga /a2Q0 and f D .fa /a2Q0 , for f D .fa /a2Q0 and g D .ga /a2Q0 in HomQ .M; N /, and 2 K. Moreover, for any triple L, M , N of representations of Q over K the composition map W HomQ .M; N / HomQ .L; M / ! HomQ .L; N /;

22


which assigns to h D .ha /a2Q0 2 HomQ .M; N /, g D .ga /a2Q0 2 HomQ .L; M / the homomorphism hg D .ha ga /a2Q0 2 HomQ .L; N /, is K-bilinear. Finally, let f D .fa /a2Q0 2 HomQ .M; N / for representations M D .Ma ; '˛ / and N D .Na ; ˛ / of Q over K. Then the kernel of f is defined as Ker f D .Ker fa ; '˛0 /, where '˛0 W Ker fs.˛/ ! Ker f t.˛/ denotes the restriction of '˛ to Ker fs.˛/ , the image of f is defined as Im f D .Im fa ; ˛0 /, where ˛0 W Im fs.˛/ ! Im f t.˛/ is the restriction of ˛ to Im fs.˛/ , and the cokernel of f is defined as Coker f D .Coker fa ; N ˛ /, where N ˛ W Coker fs.˛/ ! Coker f t.˛/ is given by N ˛ .x C Im fs.˛/ / D ˛ .x/ C Im f t.˛/ , for x 2 Ns.˛/ . Observe that Ker f is a subrepresentation of M , Im f is a subrepresentation of N , and Coker f is the factor representation of N by Im f . Moreover, if the representations M and N are finite dimensional, then the representations Ker f , Im f and Coker f are finite dimensional. Given two representations M D .Ma ; '˛ / and N D .Na ; ˛ / of Q over K their direct sum is the representation

'˛ 0 : M ˚ N D Ma ˚ Na ; 0 ˛ A representation M of Q over K is said to be indecomposable if M is nonzero and not isomorphic (as a representation of Q over K) to a direct sum L ˚ N of two nonzero representations of Q over K. We denote by RepK .Q/ the category of all K-linear representations of Q and the morphisms of representations, and by repK .Q/ the full subcategory of RepK .Q/ consisting of all finite dimensional representations. The following proposition summarizes our discussion above. Proposition 2.8. Let Q be a finite quiver and K a field. Then RepK .Q/ and repK .Q/ are abelian K-categories. Examples 2.9. Let K be a field. (a) Let Q be the quiver

1 consisting of one vertex. Then RepK .Q/ is just the category Mod K of K-vector spaces. Furthermore, the field K is a unique indecomposable representation in RepK .Q/, up to isomorphism. (b) Let Q be the quiver ˛ 2 : 1 o

Then the objects of RepK .Q/ are triples M D .M1 ; M2 ; '˛ W M2 ! M1 /, which we write briefly as '˛ M W M1 o M2 consisting of one K-linear map between vector spaces. We claim that the representations 1 K; 0; Ko 0o K Ko


23

are representatives of the isomorphism classes of all indecomposable representations in RepK .Q/. Observe that, if M is a representation in RepK .Q/ with '˛ an isomorphism, then M is isomorphic to a direct sum of copies of the representation 1 Ko K . Further, if Ker '˛ ¤ 0, ˇ then taking a K-vector space decomposition M2 D M20 ˚ Ker '˛ and '˛0 D '˛ ˇM 0 , we conclude that M is isomorphic to the 2 direct sum of representations

M1 o

0 '˛

M20 ˚ 0 o

Ker '˛ :

Finally, if Im '˛ ¤ M1 , then taking a K-vector space decomposition M1 D M10 ˚ Im '˛ , we conclude that M is isomorphic to the direct sum of representations

M10 o

0 ˚ Im '˛ o

'˛

M1 :

(c) Let Q be the quiver 1 d

˛.

Then the category RepK .Q/ is the category of endomorphisms of K-vector spaces V1 D V g

'D'˛ :

In particular, the classification problem of objects in repK .Q/ up to isomorphism is equivalent to the classification of matrices in Mn .K/, n 1, up to conjugation, and hence to Frobenius normal forms of finite square matrices over K (see [Coh2], Chapter 11, for details). In particular, if K is algebraically closed, then the following representations K n f Jn ./ ; where Jn ./ is the Jordan block 2 3 1 0 0 0 60 1 0 07 6 7 60 0 1 07 6 7 Jn ./ D 6 : : : : : : : : : ::: 7 6 :: :: :: 7 6 7 40 0 0 15 0 0 0 0 for n 1, 2 K, give representatives of the isomorphism classes of all finite dimensional indecomposable representations of Q over K. We note that this has been proved by C. Jordan in 1870 [Jor]. (d) Let Q be the quiver $ o ˇ ˛ : 1 2

24


Then the objects of RepK .Q/ are quadruples M D M1 ; M2 ; '˛ W M1 ! M1 ; 'ˇ W M2 ! M1 ; denoted briefly by (

'˛

'ˇ

M1 o

M2 ;

consisting of two vector spaces M1 , M2 , and two K-linear maps '˛ and 'ˇ . Moreover, a morphism f W M ! N of two representations consists of two K-linear maps f1 W M1 ! N1 and f2 W M2 ! N2 such that f1 '˛ D ˛ f1 and f1 'ˇ D ˇ f2 , where N D .N1 ; N2 ; ˛ W N1 ! N1 ; ˇ W N2 ! N1 /, or equivalently the diagram of K-linear maps ( 'ˇ '˛ M1 o M2 f1

f2

( N1 o

˛

N2

ˇ

is commutative. In particular, for the representations XW

0

%

Ko

1

K;

YW

01 00

(

1 0

K o 2

K;

ZW

01 00

(

K o 2

0 1

K

in repK .Q/, we have a morphism f W X ! Y of representations given by the commutative diagram in mod K %

0

01 00

Ko

1

1 0 1 0

( K2 o

K 1

K

with Ker f D 0 (hence f is a monomorphism), and a morphism g W Z ! X of representations given by the commutative diagram in mod K

01 00

(

K2 o

0

01

% Ko

0 1

K

1

1

K

with Im g D X (hence g is an epimorphism). Observe also that X, Y and Z are indecomposable representations and Coker f and Ker g are isomorphic to the representation ' 0 0: Ko


25

Consider also the representation M in repK .Q/ of the form

01 00

(

K2 o

10 01

K 2:

We determine the endomorphism K-algebra EndQ .M /. Every endomorphism in EndQ .M / is given by two matrices a12 b12 b a and B D 11 A D 11 a21 a22 b21 b22 in M2 .K/ satisfying the conditions 0 1 0 1 ADA 0 0 0 0

A

and

Then A D B and a11 D a22 , a21 D K-algebras ² EndQ .M / Š 0

1 0 1 0 D B: 0 1 0 1

0. Therefore, we obtain isomorphisms of ³ ˇ ˇ ; 2 K Š KŒx=.x 2 /:

Moreover, a basis of the K-vector space EndQ .M / is formed by the identity morphism idM of M and the morphism h given by the commutative diagram in mod K of the form 10 ( 01 01 00 K2 K2 o

01 00

01 00

( K2 o

10 01

01 00

K2 .

Observe also that h2 D 0, and Ker h, Im h and Coker h are isomorphic to X . In particular, h is neither a monomorphism nor an epimorphism in repK .Q/. (e) Let Q be the Kronecker quiver o 1 o

˛ ˇ

2 :

Then the objects of RepK .Q/ are the pairs M1 oo

'˛ 'ˇ

M2

of K-linear maps between two K-vector spaces. For K D C, the problem of a classification of the representations in repC .Q/ (up to isomorphism) has been solved by L. Kronecker [Kro]. We only note that the family of representations o Kn o

Jn .0/ In

K n;

26


for n 1, provides an infinite family of pairwise nonisomorphic indecomposable representations in repK .Q/. We refer to [SS1], XI.4, for a complete description of the indecomposable representations in repK .Q/ for an algebraically closed field K. Let Q D .Q0 ; Q1 ; s; t / be a finite quiver and I an admissible ideal of the path algebra KQ of Q over a field K. Moreover, let M D .Ma ; '˛ /a2Q0 ;˛2Q1 be a representation in RepK .Q/. For any path w D ˛1 ˛2 : : : ˛l of length l 1 in Q, we define the K-linear map 'w D '˛l '˛l1 : : : '˛2 '˛1 W Ms.˛1 / ! M t.˛l / called the evaluation map of M on the path w. Then for a K-linear combination %D

r X

i wi

iD1

of paths in Q with a common source a and a common target b, we define the K-linear map r X i 'wi W Ma ! Mb : '% D iD1

A representation M in RepK .Q/ is said to be bound by I , or satisfying the relations of I , if we have '% D 0 for all relations % 2 I . It follows from Lemma 1.6 that the ideal I is generated, as a two-sided ideal, by a finite set f%1 ; : : : ; %r g of relations. Hence every relation % 2 I is of the form %D

s X

j uj j vj

j D1

where j 2 K, uj , vj are paths in Q (possibly trivial), and j 2 f%1 ; : : : ; %r g, for any j 2 f1; : : : ; sg. Since '% D

s X

j 'vj 'j 'uj ;

j D1

we conclude that a representation M in RepK .Q/ is bound by I if and only if '%1 D 0; : : : ; '%r D 0. For a bound quiver .Q; I /, we denote by RepK .Q; I / (respectively, repK .Q; I /) the full subcategory of RepK .Q/ (respectively, repK .Q/) consisting of the representations of Q bound by I . It follows immediately from the proof of Proposition 2.8 that RepK .Q; I / and repK .Q; I / are abelian K-categories. This follows also from the following theorem.


27

Theorem 2.10. Let A D KQ=I , where Q is a finite quiver, K a field, and I is an admissible ideal of KQ. Then there exists a K-linear equivalence of categories F W Mod A ! RepK .Q; I /

which restricts to a K-linear equivalence of categories F W mod A ! repK .Q; I /:

Proof. We construct a K-linear functor F W Mod A ! RepK .Q; I / and its quasiinverse functor G W RepK .Q; I / ! Mod A. Let Q D .Q0 ; Q1 ; s; t /. For each vertex a 2 Q0 , the idempotent "a D .aka/ of KQ gives the idempotent ea D "a C I of A. Let M be a module in Mod A. We associate to M the K-linear representation F .M / D .Ma ; '˛ /a2Q0 ;˛2Q1 of Q bound by I as follows. For each vertex a 2 Q0 , we set Ma D M ea , and, for each arrow ˛ 2 Q1 , we define '˛ W Ms.˛/ ! M t.˛/ to N where x 2 Ms.˛/ , and ˛N D ˛ C I . be the K-linear map defined by '˛ .x/ D x ˛, Observe that ˛N D es.˛/ ˛e N t.˛/ , and hence x ˛NPD xes.˛/ ˛e N t.˛/ 2 Me t.˛/ D M t.˛/ . We show that F .M / is bound by I . Let % D m w be a relation from a vertex i i iD1 a to a vertex b in I , and wi D ˛i;1 ˛i;2 : : : ˛i;li for i 2 f1; : : : ; mg. Then we obtain, for an element x 2 Ma , the equalities '% .x/ D

m X

i 'wi .x/ D

iD1

Dx

m X

m X

i '˛i;li : : : '˛i;1 .x/ D

iD1

m X

i .x ˛N i;1 : : : ˛N i;li /

iD1

i ˛N i;1 : : : ˛N i;li D x %N D x0 D 0:

iD1

Let f W M ! N be a homomorphism in Mod A, and F .M / D .Ma ; '˛ /, F .N / D .Na ; ˛ /. We define the morphism F .f / W F .M / ! F .N / of representations of Q over K. For a vertex a 2 Q0 and x D xea 2 Mea D Ma , we have f .x/ D f .xea / D f .x/ea 2 Nea D Na , because f is a homomorphism of right A-modules. Hence the restriction fa of f to Ma gives a K-linear map fa W Ma ! Na . We set F .f / D .fa /a2Q0 . For each arrow ˛ 2 Q1 and x 2 Ms.˛/ , we have the equalities f t.˛/ '˛ .x/ D f t.˛/ .x ˛/ N D f .x ˛/ N D f .x/˛N D fs.˛/ .x/˛N D

˛ fs.˛/ .x/:

Therefore, f t.˛/ '˛ D ˛ fs.˛/ for any arrow ˛ 2 Q1 , and hence F .f / is a morphism in RepK .Q; I /. Clearly, F W Mod A ! RepK .Q; I / is a K-linear functor and restricts to a K-linear functor F W mod A ! repK .Q; I /. We define now a K-linear functor G W RepK .Q; I / ! Mod A which is quasi'˛ / be a representation in RepK .Q; I /. Consider inverse of F . Let M D .Ma ;L the K-vector space G.M / D a2Q0 Ma . We define first a structure of the right

28


KQ-module on G.M /. Let x D .xa /a2Q0 belong to G.M /, and w be a path in Q. Then xw is defined as follows. If w D "a is a trivial path, we set xw D x"a D xa . For a path w D ˛1 ˛2 : : : ˛l of length l 1 in Q, consider the K-linear map ' w W '˛l : : : '˛1 W Ms.˛1 / ! M t.˛l / , and define xw to be the element of G.M / D L a2Q0 Ma whose only nonzero component is .xw/ t.˛l / D 'w .xs.˛1 / / 2 M t.˛l / . The multiplication of elements of G.M / by paths of Q extends in an obvious way to a right KQ-module. Since the representation M of Q is bound by I , for each relation % in I , and each x 2 G.M /, we have x% D 0. Clearly, then G.M /I D 0. Therefore, G.M / is a right A-module by the formula x.v CI / D xv for x 2 G.M / and v 2 KQ. Let now f D .fa /a2Q0 be a morphism from M D .Ma ; '˛ / to N D .Na ; ˛ / in RepK .Q; I /. Consider the K-linear homomorphism M M M fa W Ma ! Na : G.f / D a2Q0

a2Q0

a2Q0

We show that G.f / W G.M / ! G.N / is a homomorphism of right A-modules. Indeed, take x D xa 2 Ma and wN D w C I for a path w in Q with source a 2 Q0 and target b 2 Q0 . Then we have the equalities G.f /.x w/ N D G.f /.xa w/ N D fb 'w .xa / D

w fa .xa /

D fa .xa /wN D G.f /.x/w: N

Clearly, then we have G.f /.x w/ N D G.f /.x/wN for any element x 2 G.M / and wN 2 KQ=I , and our claim follows. Observe that if M belongs to repK .Q; I / then G.M / belongs to mod A, because the quiver Q is finite. Consequently, G W RepK .Q; I / ! Mod A is a K-linear functor which restricts to a K-linear functor G W repK .Q; I / ! mod A. A standard checking shows that we have equivalences of functors GF ! 1Mod A and F G ! 1RepK .Q;I / . We note that the K-linear equivalence of categories F W Mod A ! RepK .Q; I /, described above, carries the A-submodules (respectively, A-factor modules) into K-linear subrepresentations (respectively, factor representations) of .Q; I /. Corollary 2.11. Let Q be a finite acyclic quiver, K a field and A D KQ. Then there exists a K-linear equivalence of categories F W Mod A ! RepK .Q/

which restricts to a K-linear equivalence of categories F W mod A ! repK .Q/:

Proof. Since Q is a finite and acyclic quiver, KQ is a finite dimensional K-algebra, by Lemma 1.3 (ii). Then the statement follows from Theorem 2.10 for I D 0.


29

Examples 2.12. (a) Let Q be the quiver 1 d

˛

m and I D RQ , for some m 2, in KQ. Then RepK .Q; I / is the full subcategory of RepK .Q/ consisting of all representations

V1 f

'˛

satisfying the condition '˛m D 0. In particular, the representations Kr f

Jr .0/ ;

for r 2 f1; : : : ; mg, given by the nilpotent Jordan blocks Jr .0/ of degree r m, form a complete set of representatives of the isomorphism classes of indecomposable representations in repK .Q; I /. Since KQ=I D KŒx=.x m /, these representations correspond, via the K-linear equivalence G W repK .Q; I / ! mod KQ=I described in Theorem 2.10, to the indecomposable (right) KŒx=.x m /-modules KŒx=.x r /, r 2 f1; : : : ; mg. (b) Let Q be the quiver $ o ˇ ˛ 1 2 and I the ideal of KQ generated by ˛ 2 . Then I is an admissible ideal of KQ, and the bound quiver algebra A D KQ=I is isomorphic to the matrix algebra KŒx=.x 2 / 0 BD KŒx=.x 2 / K (compare Example 1.7 (d)). A simple analysis of indecomposable objects in repK .Q; I /, invoking Examples 2.9 (b) and 2.12 (a), shows that the following 7 representations 0

'

Ko

0;

01 00

01 00

(

(

0

K2 o

K2 o

'

Ko

0; 0 1

K;

1

01 00

01 00

$

K;

( (

K2 o

1 0

K2 o

0o

10 01

K;

K;

K2

form a complete set of representatives of the isomorphism classes of indecomposable representations in repK .Q; I /. Then, applying Theorem 2.10, we conclude that the number of pairwise nonisomorphic finite dimensional indecomposable right B-modules is equal to 7.

30


3 The Jacobson radical Let K be a field. A right (respectively, left) ideal I of a K-algebra A is called maximal if I is maximal, with respect to inclusion, in the set of all proper right (respectively, proper left) ideals of A. We observe first that a K-algebra A admits maximal right (respectively, maximal left) ideals, since the family of right ideals (respectively, left ideals) of A different from A contains the zero ideal 0, and hence is nonempty. The Jacobson radical (briefly, radical) rad A of a K-algebra A is the intersection of all the maximal right ideals of A. It follows from the lemma below that rad A is the intersection of all the maximal left ideals of A, and consequently rad A is a two-sided ideal of A. Lemma 3.1. Let A be a K-algebra. The following conditions are equivalent. (i) a 2 rad A. (ii) For any b 2 A, the element 1A ab has a right inverse. (iii) For any b 2 A, the element 1A ab has a two-sided inverse. (iv) For any b 2 A, the element 1A ba has a two-sided inverse. (v) For any b 2 A, the element 1A ba has a left inverse. (vi) a belongs to the intersection of all maximal left ideals of A. Proof. We abbreviate 1 D 1A . We prove first that (i) and (ii) are equivalent. Let b 2 A. Suppose that the element 1 ab has no right inverse in A. Then there exists a maximal right ideal I of A such that 1 ab 2 I , since .1 ab/A is a right ideal of A different from A. Because a 2 rad A I , we obtain ab 2 I , and hence 1 2 I , a contradiction. This shows that 1 ab has a right inverse. Conversely, assume a … rad A, and let I be a maximal right ideal of A such that a … I . Then I C aA is a right ideal of A containing I as a proper subset, and hence A D I C aA. Thus there exist x 2 I and b 2 A such that 1 D x C ab. But then x D 1 ab 2 I has no right inverse, because I ¤ A. The equivalence of (v) and (vi) can be proved in a similar way. The equivalence of (iii) and (iv) follows from the following implications: (a) if .1 cd /x D 1, then .1 dc/.1 C dxc/ D 1; (b) if y.1 cd / D 1, then .1 C dyc/.1 dc/ D 1; for c; d; x; y 2 A. We prove now that (ii) implies (iii). Let b 2 A. By assumption (ii), there exists c 2 A such that .1ab/c D 1. Then c D 1a.bc/. Applying (ii) to c, we obtain

3. The Jacobson radical

31

that 1 D cd for some d 2 A, and hence 1 D .1a.bc//d D d Cabcd D d Cab. Therefore d D 1 ab and c is a left inverse of 1 ab. The implication (v) ) (iv) follows in a similar way. Clearly, (iii) implies (ii) and (iv) implies (v). The lemma is proved. Corollary 3.2. Let A be a K-algebra. Then rad A is a two-sided ideal of A and rad.A= rad A/ D 0. Proof. The fact that rad A is a two-sided ideal follows from the equivalence of (i) and (vi) in Lemma 3.1. For the second statement, observe that the maximal right ideals of A= rad A are the ideals of the form I C rad A for maximal right ideals I of A (containing rad A), and hence indeed rad.A= rad A/ D 0. In fact, we will show in Section 6 (Lemma 6.11) that rad A is also the intersection of all maximal two-sided ideals of A. Let I be a two-sided ideal of a finite dimensional K-algebra A. For an integer m 1, we denote by I m the two-sided ideal of A consisting of all finite sums of elements of the form x1 x2 : : : xm , where x1 ; x2 ; : : : ; xm 2 I . The ideal I is said to be nilpotent if I m D 0 for some m 1. Observe that then every element x of I is nilpotent, that is, x r D 0 for some r 1. We will show that the Jacobson radical of a finite dimensional K-algebra is nilpotent. It will follow from the following useful lemma, known as Nakayama’s lemma. Lemma 3.3. Let A be a K-algebra, M be a finitely generated right A-module, and I rad A be a two-sided ideal of A. If MI D M , then M D 0. Proof. Assume MI D M and M is generated by elements m1 ; : : : ; mr , that is, M D m1 AC Cms A. We proceed by induction on r. If r D 1, then m1 A D m1 I implies that m1 D m1 x1 for some x1 2 I . Hence m1 .1 x1 / D 0, and so m1 D 0, because 1 x is right invertible, by Lemma 3.1. Therefore, M D 0. Assume that r 2. Then M D MI D m1 I C C mr I implies that m1 D m1 x1 C m2 x2 C C mr xr for some x1 ; : : : ; xr 2 I . Hence m1 .1 x1 / D m2 x2 C C mr xr , and so m1 D m2 y2 C C mr yr 2 m2 A C C mr A, for y2 D x2 z; : : : ; yr D xr z, where z is the right inverse of 1 x1 . Therefore, M D m2 A C C mr A, and, by induction, we obtain M D 0. Corollary 3.4. Let A be a finite dimensional K-algebra. Then rad A is a nilpotent ideal of A. Proof. We have the chain of two-sided ideals of A, A rad A .rad A/2 .rad A/i .rad A/iC1 : Since dimK A is finite, there exists m 1 such that .rad A/m D .rad A/mC1 . Then .rad A/m D .rad A/m rad A, and, applying Lemma 3.3, we conclude that .rad A/m D 0.

32


We will determine now the radical of the bound quiver algebra KQ=I of a bound quiver .Q; I /. The following lemma will be essential. Lemma 3.5. Let A be a finite dimensional K-algebra and I be a nilpotent two-sided ideal of A. Then (i) I rad A. (ii) If the algebra A=I is isomorphic to a product F1 Fn of division K-algebras F1 ; : : : ; Fn , then I D rad A. Proof. (i) Since I is a nilpotent ideal of A, we have I m D 0 for some m 1. Let x 2 I and a be an element of A. Then ax 2 I , .ax/m D 0, and the equality .1A ax/ 1A C ax C .ax/2 C C .ax/m1 D 1A holds, and hence x 2 rad A, by Lemma 3.1. Therefore I rad A. (ii) Assume A=I Š F1 Fn for some division K-algebras F1 ; : : : ; Fn . Clearly, rad.F1 Fn / D 0, because F1 Fi1 0FiC1 Fn , for i 2 f1; : : : ; ng, are maximal right ideals of F1 Fn , and consequently rad.A=I / D 0. Consider the canonical surjective K-algebra homomorphism W A ! A=I such that .a/ D a C I for any a 2 A. We claim that .rad A/ D rad.A=I /. Indeed, if a 2 rad A and .b/ D b C I , with b 2 A, is an arbitrary element of A=I , then c.1A ba/ D 1A for some element c 2 A, by Lemma 3.1. Hence, .c/.1A=I .b/.a// D 1A=I and consequently .a/ is an element of rad.A=I /. Since rad.A=I / D 0, we obtain .a/ D 0A=I , and so a 2 I . This shows that rad A I , and hence I D rad A. Lemma 3.6. Let Q be a finite quiver, I an admissible ideal of KQ, and RQ the arrow ideal of KQ. Then rad KQ=I D RQ =I . m 2 Proof. Since the ideal I is admissible, we have RQ I RQ for some m m 2. Consider the canonical surjective K-algebra homomorphism f W KQ=RQ ! m m KQ=I . Observe that f .RQ =RQ / D RQ =I . Obviously, RQ =RQ is a nilpotent m ideal of KQ=RQ , and hence RQ =I is a nilpotent ideal of KQ=I . Moreover, we have canonical isomorphisms of algebras Y ı .KQ=I / .RQ =I / ! KQ=RQ ! Ka ; a2Q0

where Ka D K for any vertex a 2 Q0 . Applying Lemma 3.5, we conclude that rad KQ=I D RQ =I . Corollary 3.7. Let Q be a finite acyclic quiver. Then rad KQ D RQ . A K-algebra A is said to be local if A has a unique maximal right ideal, or equivalently, A has a unique maximal left ideal, as it is shown in the lemma below.


33

Lemma 3.8. Let A be a K-algebra. The following conditions are equivalent. (i) A is a local algebra. (ii) A has a unique maximal left ideal. (iii) rad A consists of all noninvertible elements of A. (iv) The set of all noninvertible elements of A is a two-sided ideal of A. (v) For any a 2 A, one of the elements a or 1A a is invertible. (vi) A= rad A is a division algebra. Proof. We abbreviate 1 D 1A and 1N D 1A= rad A . We prove first that (i) implies (iii). Assume A is a local algebra. Then rad A is a unique maximal right ideal of A. Hence a 2 rad A if and only if a has no right inverse. We claim that every right inverse element a of A is invertible. Indeed, if ab D 1, then .1 ba/b D 0. Observe that then b … rad A, because otherwise the element .1 ba/ is invertible, by Lemma 3.1, and then b D 0, a contradiction. Thus b has a right inverse, and consequently 1 ba D 0. This shows that a has a left inverse, and so a is invertible. Therefore, we have proved that a 2 rad A if and only if a has no right inverse, if and only if a is not invertible. The fact that (ii) implies (iii) follows in a similar way. Clearly, (iii) implies (iv). We show now that (iv) implies (v). Let I be a two-sided ideal of A consisting of all noninvertible elements, and a 2 A. If a and 1 a are noninvertible, then a and 1 a belong to I , and consequently 1 D a C .1 a/ 2 I , a contradiction. Hence, a or 1 a is invertible. Assume (v) hold. We claim that (i) hold. Indeed, suppose I and J are different maximal right ideals of A. Then I C J is a right ideal of A with I as proper subset of I C J , and hence I C J D A. Thus 1 D a C b for some a 2 I and b 2 J . By (v) one of the elements a 2 I or 1 a D b 2 J is invertible, which is impossible because I and J are different from A. Similarly we show that (v) implies (ii). Consider the canonical K-algebra homomorphism W A ! A= rad A, .a/ D a C rad A, for a 2 A. Assume (iii) holds. We show that A= rad A is a division algebra, and so (vi) holds. Let .a/ D a C rad A be a nonzero element of A= rad A. Then a … rad A, and hence a is an invertible element of A, by (iii). Hence ba D 1 D ab for some b 2 A, and consequently we obtain .b/.a/ D .ba/ D 1N D .ab/ D .a/.b/, that is, .a/ is an invertible element of A= rad A. Assume now that (vi) holds. We show that rad A is a maximal right ideal of A, and hence (i) holds. Suppose rad A is properly contained in a maximal right ideal I of A, and a 2 I n rad A. Then .a/ is a nonzero element of A= rad A, and hence .a/ is invertible. Thus .a/.b/ D 1N for some b 2 A. Hence ab C rad A D 1 C rad A, which implies that 1 D ab C x for some x 2 rad A. Then rad A I and ab 2 I imply that 1 2 I , a contradiction with I ¤ A.

34


Corollary 3.9. Let A be a finite dimensional K-algebra. Then A is a local algebra if and only if every element of A is nilpotent or invertible. Proof. Assume that a is a nilpotent element of A, say am D 0 for some m 1. Observe that a is noninvertible. Indeed, if ab D 1A for some b 2 A, then we obtain 0 D am b m D a.a.: : : a.ab/b : : : /b/b D 1A , a contradiction. Moreover, we have the equalities .1A C a C C am1 /.1A a/ D 1A D .1A a/.1A C a C C am1 /; and hence 1A a is an invertible element of A. Further, it follows from Corollary 3.4 that rad A is a nilpotent ideal of A, and hence every element of rad A is nilpotent. Therefore, the asserted equivalence follows from Lemma 3.8. Examples 3.10. (a) Let A D KŒx=.x m /, for some m 1. Then the principal ideal .x/ N of A generated by xN D x C .x m / is a unique maximal ideal of A, and consequently rad A D .x/. N In particular, A is a commutative local K-algebra. (b) Let Q be a finite quiver, K a field, I an admissible ideal of KQ, and A D KQ=I . It follows from Lemma 3.6 that rad A D RQ =I and Q Q A= rad A Š K , where K D K for any a 2 Q . Observe also that a a 0 a2Q0 a2Q0 Ka is a division algebra if and only if jQ0 j D 1. Therefore, A is a local algebra if and only if Q has only one vertex. (c) Let F be a division K-algebra, n 2, and A be the K-subalgebra 2 3 F 0 0 6F F 0 7 6 7 Tn .F / D 6 : :: : : :: 7 4 :: : :5 : F F F of the full matrix algebra Mn .F /, consisting of all triangular matrices Œaij 2 Mn .F / with aij D 0 for all 1 i < j n. Denote by In .F / the ideal of Tn .F / consisting of all matrices Œaij 2 Mn .F / with aij D 0 for all 1 i j n. Then In .F /n D 0 and Tn .F /=In .F / is isomorphic to the product F F of n copies of F . Applying Lemma 3.5, we conclude that rad A D In .F /. Moreover, A is not a local algebra. Let ƒn .F / be the K-subalgebra of Tn .F / consisting of all matrices Œaij with a11 D a22 D D ann . Then again rad ƒn .F / D In .F / but ƒn .F /= rad ƒn .F / Š F is a division K-algebra. Therefore, ƒn .F / is a local K-algebra. (d) Let ƒ be the matrix K-algebra KŒx=.x 3 / 0 ƒD KŒx=.x 3 / K over a field K, considered in Example 1.7 (d), and .x/ N 0 J D ; KŒx=.x 3 / 0


35

where .x/ N is the principal ideal of KŒx=.x 3 / generated by xN D x C x 3 . Then J is a two-sided ideal of ƒ such that J 4 D 0 and ƒ=J ! K K. Hence, applying Lemma 3.5, we obtain that J D rad ƒ. We exhibit now an important class of local algebras. Lemma 3.11. Let Q be a finite quiver, K a field, I and admissible ideal of KQ, and A D KQ=I . Moreover, let a be a vertex of Q0 and ea D "a C I the coset of the trivial path "a of Q at a. Then ea Aea is a finite dimensional local K-algebra. Proof. Clearly, ea Aea is a finite dimensional K-algebra with ea the identity 1eAe of eAe. Moreover, we have a canonical isomorphism of K-algebras ea Aea D ."a C I /.KQ=I /."a C I / ! "a .KQ/"a ="a I "a ;

where "a .KQ/"a is the K-algebra whose underlying K-vector space has as its basis the set of all (oriented) cycles in Q throughout the vertex a, and clearly "a is the identity of "a .KQ/"a . Moreover, since I is an admissible ideal of KQ, m we have RQ I for some m 2. Consider the two-sided ideal "a RQ "a of "a .KQ/"a generated by all nontrivial cycles around a. Then we have ."a RQ "a /m m "a RQ "a "a I "a and hence "a RQ "a ="a I "a is a nilpotent two-sided ideal of "a .KQ/"a ="a I "a . Further, we have canonical isomorphisms of K-algebras ı ."a .KQ/"a ="a I "a / ."a RQ "a ="a I "a / ! K."a C "a I "a / ! K: Applying Lemma 3.5, we conclude that "a RQ "a ="a I "a D rad ."a .KQ/"a ="a I "a / : Then it follows from Lemma 3.8 that "a .KQ/"a ="a I "a is a local algebra. Finally, we conclude that ea Aea is a local algebra and rad ea Aea D ea .rad A/ea . An element e of a K-algebra A is called an idempotent if e 2 D e. Observe that then 1A e is also an idempotent of A and e.1A e/ D 0A D .1A e/e, that is, the idempotents e and 1A e are orthogonal. In general, two idempotents e and f of A are called orthogonal if ef D 0A D f e. Moreover, an idempotent e of A is said to be central if ea D ae for all elements a of A. Every K-algebra A has always two trivial idempotents 0A and 1A , which are central idempotents of A. The following lemma provides a useful lifting property of idempotents modulo the nilpotent ideals. Lemma 3.12. Let A be a finite dimensional K-algebra, B D A=I for a nilpotent ideal I of A and W A ! B the canonical surjective K-algebra homomorphism. Then the following assertions hold.

36


(i) For any pairwise orthogonal idempotents f1 ; : : : ; fn in B there exist pairwise orthogonal idempotents e1 ; : : : ; en in A with .ei / D fi for i 2 f1; : : : ; ng. (ii) Let e be an idempotent in A such that .e/ D f1 C C fn for pairwise orthogonal idempotents f1 ; : : : ; fn in B. Then there exist pairwise orthogonal idempotents e1 ; : : : ; en in A such that e D e1 C C en and .ei / D fi for i 2 f1; : : : ; ng. Proof. Let I m D 0A for a positive integer m, and for elements x1 ; : : : ; xn in A we denote by hx1 ; : : : ; xn i the set of K-linear combinations of elements of the form x1r1 : : : xnrn for nonnegative integers r1 ; : : : ; rn . We shall show (i) by induction on n in the following stronger form: For any pairwise orthogonal idempotents f1 ; : : : ; fn in B with fi D .xi / for some xi 2 A and i 2 f1; : : : ; ng, there exist pairwise orthogonal idempotents e1 ; : : : ; en such that .ei / D fi and ei 2 hx1 ; : : : ; xn i for i 2 f1; : : : ; ng. First consider the case n D 1. Let f D .x/ D x C I for some x 2 A. Then .x/ D f D f 2 D .x 2 / implies x x 2 2 I , and hence .x x 2 /m D 0A . On the other hand, by Newton’s binomial x 2 /m D x m x mC1 y, where y D formula, we have .x Pm i1 m i1 m x . Then we obtain x D x mC1 y and xy D yx. We claim iD1 .1/ i that the element e D .xy/m is a required idempotent of A with .e/ D f . Observe first that e 2 hxi and e D .xy/m D x m y m D x mC1 y mC1 D D x 2m y 2m D ..xy/m /2 D e 2 ; and so e is an idempotent. Moreover, we have the equalities x x m D x 1 x m1 D x.1 x/ 1 C x C C x m2 D x x 2 1 C x C C x m2 : Hence x x 2 2 I forces x x m 2 I , and so .x/ D .x m /. Then we obtain the equalities f D .x/ D .x m / D x mC1 y D x mC1 .y/ D .x m / .x/.y/ D .x/.x/.y/ D x 2 .y/ D .x/.y/ D .xy/; and hence the required claim .e/ D ..xy/m / D ..xy//m D f m D f: Assume that n > 1 and the assertion is true for n 1. Let f1 ; : : : ; fn be pairwise orthogonal idempotents in B and fi D .xi /, for some xi 2 A, i D 1; : : : ; n. It follows from the first part of the proof that there is an idempotent an in A with fn D .an / and an 2 hxn i. Moreover, by induction


37

hypothesis, there exist pairwise orthogonal idempotents e1 ; : : : ; en1 in A such that .ei / D fi and ei 2 hx1 ; : : : ; xn1 i for i 2 f1 : : : ; n 1g. Let a D e1 C C en1 ; an0 D an aan an a C aan a: Then, for i 2 f1; : : : ; n 1g, we have an0 ei D 0A D ei an0 , because aei D ei D ei a. Further, we have the equalities .a/.an / D .f1 C Cfn1 /fn D 0B ; .an /.a/ D fn .f1 C Cfn1 / D 0B ; and hence .an0 / D .an / .aan / .an a/ C .aan a/ D .an / .a/.an / .an /.a/ C .a/.an /.a/ D .an / D fn : Therefore there is an idempotent en in A such that .en / D fn and en 2 han0 i, which implies in particular that en ei D 0A D ei en for all i 2 f1; : : : ; n 1g. Moreover, en 2 hx1 ; : : : ; xn i, because an0 2 ha; an i, a 2 he1 ; : : : ; en1 i and an 2 hxn i, and hence an0 2 he1 ; : : : ; en1 ; xn i hx1 ; : : : ; xn1 ; xn i. Thus we have proved (i) in the stronger form. (ii) Assume that e 2 D e 2 A, .e/ D f1 C C fn and fi D .ai / for ai 2 A and i 2 f1; : : : ; ng, where f1 ; : : : ; fn are pairwise orthogonal idempotents in B. Let di D eai e for i 2 f1; : : : ; ng. Then, for i 2 f1; : : : ; ng, we obtain .di / D .e/fi .e/ D .f1 C C fn /fi .f1 C C fn / D fi : It follows from (i) in the stronger form that there are pairwise orthogonal idempotents e1 ; : : : ; en in A such that .ei / D fi and ei 2 hd1 ; : : : ; dn i for i 2 f1; : : : ; ng. In particular, we have ei 2 eAe, because di 2 eAe for any i 2 f1; : : : ; ng. Now let d D e1 C C en : Clearly then d 2 D d . Further, because d 2 eAe, we have ed D d D de and hence .e d /2 D e 2 ed de C d 2 D e d . Moreover, we conclude that .e d / D

n X iD1

fi

n X iD1

.ei / D

n X iD1

fi

n X

fi D 0B :

iD1

Thus we have proved that e d is an idempotent belonging to I , which implies that e d D 0A , because 0A is the only idempotent in the nilpotent ideal I of A. Therefore e D e1 C C en as desired. A K-algebra is said to be indecomposable if A is not isomorphic to a product B C of two K-algebras B and C .

38


Lemma 3.13. Let A be a local K-algebra. Then A is an indecomposable Kalgebra. Proof. Assume there exists an isomorphism of K-algebras f W B C ! A. Observe that for every pair of maximal right ideals M of B and N of C the products M C and B N are maximal right ideals of B C , and such ideals exhaust all maximal right ideals of B C . Then we conclude that rad.B C / D ..rad B/ C / \ B .rad C / D .rad B/ .rad C /: Hence we have an isomorphism of K-algebras .B C /= rad.B C / ! .B= rad B/ .C = rad C /: Observe that the algebra .B= rad B/ .C = rad C / is not a division algebra, because .1B C rad B; 0C C rad C / is its nonzero and noninvertible element. Hence .B C /= rad.B C / is not a division algebra. Since the isomorphism f W B C ! A induces an isomorphism of K-algebras .B C /= rad.B C / ! A= rad A, we conclude that A= rad A is not a division algebra. Applying now Lemma 3.8 we obtain that A is not a local algebra. Lemma 3.14. A K-algebra A is indecomposable if and only if the unique central idempotents of A are the trivial idempotents 0A and 1A . Proof. Assume there exists an isomorphism of K-algebras f W B C ! A. Then .1B ; 0C / and .0B ; 1C / are nontrivial central idempotents of B C , and hence f ..1B ; 0C // and f ..0B ; 1C // are nontrivial central idempotents of A whose sum is 1A D f ..1B ; 1C //. Conversely, assume that e is a nontrivial central idempotent of A. Then 1A e is also a nontrivial central idempotent of A, eAe D eA D Ae and .1A e/A.1A e/ D .1A e/A D A.1A e/ are K-algebras, with the identity elements e and 1A e, respectively. Moreover, the map g W eAe .1A e/A.1A e/ ! A given by g.exe; .1A e/y.1A e// D exe C .1A e/y.1A e/, for x; y 2 A, is an isomorphism of K-algebras. Proposition 3.15. Let Q be a finite quiver, K a field and I an admissible ideal of KQ. Then the bound quiver algebra KQ=I is indecomposable if and only if the quiver Q is connected. Proof. It follows from Lemmas 1.3 and 1.5 that the cosets ea D "a C I of the trivial paths "a at the vertices a 2 Q0 of Q form P a family of pairwise orthogonal idempotents of KQ=I such that 1KQ=I D a2Q0 ea . Moreover, it follows from Lemma 3.11 that ea Aea , a 2 Q0 , are local K-algebras. Then, applying Lemma 3.13, we conclude that the algebras ea Aea , a 2 Q0 , are indecomposable.


39

Let e be a central idempotent of KQ=I . Since ea eeb D e.ea eb / D 0KQ=I for all a ¤ b in Q0 , we obtain that X X e D 1KQ=I e1KQ=I D ea e eb D

X a;b2Q0

a2Q0

ea eeb D

X

b2Q0

ea eea :

a2Q0

Observe that, for any a 2 Q0 , ea eea is a central idempotent of the algebra ea Aea , and so ea ee Pa D ea or ea eea D 0eAe D 0A , by Lemma 3.14. Therefore, e is of the form e D a20 ea , for a subset 0 of Q0 . Moreover, for any arrow ˛ 2 Q1 , we have X X "a ˛ C I D e.˛ C I / D .˛ C I /e D ˛"a C I; a20

a20

which shows that the source s.˛/ of ˛ and the target t .˛/ of ˛ belong simultaneously to 0 or Q0 n 0 . Hence, e is a nontrivial central idempotent of KQ=I if and only if Q is a disjoint union of two subquivers with the nonempty sets 0 and Q0 n 0 of vertices. Applying Lemma 3.14, we conclude that the K-algebra KQ=I is indecomposable if and only if the quiver Q is connected. Proposition 3.16. Let A be a finite dimensional K-algebra over a field K. Then the following statements hold. (i) There are pairwise orthogonal central idempotents e1 ; : : : ; es of A such that 1A D e1 C C es , B1 D e1 A; : : : ; Bs D es A are indecomposable Kalgebras and two-sided ideals of A, and A D B1 ˚ ˚ Bs as two-sided ideals of A. (ii) Let B1 ; : : : ; Bs and B10 ; : : : ; B t0 be indecomposable K-algebras and two-sided ideals of A such that B1 ˚ ˚ Bs D A D B10 ˚ ˚ B t0 as two-sided ideals of A. Then s D t and there exists a permutation of f1; : : : ; sg such that Bi D B0 .i/ for each i 2 f1; : : : ; sg. Proof. (i) It follows from Lemma 3.14 that A is indecomposable if and only if the identity 1A of A is the unique nonzero central idempotent of A. Then the claim follows by induction on dimK A. (ii) Observe that, for the given decompositions B1 ˚ ˚Bs D A D B10 ˚ ˚ 0 B t , there exist central idempotents e1 ; : : : ; es and e10 ; : : : ; e t0 of A such that e1 ; : : : ; es are pairwise orthogonal with 1A D e1 C Ces and B1 D e1 A; : : : ; Bs D es A, and e10 ; : : : ; e t0 are pairwise orthogonal with 1A D e10 C Ce t0 and B10 D e10 A; : : : ; B t0 D

40


e t0 A. Moreover, since B1 ; : : : ; Bs ; B10 ; : : : ; B t0 are indecomposable K-algebras, e1 ; : : : ; es ; e10 ; : : : ; e t0 are the unique central idempotents of B1 ; : : : ; Bs ; B10 ; : : : ; B t0 , respectively. Further, for each i 2 f1; : : : ; sg and j 2 f1; : : : ; tg, ei ej0 is a central idempotent of A. Hence, for each i 2 f1; : : : ; sg, the equality ei D ei e10 C Cei e t0 forces that ei D ei ej0 D ej0 for exactly one j 2 f1; : : : ; tg. This shows that s D t and 0 0 , and hence Bi D B.i/ , there is a permutation of f1; : : : ; sg such that ei D e.i/ for any i 2 f1; : : : ; sg. It follows from the above proposition that, for a finite dimensional K-algebra A over a field K, there is a unique decomposition A D B1 ˚ ˚ Bs of A into a direct sum of two-sided ideals B1 D e1 A; : : : ; Bs D es A, for central idempotents e1 ; : : : ; es of A with 1A D e1 C C es , and moreover B1 ; : : : ; Bs are indecomposable K-algebras. The indecomposable K-algebras B1 ; : : : ; Bs in this decomposition are called the blocks of A. Observe that for every indecomposable module M in mod A we have the induced decomposition M D Me1 ˚ ˚ M es in mod A, and hence M D M ei for exactly one i 2 f1; : : : ; sg and Mej D 0 for j 2 f1; : : : ; sg n fi g. Clearly, then M is an indecomposable Bi -module, and we say that M belongs to the block Bi of A. We end this section with the following lemma. Lemma 3.17. Let A and B be finite dimensional K-algebras over a field K, f W A ! B a surjective homomorphism of K-algebras, and I D Ker f . Then the following statements hold. (i) f induces the inclusion preserving bijection ˚

right ideals of A containing I o

/ ˚right ideals of B :

(ii) f induces the inclusion preserving bijection ˚

left ideals of A containing I o

/ ˚left ideals of B :

(iii) f induces the inclusion preserving bijection ˚ two-sided ideals of A containing I o

/ ˚two-sided ideals of B :

(iv) If I is nilpotent then f .rad A/ D rad B and f 1 .rad B/ D rad A. Proof. (i) Let J be a right ideal of B. Then f 1 .J / is a right ideal of A, because f .xa/ D f .x/f .a/ 2 J for x 2 J and a 2 A. Moreover, I D f 1 .f0B g/ f 1 .J /. Further, for a right ideal L of A, f .L/ is a right ideal of B, because B D f .A/. Finally, for right ideals L and N of A containing I , the equality

4. The Krull–Schmidt theorem

41

f .L/ D f .N / implies L D N . Indeed, for x 2 L, there exists y 2 N such that f .x/ D f .y/, and hence y x 2 Ker f D I , so x 2 N , since I N . This shows that L N . Similarly, we obtain that N L. Hence indeed L D N . This completes the proof of (i). The proofs of (ii) and (iii) are similar. (iv) It follows from (i) that f induces a bijection between the set of all maximal right ideals of A containing I and the set of all maximal right ideals of B. Moreover, since I is a nilpotent ideal of A, applying Lemma 3.5, we obtain I rad A. Therefore, f .rad A/ D rad B and f 1 .rad B/ D rad A.

4 The Krull–Schmidt theorem In this section we show that every finite dimensional module over a finite dimensional K-algebra A has a unique decomposition into a direct sum of indecomposable modules. Let K be a field. Let A be a finite dimensional K-algebra. A homomorphism u W L ! M in mod A is called a section if there exists a homomorphism v W M ! L in mod A such that vu D idL . Further, a homomorphism r W M ! N in mod A is said to be a retraction if there exists a homomorphism s W N ! M in mod A such that rs D idN . Observe that every section is a monomorphism and every retraction is an epimorphism. The following lemma provides examples of sections and retractions. Lemma 4.1. Let u W L ! M and v W M ! N be two homomorphisms in mod A such that vu is an isomorphism in mod A. Then u is a section and v is a retraction in mod A. Proof. Let w W N ! L be the inverse homomorphism of vu in mod A. Then idL D w.vu/ D .wv/u and idN D .vu/w D v.uw/, and hence u is a section and v is a retraction. Lemma 4.2. Let f W M ! N and g W N ! M be two homomorphisms in mod A such that gf D idM . Then N D Im f ˚ Ker g. Proof. For an element y 2 N , we have y fg.y/ 2 Ker g, and hence y 2 Im f C Ker g. This shows N D Im f C Ker g. Moreover, if y 2 Im f \ Ker g, then y D f .x/ for some x 2 M and x D gf .x/ D g.y/ D 0, and so y D 0. Hence Im f \ Ker g D 0, and the claim follows. The following lemma is known as the Fitting lemma. Lemma 4.3. Let A be a K-algebra, M a finite dimensional right A-module, and f 2 EndA .M /. Then there exists a natural number n such that M D Ker f n ˚ Im f n .

42


Proof. We have two chains of A-submodules of M : Ker f Ker f 2 Ker f i Ker f iC1 ; Im f Im f 2 Im f i Im f iC1 : Since dimK M is finite, there exists a natural number n such that Ker f n D Ker f nCk and Im f n D Im f nCk for all k 0. We show that M D Ker f n ˚ Im f n . Let x 2 M . Then f n .x/ D f 2n .y/ for some y 2 M . Hence x f n .y/ 2 Kerf n , and we obtain x D .x f n .y// C f n .y/ 2 Ker f n C Im f n . Moreover, if z 2 Ker f n \ Im f n , then z D f n .x/, for some x 2 M , and 0 D f n .z/ D f 2n .x/. Then Ker f n D Ker f 2n implies z D f n .x/ D 0. Therefore, Ker f n \ Im f n D 0, and the claim follows. Lemma 4.4. Let A be a finite dimensional K-algebra and M be a finite dimensional right A-module. Then M is an indecomposable A-module if and only if EndA .M / is a local K-algebra. Proof. Assume that M is an indecomposable A-module. Take an endomorphism f 2 EndA .M /. It follows from Lemma 4.3 that M has a decomposition M D Ker f n ˚Im f n , for some n 1, as a right A-module. Since M is indecomposable, we have either M D Ker f n and Im f n D 0, or Ker f n D 0 and M D Im f n . Therefore, f is either a nilpotent or an invertible element of EndA .M /. Applying Corollary 3.9 we conclude that EndA .M / is a local K-algebra. Conversely, assume that EndA .M / is a local K-algebra. Suppose M decomposes as M D M1 ˚ M2 with M1 , M2 nonzero A-submodules of M . Denote by u1 W M1 ! M and u2 W M2 ! M the canonical embeddings, and by p1 W M ! M1 and p2 W M ! M2 the canonical projections. Then idM D u1 p1 Cu2 p2 . It follows from Lemma 3.8 that one of the endomorphisms u1 p1 and u2 p2 D idM u1 p1 is an invertible element of EndA .M /. This is impossible since Im u1 p1 D M1 ¤ M and Im u2 p2 D M2 ¤ M . Therefore M is an indecomposable A-module. The following exchange property of decompositions of modules into direct sums of submodules is very useful. Lemma 4.5. Let A be a finite dimensional K-algebra, M a finite dimensional right A-module, and X1 an A-submodule of M . Assume M D Y1 ˚ Y2 is a decomposition of M into a direct sum of A-submodules Y1 , Y2 , and the restriction of the canonical projection p W M ! Y1 , for the decomposition M D Y1 ˚ Y2 , to X1 is an isomorphism from X1 to Y1 . Then we have a direct sum decomposition M D X1 ˚ Y2 :

4. The Krull–Schmidt theorem

43

Proof. We first show that Y1 X1 C Y2 . Take y1 2 Y1 . Since p.X1 / D Y1 by assumption, there exists x1 2 X1 such that y1 D p.x1 /. Moreover, x1 D y1 C y2 for some y2 2 Y2 , because M D Y1 ˚ Y2 . Then y1 D x1 y2 2 X1 C Y2 . Hence Y1 X1 C Y2 , and then M D Y1 C Y2 X1 C Y2 M . Therefore, M D X1 C Y2 . We claim that X1 \ Y2 D 0, and hence M D X1 ˚ Y2 . Indeed, take an element x 2 X1 \ Y2 . Then p.x/ D 0, because x 2 Y2 . This forces x D 0 because by assumption the restriction of p to X1 is an isomorphism. Hence, indeed X1 \ Y2 D 0. The following theorem has been proved by W. Krull [Kru] and O. Schmidt [Schm] (for groups with operators) on the basis of a theorem of R. Remak [Rem] about the uniqueness of decompositions of finite groups as direct products of indecomposable subgroups. A proof of the Krull–Schmidt theorem invoking local algebras has been proposed by G. Azumaya [Azu1] in 1950. Theorem 4.6. Let A be a finite dimensional K-algebra over a field K. (i) Every module M in mod A has a decomposition M D M1 ˚ ˚ Mm ; where M1 ; : : : ; Mm are indecomposable A-submodules of M . (ii) Let M1 ; : : : ; Mm and N1 ; : : : ; Nn be indecomposable modules in mod A such that there exists an isomorphism of A-modules M1 ˚ ˚ Mm Š N1 ˚ ˚ Nn : Then m D n and there exists a permutation of f1; : : : ; ng such that Mi Š N.i/ for each i 2 f1; : : : ; ng. Proof. (i) This follows because dimK M is finite for any module M in mod A. (ii) We proceed by induction on m. For m D 1, the module N1 ˚ ˚ Nn is indecomposable, and hence n D 1 and M1 Š N1 . Assume that m 2. Let f W M1 ˚ ˚ Mm ! N1 ˚ ˚ Nn be an isomorphism of A-modules and g its inverse. Then we obtain M1 ˚ ˚ Mm D N10 ˚ ˚ Nn0 ; where Nj0 D g.Nj / for each j 2 f1; : : : ; ng. Denote by uj W Nj0 ! M

and pj W M ! Nj0 ;

j 2 f1; : : :L ; ng, the canonical embeddings and projections given by the Lmdecomposin 0 0 0 tion M D j D1 Nj . Moreover, let M D M1 ˚ M1 , where M1 D iD2 Mi , and

44


let u W M1 ! M , u0 W M10 ! M , p W M ! M1 , p 0 W M ! M10 be the associated embeddings and projections, respectively. We obtain the equalities idM1 D pu D p idM u D p

n X

n X uj pj u D puj pj u:

j D1

j D1

Since M1 is indecomposable, the algebra EndA .M1 / is local, by Lemma 4.4. Then it follows from Lemma 3.8 that the Jacobson radical rad EndA .M1 / of EndA .M1 / consists of all noninvertible endomorphisms. Therefore, one of the endomorphisms puj pj u 2 EndA .M1 /, j 2 f1; : : : ; ng, is an isomorphism. Without loss of generality, we may assume that v D pu1 p1 u is an isomorphism. In particular, it follows from Lemma 4.1 that ˛ D p1 u W M1 ! N10 is a section and ˇ D pu1 W N10 ! M1 is a retraction in mod A. Since N10 is indecomposable, applying Lemma 4.2, we conclude that ˛ W M1 ! N10 and ˇ W N10 ! M1 are isomorphisms. L Let N100 D jnD2 Nj0 . Then we have two decompositions M1 ˚ M10 D M D N10 ˚ N100 of M into direct sums of A-submodules. Moreover, the restriction ˛ D p1 u of the projection p1 W M ! N10 , for the decomposition M D N10 ˚ N100 , to M1 is an isomorphism. Then it follows from Lemma 4.5 that there is a decomposition M D M1 ˚ N100 of M into a direct sum of A-modules. Obviously then we have dimK M10 D dimK N100 . Denote by p100 W M ! N100 the projection for the decomposition M D M1 ˚N100 . We claim that the composition f D p100 u0 W M10 ! N100 is an isomorphism of right A-modules. Since M10 and N100 have the same dimension over K, it is enough to show that f is a monomorphism. Take x 2 M10 such that f .x/ D 0. Then p100 .u0 .x// D 0 forces u0 .x/ 2 Ker p100 D M1 . On the other hand, u0 .x/ 2 M10 and M1 \ M10 D 0, because M D M1 ˚ M10 . Hence u0 .x/ D 0, and so x D 0, since u0 is a monomorphism. Therefore, indeed f is a monomorphism. Summing up we have proved that there exists an isomorphism of right Amodules m n M M Mi D M10 ! N100 D Nj0 ; iD2

j D2

and the wanted claim follows by the induction hypothesis.

Let A be a K-algebra and M be a right A-module. Then a right A-submodule X of M is said to be a direct summand of M if M D X ˚ Y for a right A-submodule Y of M .

5. Semisimple modules

45

Proposition 4.7. Let A be a finite dimensional K-algebra over a field K, M a finite dimensional right A-module, and M1 ; : : : ; Mm pairwise nonisomorphic indecomposable direct summands of M . Then M1 C C Mm is a direct sum and a direct summand of M . Proof. We proceed by induction on m. For m D 1, the claim is obvious because M1 is a direct summand of M . Assume that m 2 and M1 C C Mm1 is a direct sum and a direct summand of M . Then we have two decompositions of M , M1 ˚ ˚ Mm1 ˚ X D M D Mm ˚ Y into a direct sum of A-submodules, for some A-submodules X and Y of M . Let X D X1 ˚ ˚ Xr

and

Y D Y1 ˚ ˚ Ys

be decompositions of X and Y into direct sums of indecomposable A-submodules X1 ; : : : ; Xr and Y1 ; : : : ; Ys , respectively. Hence, we have two decompositions of M , M D M1 ˚ ˚ Mm1 ˚ X1 ˚ ˚ Xr ; M D Mm ˚ Y1 ˚ ˚ Ys ; into direct sums of indecomposable A-submodules. Since, by assumption, Mm © Mi for i 2 f1; : : : ; m 1g, it follows from Theorem 4.6 (and its proof) that r 1, s D r C .m 2/ 1, and there exists j 2 f1; : : : ; rg such that the restriction of the canonical projection pj W M ! Xj to Mm is an isomorphism. We may assume, without loss of generality, that j D 1. Applying now Lemma 4.5 we obtain a decomposition M D M1 ˚ ˚ Mm1 ˚ Mm ˚ X2 ˚ ˚ Xr of M into a direct sum of A-submodules. In particular, M1 C C Mm is a direct sum and a direct summand of M .

5 Semisimple modules Let A be a K-algebra over a field K. A right A-module S is said to be simple if S is nonzero and any A-submodule of S is either 0 or S . A direct sum of simple right A-modules is called a semisimple A-module. The aim of this section is to describe the basic properties and structure of finite dimensional semisimple modules over finite dimensional K-algebras. We start with the important lemma proved by I. Schur in his dissertation [Schu1] from 1901, which greatly simplified the representation theory of finite groups developed by F. G. Frobenius in 1880s and 1890s.

46


Lemma 5.1. Let A be a K-algebra and S, T two simple right A-modules. Then (i) HomA .S; T / D 0 unless S Š T , (ii) EndA .S / is a division K-algebra. Proof. (i) Assume f W S ! T is a nonzero homomorphism of A-modules. Then Ker f is a proper A-submodule of S , and hence Ker f D 0. Further, Im f is a nonzero A-submodule of T , and hence Im f D T . Therefore, f is an isomorphism. (ii) For S D T , it follows from (i) that every nonzero endomorphism in EndA .S / is an isomorphism, and so EndA .S / is a division K-algebra. Corollary 5.2. Let A be a K-algebra over an algebraically closed field K and S be a finite dimensional simple right A-module. Then EndA .S / Š K. Proof. Since S is a finite dimensional K-vector space, EndA .S / is a finite dimensional K-algebra. Take a nonzero endomorphism f 2 EndA .S /. Then f is an isomorphism, and hence a K-linear automorphism of the K-vector space S. Because K is algebraically closed, f has a nonzero eigenvalue 2 K, and then f idS 2 EndA .S / is an endomorphism with nonzero kernel. Hence f D idS , since S is a simple A-module. Therefore, the K-algebra homomorphism ' W K ! EndA .S /, given by './ D idS , is an isomorphism. We note also the following simple fact. Lemma 5.3. Let A be a K-algebra and M be a nonzero module in mod A. Then M contains a simple right A-submodule S . Proof. Take a nonzero A-submodule S of M of minimal dimension over K.

The following lemma provides a characterization of semisimple modules. Lemma 5.4. Let A be a K-algebra and M be a nonzero module in mod A. Then M is a semisimple A-module if and only if for any A-submodule N of M there exists an A-submodule L of M such that M D L ˚ N . Proof. Assume M is a semisimple A-module, say M D S1 ˚ ˚ Sn , where S1 ; : : : ; Sn are simple A-modules. Let N be a nonzero A-submodule of M . Let fSi1 ; : : : ; Sir g be a maximal family in the set fS1 ; : : : ; Sn g such that the intersection of N with the module L D Si1 ˚ ˚ Sir is zero. Then for any t 2 f1; : : : ; ng n fi1 ; : : : ; ir g, we have N \ .L C S t / ¤ 0, and hence .N C L/ \ S t ¤ 0, or equivalently, S t L C N , because S t is a simple module. Therefore, we obtain M L C N , and consequently M D L ˚ N . The converse implication follows from Lemma 5.3 by induction on dimK M .


47

Corollary 5.5. Let A be a K-algebra and M be a semisimple module in mod A. Then the following holds true. (i) Every nonzero A-submodule of M is semisimple. (ii) Every nonzero factor A-module of M is semisimple. Proof. (i) Let N be a nonzero right A-submodule of M . Since M is a semisimple right A-module, it follows from Lemma 5.4 that there exists a right A-submodule L of M such that M D L ˚ N . Take now a right A-submodule Z of N . Then Z is a right A-submodule of M and, applying Lemma 5.4 again, we conclude that M D Y ˚ Z for a right A-submodule Y of M . Let X D N \ Y . Obviously X is a right A-submodule of N such that X \ Z D N \ .Y \ Z/ D 0. We claim that N D X CZ, and consequently N D X ˚Z. Let n 2 N . Since N M D Y ˚Z, we have n D y C z for some y 2 Y and z 2 Z. Then y D n z 2 N , because Z N , and so y 2 N \ Y . Hence, indeed N X C Z, and then N D X C Z. Therefore, Z is a direct summand of N . Thus, by Lemma 5.4, N is a semisimple module in mod A. (ii) Let N be a proper right A-submodule of M and V D M=N the associated factor module. Since M is a semisimple module in mod A, by Lemma 5.4, we have M D L ˚ N for a right A-submodule L of N . Moreover, N ¤ M forces L ¤ 0, and hence L is a semisimple module in mod A, by (i). Finally, observe that the canonical epimorphism W M ! M=N , .m/ D m C M for any m 2 M , induces an isomorphism of right A-modules L ! V . Therefore, V is a semisimple module in mod A. We describe now the structure of finite dimensional modules over the K-algebras A= rad A. In particular, we will show that all such modules are semisimple. Let A be a K-algebra. Two idempotents e1 ; e2 2 A are said to be orthogonal if e1 e2 D 0A D e2 e1 . An idempotent e 2 A is said to be primitive if e is nonzero and cannot be written as a sum e1 C e2 , where e1 and e2 are nonzero orthogonal idempotents of A. Recall that 0A and 1A are called the trivial idempotents of A. Lemma 5.6. Let A be a K-algebra and x a nonzero idempotent of A. The following statements hold. (i) Let e 2 xA be an idempotent of A different from 0A and x such that ex D e D xe. Then x e is an idempotent of A, different from 0A and x, orthogonal to e, and there is a nontrivial decomposition xA D eA ˚ .x e/A of xA into a direct sum of right A-submodules. (ii) Assume that xA D M ˚ N is a nontrivial decomposition of xA into a direct sum of right A-submodules. Then there is a pair e; f of orthogonal idempotents of A, different from 0A and x, such that x D eCf , ex D e D xe, f x D f D xf , and M D eA, N D fA.

48


Proof. (i) Let e 2 xA be an idempotent of A different from 0A and x such that ex D e D xe. Then .x e/2 D x 2 xe ex C e 2 D x 2 e e C e D x e and e.x e/ D ex e 2 D e e D 0A , .x e/e D xe e 2 D e e D 0A . Hence, x e is an idempotent of A different from 0A and x, orthogonal to e, and eA, .x e/A are nonzero right A-submodules of xA. For any a 2 A, we have xa D ea C .x e/a 2 eA C .x e/A, and hence xA D eA C .x e/A. Moreover, if y 2 eA \ .x e/A, then y D eb D .x e/c for some b; c 2 A, and then y D .x e/c D .x e/.x e/c D .x e/.eb/ D .xe e/b D 0A , because xe D e. Therefore, by Lemma 2.6, we have a nontrivial decomposition xA D eA ˚ .x e/A of xA into a direct sum of right A-submodules. (ii) We write x D e C f , with e 2 M and f 2 N . Clearly then eA M and fA N . Moreover, for any a 2 A we have xa D ea C f a 2 eA C fA, and hence xA D eACfA. Since xA D M ˚N we then conclude that M D eA and N D fA. Observe also that there is a nontrivial decomposition AA D xA ˚ .1A x/A of AA into a direct sum of right A-submodules. Indeed, 1A D x C .1A x/ implies AA D xA C .1A x/A and xA \ .1A x/A D 0A , because xa D .1A x/b for a; b 2 A forces xa D x.xa/ D x.1A x/b D .x x 2 /b D 0A , so the claim follows from Lemma 2.6. Therefore, we obtain a decomposition AA D eA ˚ fA ˚ .1A x/A of AA into a direct sum of A-submodules. Since 1A D xC.1A x/ D eCf C.1A x/, we have e D 1A e D e 2 C f e C .1A x/e and f D 1A f D ef C f 2 C .1A x/f , and consequently e D e 2 , f e D 0A , .1A x/e D 0A , ef D 0A , f 2 D f , .1A x/f D 0A . In particular, we obtain that e D xe and f D xf . Further, x D e C f then implies ex D e 2 C ef D e 2 D e and f x D f e C f 2 D f 2 D f . Therefore, e and f are orthogonal idempotents of A, with ex D e D xe and f x D f D xf . Finally, xA D M ˚ N with M D eA and N D fA nonzero A-submodules of AA , so e and f are different from 0A and x. We have the following immediate consequences of Lemma 5.6. Lemma 5.7. Let A be a K-algebra. The following statements hold. (i) If e is a nontrivial idempotent of A, then there is a nontrivial decomposition AA D eA ˚ .1A e/A of AA into a direct sum of right A-submodules. (ii) Assume that AA D M ˚ N is a nontrivial decomposition of AA into a direct sum of right A-submodules. Then there is a pair e; f of nontrivial orthogonal idempotents of A such that 1A D e C f and M D eA, N D fA. For two idempotents e and x of a K-algebra A, we say that e is a summand of x, and write e x, if ex D e D xe. Then it follows from Lemma 5.6 that, if e x, then x D e C f where f is an idempotent of A, orthogonal to e, and a summand of x. Clearly, every idempotent e of A is a summand of the trivial idempotent 1A . Corollary 5.8. Let A be a K-algebra and e 2 A an idempotent. Then e is primitive if and only if the right A-module eA is indecomposable.


49

Corollary 5.9. Let A be a finite dimensional K-algebra. Then there is a decomposition AA D e1 A ˚ ˚ en A of AA into a direct sum of indecomposable right ideals, where e1 ; : : : ; en are pairwise orthogonal primitive idempotents of A such that 1A D e1 C C en . Moreover, every decomposition of AA into a direct sum of indecomposable right A-submodules is of this form. Corollary 5.10. Let A be a finite dimensional K-algebra and e1 ; : : : ; en be a set of pairwise orthogonal primitive idempotents of A with 1A D e1 C Cen . Then there is a decomposition AA D e1 A˚ ˚en A of A into a direct sum of indecomposable right A-modules. Corollary 5.11. Let A be a finite dimensional K-algebra and e a primitive idempotent of A. Then there are pairwise orthogonal primitive idempotents e1 ; : : : ; en of A such that 1A D e1 C C en and e1 D e. Proposition 5.12. Let A be a finite dimensional K-algebra and B D A= rad A. The following statements hold. (i) Every nonzero right ideal I of B is a direct sum of simple right ideals of B of the form eB, where e is a primitive idempotent of B. In particular, the right B-module BB is semisimple. (ii) Any nonzero module M in mod B is isomorphic to a direct sum of simple right ideals of B of the form eB, where e is a primitive idempotent of B. Proof. (i) Assume I is a nonzero right ideal of B. It follows from Lemma 5.3 that I contains a simple right B-submodule S . Consider the right B-submodule S 2 of S consisting of all finite sums of elements of the form xy, with x; y 2 S . We claim that S 2 ¤ 0. Indeed, suppose that S 2 D 0. Then for any x 2 S and b 2 B we have .1B xb/.1B C xb/ D 1B , and so x 2 rad B, by Lemma 3.1. Consequently, S rad B D 0, by Corollary 3.2, a contradiction. Hence S 2 is a nonzero submodule of S, and so S 2 D S, because S is simple. Then there exists x 2 S with xS ¤ 0, and hence S D xS, because xS is a right B-submodule of S. In particular, x D xe for some e 2 S. Consider the homomorphism f W S ! S of right B-modules given by f .y/ D xy for y 2 S . Since f .e/ D xe D x ¤ 0B , f is a nonzero homomorphism, and consequently an isomorphism, by Lemma 5.1. Further, we have f .e 2 e/ D f .e 2 / f .e/ D xee xe D xe xe D 0B ; and hence e 2 e D 0B . Therefore, e is a nonzero idempotent of B, and S D eB. It follows from Lemma 5.7 that BB D eB ˚.1B e/B, and hence I D S ˚.1B e/I . Because dimK .1B e/I < dimK I , the statement (i) follows by induction on the dimension of I . (ii) Let M be a nonzero module in mod B. Let m1 ; : : : ; mr be elements of M such that M D m1 B C C mr B. Consider the surjective epimorphism of

50


right B-modules ' W B r ! M given by '..b1 ; : : : ; br // D m1 b1 C C mr br for .b1 ; : : : ; br / 2 B r . Then M Š B r = Ker '. It follows from (i) that B r is a semisimple right B-module which is a direct sum of simple right ideals of the form eB, where e is a primitive idempotent of B. Further, by Lemma 5.4, there exists a right B-submodule L of B r such that B r D L ˚ Ker '. Applying Theorem 4.6, we conclude that L and Ker ' are direct sums of simple right ideals of the form eB for primitive idempotents e of B. Therefore, the module M , isomorphic to L, is isomorphic to a direct sum of simple right ideals of B of the form eB, where e is a primitive idempotent of B. Let A be a finite dimensional K-algebra and M be a module in mod A. The Jacobson radical (briefly, radical) rad M of M is the intersection of all the maximal right A-submodules of M . Observe that the radical rad AA of the right A-module AA is the radical rad A of the algebra A. We will describe basic properties of the radicals of modules as well as provide another characterization of semisimple modules. Proposition 5.13. Let A be a finite dimensional K-algebra and M , N be modules in mod A. The following statements hold. (i) An element m 2 M belongs to rad M if and only if f .m/ D 0 for any f 2 HomA .M; S / and any simple right A-module S . (ii) rad.M ˚ N / D rad M ˚ rad N . (iii) If f 2 HomA .M; N /, then f .rad M / rad N . (iv) rad M D M rad A. (v) If L C rad M D M for a right A-submodule L of M , then L D M . Proof. (i) Observe that if L is a maximal right A-submodule of M then M=L is a simple right A-module, by Lemma 3.17 (i), and there is a canonical epimorphism p W M ! M=L with Ker p D L. Conversely, for a nonzero homomorphism f 2 HomA .M; S /, where S is a simple right A-module, Ker f is a maximal right A-submodule of M . Then the claim (i) follows. The statements (ii) and (iii) are direct consequences of (i). We prove now that (iv) also holds. For any element m 2 M , consider the homomorphism fm W A ! M of right A-modules defined by fm .a/ D ma for a 2 A. Then it follows from (iii) that for a 2 rad A we have ma D fm .a/ 2 fm .rad A/ rad M , and consequently M rad A rad M . We show that also rad M M rad A holds. Observe first that .M=M rad A/ rad A D 0. This allows us to consider M=M rad A as a right module over the algebra A= rad A by the multiplication .m C M rad A/.a C rad A/ D ma C M rad A;


51

for m 2 M and a 2 A. It follows from Proposition 5.12 that every module in mod.A= rad A/ is semisimple. Therefore, the right .A= rad A/-module M=M rad A is a direct sum of simple .A= rad A/-modules. The radical rad S of every simple module S is trivially zero. Then (ii) implies that the radical of any semisimple module in mod A is also zero. Hence we conclude that rad.M=M rad A/ D 0. Consider now the canonical epimorphism W M ! M=M rad A of right A-modules. Applying (iii) we obtain that .rad M / rad.M=M rad A/ D 0, and consequently rad M Ker D M rad A. This shows that rad M D M rad A. For (v), let L be a right A-submodule of M with L C rad M D M . Suppose L ¤ M . Then L is contained in a maximal right A-submodule X of M , and we obtain M D L C rad M X ¤ M , a contradiction. Hence L D M . Corollary 5.14. Let A be a finite dimensional K-algebra and M be a module in mod A. The following statements hold. (i) The right A-module M= rad M is semisimple and it is a right module over the K-algebra A= rad A. (ii) If L is a right A-submodule of M such that M=L is a semisimple A-module, then rad M L. Proof. (i) We know from Proposition 5.13 that rad M D M rad A. Therefore, .M= rad M / rad A D 0 and M= rad M is a right A= rad A-module by the action .m C rad M /.a C rad A/ D ma C rad M , for m 2 M and a 2 A. Since, by Proposition 5.12, every module in mod.A= rad A/ is semisimple, we conclude that M= rad M is a semisimple right A= rad A-module, and consequently also a semisimple right A-module. (ii) Assume that L is a right A-submodule of M such that M=L is a semisimple A-module. Let W M ! M=L be the canonical epimorphism, .m/ D m C L for m 2 M . Then it follows from Proposition 5.13 (iii) that .rad M / rad.M=L/ D 0, because M=L is semisimple. Hence rad M Ker D L. Corollary 5.15. Let A be a finite dimensional K-algebra and M be a module in mod A. Then M is a semisimple module if and only if rad M D 0. Proof. Observe that if S is a simple right A-module, then S ¤ S rad A, by Lemma 3.3, and consequently S rad A D 0. Then rad S D S rad A D 0. Let M be a semisimple A-module. Then M D S1 ˚ ˚ Sr , where S1 ; : : : ; Sr are simple right A-modules. Applying Proposition 5.13 (ii), we then obtain rad M D rad S1 ˚ ˚ rad Sr D 0. Conversely, assume that rad M D 0. Then, by Corollary 5.14 (i), M D M= rad M is a semisimple A-module. The following proposition describes the relation between the indecomposable direct summands of a finite dimensional K-algebra A, regarded as a right A-module, and the indecomposable direct summands of its factor algebra B D A= rad A,

52


regarded as a right B-module, observed already by T. Nakayama in his paper [Nak1] from 1938. Proposition 5.16. Let A be a finite dimensional K-algebra, B D A= rad A, e a nonzero idempotent of A and eN D e C rad A the associated idempotent of B. The following conditions are equivalent. (i) eA is an indecomposable right A-module. (ii) e rad A is a unique maximal right A-submodule of eA. (iii) eB N is a simple right B-module. Proof. It follows from Proposition 5.13 (iv) that e rad A D rad eA. Then e rad A is a maximal right A-submodule of eA if and only if rad eA is a unique maximal right A-submodule of eA. Since eB N Š eA=e rad A, the conditions (ii) and (iii) are equivalent (see Lemma 3.17). Moreover, (iii) implies (i). Indeed, if eA D X ˚ Y for nonzero right A-submodules X and Y of eA then, by Lemma 5.6, e D e1 C e2 for two orthogonal idempotents e1 and e2 different from e, and X D e1 A and Y D e2 A. But then eB N D eN1 B ˚ eN2 B, with eN1 D e1 C rad A and eN2 D e2 C rad A, is a decomposition of eB N into a direct sum of nonzero right B-submodules, a contradiction because eB N is a simple right B-module. Assume now that (i) holds, that is, eA is indecomposable. Then, by Corollary 5.8, e is a primitive idempotent of A. We claim that eB N is a simple right B-module, and hence (iii) holds. Suppose that eB N is not simple. Then it follows from Proposition 5.12 and Lemma 5.6 that eB N D f1 B ˚ f2 B, where f1 ; f2 are nonzero idempotents of B such that eN D f1 C f2 and f1 f2 D 0B D f2 f1 . Applying Lemma 3.12, we conclude that there are idempotents e1 , e2 in A such that e D e1 Ce2 , e1 e2 D 0A D e2 e1 , and f1 D e1 Crad A, f2 D e2 Crad A. Moreover, applying Lemma 5.6, we obtain a decomposition eA D e1 A ˚ e2 A of eA into a direct sum of nonzero right A-submodules, which contradicts indecomposability of eA. Therefore, indeed eB N is a simple right B-module. Corollary 5.17. Let A be a finite dimensional K-algebra, e1 ; : : : ; en a set of pairwise orthogonal primitive idempotents of A with 1A D e1 C C en , and B D A= rad A. Then every simple module S in mod B is isomorphic to a right B-module ei A=ei rad A for some i 2 f1; : : : ; ng. Proof. Let eN1 D e1 C rad A; : : : ; eNn D en C rad A be the idempotents of B associated to the idempotents e1 ; : : : ; en . It follows from Proposition 5.16 that eN1 ; : : : ; eNn is a set of pairwise orthogonal primitive idempotents of B such that 1B D eN1 C C eNn and eN1 B; : : : ; eNn B are simple right B-modules. Moreover, it follows from Corollary 5.10 that we have in mod B a direct sum decomposition BB D eN1 B ˚ ˚ eNn B. Observe also that we have canonical isomorphisms of right B-modules eNi B Š ei A=ei rad A, for i 2 f1; : : : ; ng. On the other hand, it


53

follows from Proposition 5.12 that every simple right B-module S is isomorphic to a right B-module of the form f B for some primitive idempotent f of B. Moreover, by Corollary 5.11, there exists a set f1 ; : : : ; fm of pairwise orthogonal primitive idempotents of B such that 1B D f1 C C fm and f1 D f . Applying Corollary 5.10 and Proposition 5.16, we conclude that there is in mod B a direct sum decomposition BB D f1 B ˚ ˚ fm B, where f1 B; : : : ; fm B are simple right B-modules. Therefore, by the Krull–Schmidt theorem (Theorem 4.6), we obtain that f B D f1 B is isomorphic to a right B-module eNi B for some i 2 f1; : : : ; ng. Obviously then S is isomorphic to ei A=ei rad A. Summing up our discussion, we established that for a finite dimensional Kalgebra A the semisimple modules in mod A coincide with the (semisimple) modules in mod.A= rad A/, and are finite direct sums of simple right ideals of B D A= rad A of the form eB, N for primitive idempotents eN D e C rad A of B induced by primitive idempotents e of A. Let A be a finite dimensional K-algebra and M a nonzero module in mod A. We assign to M two natural semisimple A-modules top.M / D M= rad M called the top of M , and the sum soc.M / D

X

S;

S2SM

where SM is the set of all simple right A-submodules of M , called the socle of M . Observe that rad soc.M / D soc.M / rad A D 0, and hence it follows from Corollary 5.15 that soc.M / is indeed a semisimple right A-submodule of M . Therefore, top.M / is the largest semisimple factor A-module of M (see Corollary 5.14 (ii)) while soc.M / is the largest semisimple A-submodule of M . Further, if f W M ! N is a homomorphism in mod A, then f .rad M / rad N , by Proposition 5.13 (iii), and hence we obtain the induced A-homomorphism top.f / W top.M / ! top.N / given by top.f /.m C rad M / D f .m/ C rad N for m 2 M . Similarly, the restriction f to soc.M / induces an A-homomorphism soc.f / W soc.M / ! soc.N /, since f .soc.M // is a semisimple A-submodule of N , by Corollary 5.5 (i), and so is contained in soc.N /. We obtain also the following useful characterizations of epimorphisms and monomorphisms in mod A. Lemma 5.18. Let A be a finite dimensional K-algebra and f W M ! N a nonzero homomorphism in mod A. Then (i) f is an epimorphism if and only if top.f / is an epimorphism. (ii) f is a monomorphism if and only if soc.f / is a monomorphism.

54


Proof. (i) Obviously if f is an epimorphism then top.f / is an epimorphism. Conversely, assume that top.f / is an epimorphism. Then we obtain Im f Crad N D N , and hence Im f D N , by Proposition 5.13 (v). Consequently,ˇ f is an epimorphism. (ii) Clearly, if f is a monomorphism then soc.f / D f ˇsoc.M / is a monomorphism. Assume soc.f / is a monomorphism. Then we have soc.M / \ Ker f D 0, and consequently Ker f D 0, because every nonzero finite dimensional right Amodule contains a simple A-submodule, by Lemma 5.3. Hence f is indeed a monomorphism. Let A be a finite dimensional K-algebra and M a nonzero module in mod A. For each i 1, we define radi M D M.rad A/i . It follows from Proposition 5.13 (iv) that radi M D rad.radi1 M /, where rad0 M D M . Moreover, if radiC1 M D radi M for some i 0 then .radi M / rad A D radiC1 M D radi M , and hence radi M D 0, by Lemma 3.3. Further, we have radn M D 0 for some n 1, because the radical rad A is nilpotent (Corollary 3.4). Observe also that if radi M ¤ radiC1 then radi M= radiC1 A-module. Indeed, we have M M is ai semisimpleiC1 i iC1 rad rad M= rad rad A D 0, and the claim M D M.rad A/ =M.rad A/ follows from Corollary 5.15. Summing up the above results, we have the decreasing sequence of right A-submodules of M M rad M rad2 M radn1 M radn M D 0 such that the modules radi M= radiC1 M , i 2 f0; 1; : : : ; n 1g, are semisimple right A-modules. The sequence is called the radical series of M or the Loewy series of M , and the number n is said to be the Loewy length of the module M , and denoted by ``.M /. Moreover, ``.A/ is called the Loewy length of the algebra A. Therefore, the semisimple modules in mod A are the modules of the Loewy length one. The Loewy length of the zero module 0 in mod A is defined to be 0. We also note that ``.A/ D ``.AA / D ``.A A/. Corollary 5.19. Let A be a finite dimensional K-algebra and M be a module in mod A. Then ``.M / ``.A/. Proof. This follows from the equalities radi M D M.rad A/i for i 1.

The following direct consequence of Lemma 3.6 describes the Loewy length of the bound quiver algebras. Corollary 5.20. Let Q be a finite quiver and I an admissible ideal of the path algebra KQ. Then ``.KQ=I / 1 is the length of the longest path in Q which does not belong to I . In particular, we obtain the following fact. Corollary 5.21. Let Q be a finite acyclic quiver. Then ``.KQ/ 1 is the length of the longest path in Q.


55

Examples 5.22. (a) Let Q be the quiver 1 o

˛

2;

K a field, and S.1/ W K o

0, P W K o

K , S.2/ W 0 o

1

K

the indecomposable representations in repK .Q/ (see Example 2.9 (b)). Then S.1/ D soc.P / D rad P

S.2/ D top.P / D P = rad P:

and

Moreover, ``.P / D 2. (b) Let K be a field, Q the quiver $

˛

ˇ

o 1

2

and I the admissible ideal in KQ generated by ˛ 2 (see Examples 2.9 (d) and 2.12). Consider the indecomposable representation MW

01 00

(

K2 o

10 01

K2

in repK .Q; I /. Then we have the commutative diagrams

01 00

(

K2 o

01 00

10 01

( K2 o

1 0

K

10 01

01 00

(

K2 o

1 0

K2 ,

01 00

10 01

0 1

K

( K2 o

10 01

0 1

K2

of K-vector spaces, and hence YW

01 00

(

K2 o

1 0

and

K

ZW

01 00

(

K2 o

0 1

K

are maximal subrepresentations of M in repK .Q; I /. Clearly, these are the unique maximal subrepresentations of M , and consequently ( rad M D Y \ Z W 00 10 0: K2 o Moreover, S.1/ W

0

'

Ko

0

is a unique maximal subrepresentation of rad M and hence rad2 M D rad.rad M / D S.1/. Hence ``.M / D 3 and S.1/ D soc.M /. Observe also that top.M / D M= rad M D S.2/ ˚ S.2/, where $ o S.2/ W 0 K:

56


6 Semisimple algebras In this section we prove the Wedderburn theorem from 1908 [Wed] describing the structure of finite dimensional algebras over a field for which all finite dimensional modules are semisimple, and characterize the semisimple group algebras of finite groups. Let K be a field. The following lemma will be useful. Lemma 6.1. Let A be a finite dimensional K-algebra. Then there exists an isomorphism of K-algebras A ! EndA .AA /. Proof. Consider the map ' W A ! EndA .AA / given by 'a .x/ D ax for any a; x 2 A. Observe that 'a 2 EndA .AA / for any a 2 A, because 'a .xb/ D a.xb/ D .ax/b D 'a .x/b for x; b 2 A. Further, for a; b; x 2 A and 2 K, we have 'aCb .x/ D .a C b/x D ax C bx D 'a .x/ C 'b .x/; 'a .x/ D .a/x D .ax/ D 'a .x/; '0A .x/ D 0A x D 0A ; '1A .x/ D 1A x D x D idA .x/; 'ab .x/ D .ab/x D a.bx/ D 'a .bx/ D 'a .'b .x// D .'a 'b /.x/; and consequently ' W A ! EndA .AA / is a homomorphism of K-algebras. Further, ' is a monomorphism, because 'a D 'b forces a D 'a .1A / D 'b .1A / D b. Finally, let f 2 EndA .AA / and take a D f .1A /. Then for any x 2 A, we have f .x/ D f .1A x/ D f .1A /x D ax D 'a .x/, and hence f D 'a . This shows that ' is also an epimorphism. Lemma 6.2. Let F be a finite dimensional division K-algebra, n a positive integer, and Mn .F / the full n n matrix algebra over F . Then rad Mn .F / D 0. Proof. The identity matrix In of Mn .F / has the canonical decomposition In D E11 C E22 C C Enn as a sum of the diagonal elementary matrices Ei i , i 2 f1; : : : ; ng, which are obviously pairwise orthogonal primitive idempotents of Mn .F /. For each r 2 f1; : : : ; ng, we have ˚ Err Mn .F / D Œaij 2 Mn .F / j aij D 0 for i ¤ r : We claim that Sr D Err Mn .F /, r 2 f1; : : : ; ng, are simple right Mn .F /-modules. Fix r 2 f1; : : : ; ng. Let A D Œaij be a nonzero element of Sr , and hence ars ¤ 0 for some s 2 f1; : : : ; ng. For each element 2 F and t 2 f1; : : : ; ng, consider the matrix Est ./ 2 Mn .F / having coefficient on the position .s; t / and coefficient 0 elsewhere. Then, for each 2 F and t 2 f1; : : : ; rg, we obtain 1 2 Err Mn .F / D Sr ; Ert ./ D AEst ars

6. Semisimple algebras

57

and consequently AMn .F / D Sr . Therefore, indeed S1 ; : : : ; Sn are simple right Mn .F /-modules, Mn .F / is a semisimple right Mn .F /-module, and consequently rad Mn .F / D 0, by Corollary 5.15. A finite dimensional K-algebra A is called semisimple if rad A D 0. The following theorem is the Wedderburn structure theorem of semisimple algebras. Theorem 6.3. Let A be a finite dimensional K-algebra over a field K. The following conditions are equivalent. (i) A is a semisimple K-algebra. (ii) Every module in mod A is semisimple. (iii) There exist positive integers n1 ; : : : ; nr and finite dimensional division Kalgebras F1 ; : : : ; Fr such that A Š Mn1 .F1 / Mnr .Fr / as K-algebras. Proof. Let A be a semisimple K-algebra. Then rad A D 0, A= rad A D A, and hence every module in mod A is semisimple, by Proposition 5.12. Hence (i) implies (ii). Assume (ii) holds. Then AA is a semisimple module in mod A, and there exists an isomorphism of right A-modules ! S1n1 ˚ S2n2 ˚ ˚ Srnr ; AA

where S1 ; S2 ; : : : ; Sr are pairwise nonisomorphic simple right A-modules, n1 , n n2 ; : : : ; nr are positive integers, and Sj j is the direct sum Sj ˚ ˚ Sj of nj copies of Sj , for any j 2 f1; : : : ; rg. It follows from Lemma 5.1 that F1 D EndA .S1 /; F2 D EndA .S2 /; : : : ; Fr D EndA .Sr / are finite dimensional division K-algebras, and HomA .Si ; Sj / D 0 for i; j 2 f1; : : : ; rg, i ¤ j . Further, by Lemma 6.1, the map ' W A ! EndA .AA / given by 'a .x/ D ax for a; x 2 A, is an isomorphism of K-algebras. Therefore, we obtain K-algebra isomorphisms ! EndA .S1n1 ˚ S2n2 ˚ ˚ Srnr / A ! EndA .AA / ! EndA .S1n1 / EndA .S2n2 / EndA .Srnr / ! Mn1 .F1 / Mn2 .F2 / Mnr .Fr /:

Hence (ii) implies (iii). Assume now that there exists an isomorphism of K-algebras AŠ

r Y iD1

Mni .Fi /

58


for some positive integers n1 ; : : : ; nr and finite dimensional division K-algebras F1 ; : : : ; Fr . Observe that, if Mj is a maximal right ideal of Mnj .Fj /, then zj D M

jY 1

Mni .Fi / Mj

rad

r Y

Qr iD1

Mni .Fi /

iDj C1

iD1

is a maximal right ideal of

r Y

Mni .Fi /. Therefore, we obtain

r Y Mni .Fi / rad Mni .Fi / D 0;

iD1

by Lemma 6.2. Hence rad A Š rad

iD1

Qr iD1

Mni .Fi / D 0, and (iii) implies (i).

Corollary 6.4. Let A be a finite dimensional K-algebra over a field K and M a semisimple module in mod A. Then EndA .M / is a semisimple K-algebra. Proof. Since M is a semisimple module in mod A, there exists an isomorphism of right A-modules M ! S1n1 ˚ S2n2 ˚ ˚ Srnr ; where S1 ; S2 ; : : : ; Sr are pairwise nonisomorphic simple right A-modules, and n1 ; n2 ; : : : ; nr are positive integers. Then we have an isomorphism of K-algebras ! Mn1 .F1 / Mn2 .F2 / Mnr .Fr /; EndA .M /

where F1 D EndA .S1 /; F2 D EndA .S2 /; : : : ; Fr D EndA .Sr / are finite dimensional division K-algebras. Hence, it follows from Theorem 6.3 that EndA .M / is a semisimple K-algebra. A finite dimensional K-algebra A is called simple if 0 and A are unique two-sided ideals of A. As a consequence of Theorem 6.3 we obtain the following description of simple algebras. Corollary 6.5. Let A be a finite dimensional K-algebra over a field K. The following conditions are equivalent. (i) A is a simple K-algebra. (ii) There exists a positive integer n and a finite dimensional division K-algebra F such that A Š Mn .F / as K-algebras.


59

Proof. Let A be a simple K-algebra. Then A is a semisimple K-algebra, because rad A is a two-sided ideal of A, by Corollary 3.2. Moreover, A is an indecomposable K-algebra, because otherwise there exists in A a nontrivial central idempotent e (see Lemma 3.14) and eAe is a two-sided ideal of A different from 0 and A. Therefore, it follows from Theorem 6.3 that A is isomorphic to matrix K-algebra Mn .F / for some positive integer n and a finite dimensional division K-algebra F . Hence (i) implies (ii). In order to prove the implication (ii) ) (i), it is enough to show that the matrix algebra Mn .F /, for a positive integer n and a finite dimensional division K-algebra F , is a simple K-algebra. For i; j 2 f1; : : : ; ng, let Eij be the elementary matrix in Mn .F / with 1F in the position .i; j / and 0F elsewhere. To show that Mn .F / is a simple K-algebra, it is enough to show that for any nonzero element M 2 Mn .F / the two-sided ideal I of Mn .F / generated by M is Mn .F /. Let M D Œaij 2 Mn .F / and a D ars ¤ 0 for some r; s 2 f1; : : : ; ng. Then, for any i 2 f1; : : : ; ng, the element aEi i D Eir MEsi belongs to I . Since a ¤ 0, the diagonal elementary matrices E11 ; : : : ; Enn belong to I . This shows that the identity matrix In of Mn .F / belongs to I , and hence I D Mn .F /. Corollary 6.6. Let A be a finite dimensional K-algebra over a field K. Then A is a semisimple algebra if and only if A is isomorphic to a product A1 Ar of simple K-algebras A1 ; : : : ; Ar . Proposition 6.7. Let A be a finite dimensional simple K-algebra over a field K. Then any two simple modules in mod A are isomorphic. Proof. It follows from Corollary 6.5 that there exists a positive integer n and a finite dimensional division K-algebra F such that A Š Mn .F / as K-algebras. Further, by the proof of Lemma 6.2, we have a decomposition of right Mn .F /-modules Mn .F / D S1 ˚ S2 ˚ ˚ Sr ; where

˚ Sr D Err Mn .F / D Œaij 2 Mn .F / j aij D 0 for i ¤ r ;

r 2 f1; : : : ; ng, are simple right Mn .F /-modules. Observe also that, for any r 2 f1; : : : ; ng, there is the canonical isomorphism S1 ! Sr of right Mn .F /modules given by the shift of the first row of any matrix in S1 to the r-th row of the corresponding matrix in Sr . Observe also that dimK Sr D n dimK F for any r 2 f1; : : : ; ng. On the other hand, since rad Mn .F / D 0, it follows from Proposition 5.12 that every simple module in mod Mn .F / is isomorphic to a module Sr , for some r 2 f1; : : : ; ng, and hence to S1 . Since A Š Mn .F /, we conclude that any two simple modules in mod A are isomorphic. The following corollary provides an isomorphism criterion for finite dimensional modules over simple algebras.

60


Corollary 6.8. Let A be a finite dimensional simple K-algebra over a field K and M , N be nonzero modules in mod A. The following conditions are equivalent. (i) M Š N in mod A. (ii) dimK M D dimK N . Proof. Since A is a simple K-algebra, we conclude from Theorem 6.3 that every module in mod A is semisimple. Moreover, by Proposition 6.7, every simple module in mod A is isomorphic to a fixed simple right A-module S. Hence, there are isomorphisms of right A-modules M Š Sm

and

N Š Sn

for some positive integers m and n. Clearly, then we have dimK M D m dimK S and dimK N D n dimK S. The equivalence of (i) and (ii) is now obvious. We exhibit now some consequences of Theorem 6.3. Corollary 6.9. Let A be a finite dimensional K-algebra over a field K. Then there exist positive integers n1 ; : : : ; nr and finite dimensional division K-algebras F1 ; : : : ; Fr such that A= rad A Š Mn1 .F1 / Mnr .Fr / as K-algebras. Moreover, if K is an algebraically closed field, then there is an isomorphism of K-algebras A= rad A Š Mn1 .K/ Mnr .K/: Proof. This follows from Corollary 3.2, Theorem 6.3 and the fact that every finite dimensional division K-algebra F over an algebraically closed field K is isomorphic to K (see Exercise 12.38). Corollary 6.10. Let A be a finite dimensional K-algebra over an algebraically closed field K. Then A is semisimple K-algebra if and only if there is an isomorphism of K-algebras A Š Mn1 .K/ Mnr .K/; for some positive integers n1 ; : : : ; nr . For a finite dimensional K-algebra over a field K, we denote by J.A/ the intersection of all maximal two-sided ideals of A. Lemma 6.11. Let A be a finite dimensional K-algebra over a field K. Then rad A D J.A/.


61

Proof. We show first that rad A J.A/. Take a maximal two-sided ideal I of A. Then it follows from Lemma 3.17 (iii) that B D A=I is a simple K-algebra. In particular, B is a semisimple K-algebra, and hence B is a semisimple right Bmodule. But then B is a semisimple right A-module, by Corollary 5.17. Then, applying Corollary 5.15, we conclude that B rad A D 0, or equivalently, rad A I . This shows that rad A J.A/. On the other hand, it follows from Lemma 3.17 (iii), that every maximal two-sided ideal of A= rad A is of the form I = rad A for a maximal two-sided ideal I of A, and consequently J.A/= rad A D J.A= rad A/. Since A= rad A is a semisimple algebra, applying Theorem 6.3, we obtain that there exist positive integers n1 ; : : : ; nr and finite dimensional division K-algebras F1 ; : : : ; Fr such that A= rad A Š Mn1 .F1 / Mnr .Fr / as K-algebras. Then we obtain isomorphisms of K-vector spaces J.A= rad A/ Š J

r Y

r Y Mni .Fi / D J Mni .Fi /

iD1

iD1

with J Mni .Fi / D 0 for any i 2 f1; : : : ; rg, by Corollary 6.5. Hence, we conclude that J.A= rad A/ D 0, which implies rad A D J.A/. Let A be a finite dimensional K-algebra over a field K, M a module in mod A and ƒM D EndA .M /. Then M has a natural structure of a left ƒM -module given by f m D f .m/ for f 2 ƒM and m 2 M . Indeed, we have the equalities .fg/m D .fg/.m/ D f .g.m// D f .gm/ for f; g 2 ƒM and m 2 M . Consider also the opposite endomorphism algebra AM D Endƒop .M /op and the K-linear homomorphism M

rM W A ! AM given by rM .a/.m/ D ma for a 2 A and m 2 M . The map rM is well defined, because we have the equalities rM .a/.f m/ D rM .a/.f .m// D f .m/a D f .ma/ D f rM .a/.m/ for a 2 A, f 2 ƒM ; m 2 M . Moreover, rM is a homomorphism of K-algebras. The following theorem, called the density theorem for semisimple algebras, is very useful. Theorem 6.12. Let A be a finite dimensional semisimple K-algebra over a field K and M a module in mod A. The following statements hold. (i) rM W A ! AM is an epimorphism of K-algebras.

62


(ii) rM W A ! AM is an isomorphism of K-algebras if and only if every simple module in mod A is isomorphic to a direct summand of M . Proof. Since A is a semisimple K-algebra, it follows from Theorem 6.3 that there is an isomorphism of K-algebras A Š Mn1 .F1 / Mn2 .F2 / Mnr .Fr / for some positive integers n1 ; n2 ; : : : ; nr and finite dimensional division K-algebras F1 ; F2 ; : : : ; Fr . Then there exists central idempotents e1 ; e2 ; : : : ; er of A such that 1A D e1 C e2 C C er , A1 D e1 Ae1 , A2 D e2 Ae2 ; : : : ; Ar D er Aer , are blocks of K-algebras of A with A D A1 A2 Ar , and there are isomorphisms of K-algebras Ai Š Mni .Fi /, for all i 2 f1; : : : ; rg. Further, the right A-module M has a decomposition M D M1 ˚ M2 ˚ ˚ Mr ; in mod A such that Mi D M ei is a right Ai -module, for any i 2 f1; : : : ; rg. We may assume that there is s 2 f1; : : : ; rg such that Mi ¤ 0 for i 2 f1; : : : ; sg, and Mj D 0 for j 2 fs C 1; : : : ; rg (if s < r). Applying Theorem 6.3 and Proposition 6.7, we conclude that, for any i 2 f1; : : : ; sg, there exists a simple module Ui in mod Ai m with EndAi .Ui / Š Fi and a positive integer mi such that Mi Š Ui i in mod Ai . Hence, we obtain isomorphisms of K-algebras ƒM Š ƒM1 ƒM2 ƒMs and ƒMi Š Mmi .Fi /, for any i 2 f1; : : : ; sg. For each i 2 f1; : : : ; sg, we choose a op op simple module Vi in mod ƒMi . Then Endƒop .Vi / Š Fi and, by Proposition 6.7, Mi

ki

for each i 2 f1; : : : ; sg, there exists a positive integer ki such that Mi Š Vi op mod ƒMi . This leads to isomorphisms of K-algebras

in

AM Š AM1 AM2 AMs op

and AMi D Endƒop .Mi /op Š Mki .Fi /op , for any i 2 f1; : : : ; sg. Moreover, we Mi have the equalities mi

dimK Mi D dimK Ui dimK Mi D

k dimK Vi i

D mi dimK Ui D mi ni dimK Fi ; D ki dimK Vi D ki mi dimK Fi ;

and hence ki D ni , for any i 2 f1; : : : ; sg. Observe now that the K-algebra homomorphism rM W A ! AM is the K-algebra homomorphism rM W A D

r Y iD1

Ai !

s Y iD1

AMi D AM


63

given, for a D .a1 ; : : : ; ar / 2 A and m D .m1 ; : : : ; ms / 2 M , by rM .a/.m/ D ma D .m1 a1 ; : : : ; ms as /. Consider now the restriction 0 W rM

s Y

Ai !

iD1

s Y

AMi D AM

iD1

of rM to A1 As . Moreover, we note that, for i 2 f1; : : : ; sg Ji D fai 2 Ai j Mi ai D 0g is a two-sided ideal of Ai , different from Ai , because Mi ¤ 0. Since A1 ; : : : ; As are simple K-algebras, we conclude that J1 D 0; : : : ; Js D 0. 0 This shows that rM is a monomorphism of K-algebras, and hence of K-vector spaces. On the other hand, we have dimK

s Y

s s Y Y Ai D dimK Mni .Fi / D n2i dimK Fi

iD1

iD1

D

s Y

iD1

ki2 dimK Fi D dimK

iD1

D dimK

s Y

Mki .Fi /

iD1 s Y

AMi D dimK AM :

iD1 0 is an isomorphismQ of K-algebras. Clearly, then rM is an epimorphism Therefore, rM of K-algebras with Ker rM D riDsC1 Ai . Moreover, rM is an isomorphism of K-algebras if and only if r D s, or equivalently, every simple module in mod A is isomorphic to a direct summand of M . This proves (i) and (ii).

Corollary 6.13. Let A be a finite dimensional K-algebra over a field K and M a semisimple module in mod A. Then rM W A ! AM is an epimorphism of Kalgebras. Proof. Let B D A= rad A. We know from Corollary 3.2 that B is a semisimple K-algebra. Moreover, the semisimplicity of M in mod A forces M rad A D 0, by Corollary 5.15, and hence M is a semisimple module in mod B. Hence ƒM D EndA .M / D EndB .M /, and consequently AM D Endƒop .M /op D BM . Finally, M we note that the K-algebra homomorphism rM W A ! AM is the composition of the canonical K-algebra epimorphism A ! A= rad A D B with the K-algebra homomorphism rM W B ! BM . Since, by Theorem 6.12 (i), rM W B ! BM is an epimorphism of K-algebras, we conclude that rM W A ! AM is also an epimorphism of K-algebras. Corollary 6.14. Let A be a finite dimensional K-algebra over a field K, S a simple module in mod A, F D EndA .S / the associated division K-algebra, and n the positive integer such that S Š F n in mod F op . Then there is an epimorphism of K-algebras A ! Mn .F /.

64


Proof. It follows from Corollary 6.13 that the K-algebra homomorphism rS W A ! AS is an epimorphism. Since AS D EndF .S / Š EndF .F n / Š Mn .F / the claim follows. As a consequence of Theorem 6.3, we obtain another useful characterization of local finite dimensional K-algebras. Corollary 6.15. Let A be a finite dimensional K-algebra over a field K. Then A is a local algebra if and only if 0A and 1A are the unique idempotents of A. Proof. Assume A is a local algebra and e is an idempotent of A. Then 1A e is an idempotent of A and e.1A e/ D 0A . It follows from Lemma 3.8 that one of the elements e or 1A e is invertible in A. Then e.1A e/ D 0A yields e D 1A or e D 0A . Conversely, assume that 0A and 1A are the unique idempotents of A. Then it follows from Lemma 3.12 that 0A Crad A and 1A Crad A are the unique idempotents of the semisimple algebra A= rad A. Therefore, by Theorem 6.3, A= rad A is a division algebra. Applying now Lemma 3.8 we conclude that A is a local algebra. The following lemma describes the semisimple bound quiver algebras. Lemma 6.16. Let Q be a finite quiver, K a field, I an admissible ideal of KQ. The bound quiver algebra KQ=I is semisimple if and only if the set Q1 of arrows of Q is empty. Proof. It follows from Lemma 3.6 that rad.KQ=I / D RQ =I , where RQ is the arrow ideal of KQ. Moreover, since I is an admissible ideal of KQ, we have m 2 RQ I RQ for some m 2. Hence rad.KQ=I / D 0 if and only if RQ D 0, or equivalently the set Q1 of arrows of Q is empty. Q Observe that KQ=I is semisimple if and only if KQ=I Š a2Q0 Ka , where Ka D K for each vertex a 2 Q0 . Lemma 6.17. Let K be a field of characteristic p > 0, m a positive integer, and G be the cyclic group of order p m . Then the group algebra KG is not semisimple. Proof. Observe first that KG is isomorphic to the truncated ˚ polynomial algebra m m KŒx= x p . Indeed, let G D .g/ D e; g; g 2 ; : : : ; g p 1 , and consider the Kalgebra homomorphism ' W KŒx ! by '.f .x// D f .g 1/ for any KG defined polynomial f .x/ 2 KŒx. Then ' .x C 1/i D g i , for any i 2 f1; : : : ; p m 1g, m m D .g 1/p D and consequently ' is an epimorphism. Moreover, ' x p m K is of characteristic p > 0, and so ' induces g p 1 D 0, since anmepimorphism m 'N W KŒx= x p ! KG of K-algebras. Since dimK KŒx= x p D pm D jGj D dimK KG, we conclude that 'N is an isomorphism of K-algebras.


65

m We know from Example 3.10 (a) that A D KŒx= x p is a commutative local K-algebra whose Jacobson rad A is the unique maximal ideal .x/ N generated mradical by the coset xN D x C x p . In particular, rad A ¤ 0, because p 2 and m 1. Therefore, KG is not a semisimple algebra. The following theorem is known as Maschke’s theorem. H. Maschke proved in 1889 [Mas] the semisimplicity of group algebras of finite groups over the field of complex numbers. We refer to II.3 for the definition and properties of tensor products of modules. Theorem 6.18. Let G be a finite group and K be a field. Then the group algebra KG is semisimple if and only if the characteristic of K does not divide the order of G. Proof. Let p be the characteristic of K. Assume p divides the order jGj of G. Then, by the Cauchy theorem, the group G contains an element g of order p. Denote by H the cyclic subgroup of order p of G generated by g. Then the group algebra KH is a K-subalgebra of KG. Moreover, it follows from Lemma 6.17 that KH is not a semisimple K-algebra. We claim that then the algebra KG is not semisimple. Consider the canonical inclusion mapi W KH P P ! KG and the K-linear map r W KG ! KH defined by r g D g2G g h2H h h. Then i and r are homomorphisms of left and right KH -modules with ri D idKH . Moreover, denote by W KH ! S D top.KH / the canonical epimorphism of right KH -modules. We note that S is a simple right KH -module, since KH is a local K-algebra (see Lemma 3.8 and Proposition 5.16). Then we obtain the commutative diagram in mod KH KH ˝KH KG

idKH ˝r

˝idKG

S ˝KH KG

idS ˝r

/ KH ˝KH KH

˛

/ KH

ˇ

/S,

˝idKH

/ S ˝KH KH

where ˛ and ˇ are the canonical isomorphisms. Observe also that ˝ idKG is an epimorphism of right KG-modules, because the functor ˝KH KG W mod KH ! mod KG is right exact. Suppose now that KG is a semisimple K-algebra. Then, by Theorem 6.3 (Proposition 5.12), every module in mod KG is semisimple. In particular, KH ˝KH KG Š KG is a semisimple right KG-module, and, applying Lemma 5.4, we conclude that there is a decomposition KH ˝KH KG Š L ˚ N of right KG-modules, where N D Ker. ˝ idKG / and L is isomorphic to Im. ˝ idKG / D S ˝KH KG. Hence, there exists a homomorphism u W S ˝KH KG ! KH ˝KH KG of right KG-modules with . ˝ idKG /u D idS˝KH KG . Clearly, then u is also a homomorphism of right KH -modules. Consider now the homomorphism j D

66


˛.idKH ˝r/u.idS ˝i /ˇ 1 W S ! KH of right KH -modules. Then we obtain the equalities j D ˛.idKH ˝r/u.idS ˝i /ˇ 1 D ˇ. ˝ idKH /.idKH ˝r/u.idS ˝i /ˇ 1 D ˇ.idS ˝r/. ˝ idKG /u.idS ˝i /ˇ 1 D ˇ.idS ˝r/.idS ˝i /ˇ 1 D idS : On the other hand, since KH Š KŒx=.x p / with p 2, we have soc.KH / D Kg p1

p1 M

Kg i D rad KH:

iD1

Then, for any f 2 HomKH .S; KH /, we have f D 0, because Ker D rad KH . This contradiction shows that KG is not a semisimple K-algebra. Assume now that p does not divide jGj. We claim that then KG is a semisimple K-algebra. By Theorem 6.3, it is sufficient to show that every module M in mod KG is semisimple. Let M be a module in mod KG and N a KG-submodule of M . We claim that there exists a KG-submodule L of M such that M D L ˚ N . This will imply, by Lemma 5.4, that M is a semisimple right KG-module. Denote by u W N ! M the inclusion map. Then u is a monomorphism of K-vector spaces and there exists a K-linear map v W M ! N such that vu D idN . Since the order jGj of G is by our assumption an invertible element of K, we have the K-linear map f W M ! M defined by the formula X v.mg 1 /g f .m/ D jGj1 g2G

for m 2 M . Then, for any m 2 M and h 2 G, we have X f .mh/ D jGj1 v .mh/g 1 g g2G 1

D jGj

X v m hg 1 g

g2G 1

D jGj

X 1 v m gh1 g

g2G

X 1 1 D jGj1 h v m gh1 gh g2G

X D jGj1 v mt 1 t h t2G

D f .m/h; and hence f is a homomorphism of right KG-modules. Moreover, for n 2 N , we have X X f .n/ D jGj1 v ng 1 g D jGj1 ng 1 g D n; g2G

g2G

7. The Jordan–Hölder theorem

67

because ng 1 2 N for any g 2 G and then v.ng 1 / D vu.ng 1 / D ng 1 . Hence f W M ! M is a homomorphism of right KG-modules with f u D idN . Take L to be the KG-submodule Ker f of M . Applying Lemma 4.2, we conclude that M D Ker f ˚ Im u D L ˚ N . Corollary 6.19. Let G be a finite group and K a field of characteristic 0. Then the group algebra KG is semisimple.

7 The Jordan–Hölder theorem Let A be a finite dimensional K-algebra over a field K. In Section 5 we have proved that the simple modules in mod A coincide with the simple modules in mod.A= rad A/ and are of the form eA=e rad A for primitive idempotents e of A. In this section we show that every module in mod A may be obtained by iterated extensions of simple modules. We start with the following modular law. Lemma 7.1. Let L N and Q be A-submodules of a module M in mod A. Then .L C Q/ \ N D L C .Q \ N /. Proof. Obviously we have L C .Q \ N / .L C Q/ \ N . Conversely, for z D x C y with x 2 L, y 2 Q, z 2 N , we have y D z x 2 Q \ N , and hence z D x C y 2 L C .Q \ N /. Therefore, .L C Q/ \ N L C .Q \ N / also holds. The following Zassenhaus isomorphism theorem will be also crucial. Theorem 7.2. Let V U and V 0 U 0 be A-submodules of a nonzero module M in mod A. Then there exist canonical isomorphisms of right A-modules .U C V 0 / \ U 0 U \ U0 .U 0 C V / \ U Š Š : .V C V 0 / \ U 0 .U 0 \ V / C .U \ V 0 / .V 0 C V / \ U Proof. It is enough to establish the first isomorphism. Consider the inclusion map '1 W U \ U 0 ! .U C V 0 / \ U 0 ; the canonical epimorphism '2 W .U C V 0 / \ U 0 ! .U C V 0 / \ U 0 =.V C V 0 / \ U 0 ; and take ' D '2 '1 W U \ U 0 ! .U C V 0 / \ U 0 =.V C V 0 / \ U 0 : We claim that ' is an epimorphism of right A-modules. Indeed, for u0 D u C v 0 2 .U C V 0 / \ U 0 , with u 2 U and v 0 2 V 0 , we have u D u0 v 0 2 U 0 , because

68


v 0 2 V 0 U 0 . Hence, u 2 U \ U 0 and '.u/ D u C .V C V 0 / \ U 0 D .u0 v 0 / C .V C V 0 / \ U 0 D u0 C .V C V 0 / \ U 0 . Moreover, applying Lemma 7.1, we obtain Ker ' D .U \ U 0 / \ ..V C V 0 / \ U 0 / D .V C V 0 / \ U \ U 0 D .V C .V 0 \ U // \ U 0 D ..V 0 \ U / C V / \ U 0 D .V 0 \ U / C .V \ U 0 / D .U 0 \ V / C .U \ V 0 /: Therefore, ' induces an isomorphism of right A-modules .U C V 0 / \ U 0 U \ U0 U \ U0 Im ' D : D ! .U 0 \ V / C .U \ V 0 / Ker ' .V C V 0 / \ U 0

Lemma 7.3. Let M be a nonzero module in mod A and L a proper right Asubmodule of M . Then the right A-module M=L is simple if and only if, for any right A-submodule N of M with L N M , we have L D N or N D M . Proof. Consider the canonical epimorphism p W M ! M=L of right A-modules. Then, for any right A-submodule N of M with L N , p.N / D N=L is a right A-submodule of M=L. Moreover, every right A-submodule X of M=L is of the form X D p p 1 .X / and p 1 .X / is a right A-submodule of M with L D p 1 .0M C L/ p 1 .X /. Then the equivalence follows. A chain 0 D M0 M1 M2 Mm1 Mm D M of A-submodules of a nonzero module M in mod A is said to be a composition series of M if Mi =Mi1 is a simple A-module for any i 2 f1; : : : ; mg. If it is the case, then M1 =M0 ; M2 =M1 ; : : : ; Mm =Mm1 are called simple composition factors of M . Proposition 7.4. Let M be a nonzero module in mod A. Then M admits a composition series 0 D M0 M1 Mm D M . Proof. It follows from Lemma 5.3 that every nonzero module X in mod A contains a simple A-submodule. Then the existence of a composition series of M follows from Lemma 7.3, by induction on dimK M . The following theorem is known as the Jordan–Hölder theorem, after C. Jordan and O. Hölder who proved its version for finite groups. In fact, the Jordan–Hölder theorem for representations of finite groups has been proved by A. Loewy in 1903 (see [Loe1], [Loe2]). Theorem 7.5. Let A be a finite dimensional K-algebra over a field K, M a nonzero module in mod A, and 0 D M0 M1 M2 Mm1 Mm D M;

7. The Jordan–Hölder theorem

69

0 D N0 N1 N2 Nn1 Nn D M two composition series of M . Then m D n and there exists a permutation of f1; : : : ; mg such that Mi =Mi1 Š N .i/ =N .i/1 for any i 2 f1; : : : ; mg. Proof. For each i 2 f1; : : : ; mg, we have the chain Mi1 D X0;i X1;i Xj;i Xn;i D Mi of A-submodules of Mi , where Xj;i D .Nj CMi1 /\Mi for any j 2 f0; 1; : : : ; ng. Since Mi =Mi1 is a simple A-module, by Lemma 7.3, we conclude that there exists exactly one j D .i / 2 f1; : : : ; ng such that Mi1 D X0;i D X1;i D D Xj 1;i and Xj;i D Xj C1;i D D Xn;i D Mi . Similarly, for each j 2 f1; : : : ; ng, we have the chain Nj 1 D Y0;j Y1;j Yi;j Ym;j D Nj of A-submodules of Nj , where Yi;j D .Mi CNj 1 /\Nj for any i 2 f0; 1; : : : ; mg. Again, since Nj =Nj 1 is a simple A-module, there exists exactly one i D .j / 2 f1; : : : ; mg such that Nj 1 D Y0;j D Y1;j D D Yi1;j and Yi;j D YiC1;j D D Ym;j D Nj . Moreover, from Theorem 7.2, we have isomorphisms of right A-modules .N .i/ C Mi1 / \ Mi .Mi C N.i/1 / \ N.i/ Mi D Š ; Mi1 .N .i/1 C Mi1 / \ Mi .Mi1 C N.i/1 / \ N.i/ .M.j / C Nj 1 / \ Nj .Nj C M.j /1 / \ M.j / Nj D Š : Nj 1 .M.j /1 C Nj 1 / \ Nj .Nj 1 C M.j /1 / \ M.j / Since N.i/1 Yi1;.i/ Yi; .i/ N .i/ and N.i/ =N.i/1 is a simple Amodule, it follows from Lemma 7.3 that N .i/1 D Yi1;.i/ and N.i/ D Yi;.i/ , and consequently i D .i / by definition of . Hence we have Mi =Mi1 Š N .i/ =N.i/1 for all i 2 f1; : : : ; mg, and is the identity map on f1; : : : ; mg. Similarly, we have the inclusions M.j /1 Xj 1;.j / Xj;.j / M.j / and M.j / =M.j /1 is a simple A-module. Applying Lemma 7.3 again, we conclude that M.j /1 D Xj 1;.j / and M.j / D Xj;.j / , and consequently j D .j / by definition of . Thus we have Nj =Nj 1 Š M.j / =M.j /1 for all j 2 f1; : : : ; ng, and is the identity map on f1; : : : ; ng. It follows that W f1; : : : ; mg ! f1; : : : ; ng and W f1; : : : ; ng ! f1; : : : ; mg are mutually inverse maps, and hence also m D n.

70


It follows from Theorem 7.5 that the number m of modules in a composition series 0 D M0 M1 Mm D M of a nonzero module M in mod A depends only on M ; it is called the length of M and is denoted by `.M /. Moreover, for a simple module S in mod A, the number of simple composition factors Mi =Mi1 of M isomorphic to S also depends only on M ; it is called the composition multiplicity of S in M and is denoted by cS .M /. Therefore, if S1 ; : : : ; Sn is a complete set Pnof pairwise nonisomorphic simple modules in mod A, then we have `.M / D iD1 cSi .M / for any module M in mod A. The length `.0/ of the zero module 0 in mod A is defined to be 0. Lemma 7.6. Let A be a finite dimensional K-algebra, M a module in mod A, and N an A-submodule of M . Then (i) `.M / D `.N / C `.M=N /. (ii) cS .M / D cS .N / C cS .M=N / for any simple module S in mod A. Proof. It follows from Lemmas 5.3 and 7.3 that the chain 0 N M of Asubmodules of M can be saturated to a composition series 0 D M0 M1 Mm D M of M with N D Mi for some i 2 f0; 1; : : : ; mg. Moreover, then 0 D Mi =Mi MiC1 =Mi Mm =Mi D M=N is a composition series of M=N . Then the statements (i) and (ii) are direct consequences of Theorem 7.5. Lemma 7.7. Let A be a finite dimensional K-algebra, and M a nonzero module in mod A. Then we have the inequalities ``.M / `.M / dimK M: Proof. It follows from Lemmas 5.3 and 7.3 that, for the radical series M rad M rad2 M radn1 M radn M D 0 of M , there exists a composition series 0 D M0 M1 M2 Mm1 Mm D M of M , and a sequence 0 D j0 < j1 < < jn D m in f0; 1; : : : ; mg such that radnr M D Mjr for any r 2 f0; 1; : : : ; ng, and hence we obtain ``.M / D n m D `.M /. Moreover, we have dimK Mi1 < dimK Mi for any i 2 f1; : : : ; mg, and so `.M / D m dimK M . Example 7.8. Let A be the path algebra KQ of the quiver

QW

1 ^<

0

S3 W

0 _> >> >> >

0

S5 W

Ko

0 >> >

K

are pairwise nonisomorphic simple representations in repK .Q/. Consider also the subrepresentations of M K À AA1 AA XW 0

} }} ~} }

Ko

0 ^= == ==

0 ÀA AA A

0 YW 0

K

} }} }~ } 1

Ko

0 ^= == ==

0

0

72


ZW

TW

VW

K aC CC 1 0 CC K2 o { {{ }{{ 0 1 K

0 ^< >>

0 , for 2 K; 0

where W K ! K.1; 1/ is given by ./ D .1; 1/ for 2 K. The following chains are some composition series of M in repK .Q/: 0 S1 0 S1 0 S2 0 S2 0 S1 0 S1 0 S1 0 S1 0 S2 0 S2 0 S2 0 S2

0

K aC 0 CC 1 0 ~~ CC ~ 1 ~~~ 1 K `@ K2 o @@1 { @@ {{ }{{ 0 1 K K

WW

0 > {{ }{{ 0 1 K 0

1

K aC K CC 1 0 1 ~~ CC ~ 1 ~~~ 1 K `@ K2 o @@ { {{ @@ }{{ 0 1 K 0

N W

K dHH HH1 0 HH K.1; 1/ o vv vv zvv 0 1 K

0

X X ˚ S2 Z U V M;

X X ˚ S2 Z U W M;

Y S1 ˚ Y Z U V M;

Y S1 ˚ Y Z U W M;

S1 ˚ S2 X ˚ S2 Z U V M;

S1 ˚ S2 X ˚ S2 Z U W M;

S1 ˚ S2 S1 ˚ Y Z U V M;

S1 ˚ S2 S1 ˚ Y Z U W M;

S1 ˚ S2 X ˚ S2 Z U V M;

S1 ˚ S2 X ˚ S2 Z U W M;

S1 ˚ S2 S1 ˚ Y Z U V M;

S1 ˚ S2 S1 ˚ Y Z U W M:

8. Projective and injective modules

73

Observe that `.M / D 7 and cS1 .M / D 1, cS2 .M / D 1, cS3 .M / D 2, cS4 .M / D 1, cS5 .M / D 1, cS6 .M / D 1. Moreover, we have ``.M / D 4, since 3 is the length of the longest path in Q (see Corollary 5.21). We also note that dimK M D 7. Observe that T , V and W are unique maximal subrepresentations of M , and hence rad M D T \ V \ W D R. Further, N D N1 is a unique maximal subrepresentation of R, and so N D rad R. Finally, S1 ˚ S2 is a unique maximal subrepresentation of N , and hence rad N D S1 ˚ S2 . Therefore, the radical series of M is of the form M R N S1 ˚ S2 0: Observe that top.M / D M= rad M Š S3 ˚ S5 ˚ S6 and soc.M / D S1 ˚ S2 . We also note that N , 2 K, form a family of pairwise different subrepresentations of M with top.N / Š S3 and soc.N / D S1 ˚ S2 . In particular, if the field K is infinite, M admits infinitely many pairwise different composition series. Example 7.9. Let F be a finite dimensional division K-algebra over a field K, n a positive natural number and Mn .F / the full n n matrix algebra over F . It follows from Lemma 6.2 (and its proof) that Mn .F / is a semisimple algebra having a decomposition Mn .F / D S1 ˚ S2 ˚ ˚ Sn into a direct sum of simple right ideals (modules), where ˚ Sr D Err Mn .F / D Œaij 2 Mn .F / j aij D 0 for i ¤ r ; for each r 2 f1; : : : ; ng. Moreover, by Proposition 5.12 and Corollary 5.17, every simple right Mn .F /-module is isomorphic to a module Sr . In fact, for each r 2 f1; : : : ; ng, there is a canonical isomorphism S1 ! Sr of right Mn .F /modules given by the shift of the first row of S1 onto the r-th row of Sr . Therefore, the simple Mn .F /-modules S1 ; : : : ; Sn are isomorphic. Observe now that dimK S1 D n dimK F and clearly `.S1 / D 1. In particular, for the right Mn .F /module Mn .F / D S1 ˚ ˚ Sn , we have ``.Mn .F // D 1;

`.Mn .F // D n;

dimK Mn .F / D n2 dimK F:

8 Projective and injective modules The aim of this section is to describe the basic properties and structure of finite dimensional projective and injective modules over finite dimensional K-algebras. Let K be a field. Let A be a finite dimensional K-algebra. A module F in mod A is called free if F is isomorphic to a direct sum of copies of the module AA . A module P in mod A is said to be projective if, for any epimorphism h W M ! N in mod A and f 2 HomA .P; N /, there exists g 2 HomA .P; M / such that hg D f , that is,

74


making the following diagram commutative P || | | f || | ~|h /N M g

(see also Exercise 12.22). Lemma 8.1. Let P be a module in mod A. The following conditions are equivalent. (i) P is a projective module in mod A. (ii) Every epimorphism h W M ! P in mod A is a retraction. (iii) There exist a free module F and a module P 0 in mod A such that P ˚P 0 Š F . Proof. Assume that P is a projective A-module and h W M ! P is an epimorphism in mod A. Then there exists a homomorphism g W P ! M in mod A such that hg D idP , and hence h is a retraction. Thus (i) implies (ii). Assume (ii) holds. Let x1 ; : : : ; xm be a basis of the K-vector space P . Then P D x1 A C C xm A and the map v W AAm ! P , given by v.a1 ; : : : ; am / D x1 a1 C C xm am , for a1 ; : : : ; am 2 A, is an epimorphism of right A-modules. It follows from our assumption that idP D vu for some u 2 HomA .P; AAm /. Then, by Lemma 4.2, we obtain AAm D Im u ˚ Ker v. Since u is a section, it induces an isomorphism P ! Im u. Therefore, we have P ˚ P 0 Š F for F D AAm and P 0 D Ker v. Hence (ii) implies (iii). Assume (iii) holds. Then there exists a free right A-module F D y1 A ˚ ˚ ym A and homomorphisms r 2 HomA .F; P / and s 2 HomA .P; F / such that rs D idP . Let h W M ! N be an epimorphism in mod A and f 2 HomA .P; N /. Then there exist elements x1 ; : : : ; xm 2 M such that h.xi / D f r.yi / for any i 2 f1; : : : ; mg. Consider the homomorphism w W F ! M of right A-modules such that w.yi / D xi for any i 2 f1; : : : ; mg. Then we have hw D f r, and consequently hg D f for g D ws 2 HomA .P; M /. This shows that P is a projective A-module. Hence (iii) implies (i). The following proposition describes the structure of finite dimensional projective modules over finite dimensional algebras. Proposition 8.2. Let A be a finite dimensional K-algebra and e1 ; : : : ; en a set of primitive idempotents of A with 1A D e1 C C en . Then (i) AA D e1 A ˚ ˚ en A is a decomposition of AA into a direct sum of indecomposable projective right A-modules.


75

(ii) Every nonzero projective module P in mod A is a direct sum P D P1 ˚ ˚ Pm , where each module Pj , j 2 f1; : : : ; mg, is isomorphic to a module ei A with i 2 f1; : : : ; ng. Proof. It follows from Corollary 5.10 and Lemma 8.1 that e1 A; : : : ; en A are indecomposable projective modules in mod A, and hence (i) holds. For (ii), let P be a projective module and P D P1 ˚ ˚ Pm a decomposition of P into a direct sum of indecomposable A-submodules. Since P is projective, it follows from Lemma 8.1 that there exists a module P 0 such that P ˚ P 0 is isomorphic to AAt for some t 1. Decomposing P 0 D P10 ˚ ˚ Pr0 into a direct sum of indecomposable A-submodules we obtain P1 ˚ ˚ Pm ˚ P10 ˚ ˚ Pr0 Š .e1 A ˚ ˚ en A/t : Applying Theorem 4.6, we obtain that each module Pj , j 2 f1; : : : ; mg, is indeed isomorphic to a module ei A with i 2 f1; : : : ; ng. Our next aim is to associate to a module M in mod A a minimal epimorphism P .M / ! M in mod A with P .M / a projective module. We start with some useful concepts. An A-submodule L of a module M in mod A is said to be superfluous if, for every A-submodule X of M with L C X D M , the equality X D M holds. Then a nonzero epimorphism h W M ! N in mod A with Ker h a superfluous submodule of M is said to be a minimal epimorphism. It follows from Proposition 5.13 (v) that the radical rad M of any module M in mod A is a superfluous A-submodule of M , and consequently the canonical epimorphism hM W M ! M= rad M D top.M / is a minimal epimorphism. The following characterization of minimal epimorphisms in mod A is useful. Lemma 8.3. Let h W M ! N be a nonzero epimorphism in mod A. Then h is a minimal epimorphism if and only if, for any homomorphism g W L ! M in mod A with hg an epimorphism, g is also an epimorphism. Proof. Assume that h is a minimal epimorphism and g W L ! M is a homomorphism in mod A such that hg W L ! N is an epimorphism. Since hg is an epimorphism, for any m 2 M , there exists x 2 L such that h.m/ D hg.x/, and hence mg.x/ 2 Ker h. This implies that Ker hCIm g D M . By our assumption, Ker h is a superfluous A-submodule of M , and then Im g D M . Therefore, g is an epimorphism. Conversely, assume that the epimorphism h W M ! N has the stated property. Let X be an A-submodule of M such that Ker h C X D M , and consider the inclusion map g W X ! M . Then the composed homomorphism hg W X ! N is an epimorphism, and hence, by our assumption, g is an epimorphism. This implies that X D M , and consequently Ker h is a superfluous A-submodule of M . Therefore, h is a minimal epimorphism.

76


An epimorphism h W P ! M in mod A is said to be a projective cover of M if P is a projective A-module and h is a minimal epimorphism. Theorem 8.4. Let A be a finite dimensional K-algebra over a field K. (i) For any nonzero module M in mod A there exists a projective cover h W P .M / ! M: Moreover, the induced homomorphism top.h/ W top.P .M // ! top.M / of semisimple modules in mod A is an isomorphism. (ii) For any projective cover h0 W P 0 ! M of a nonzero module M in mod A there exists an isomorphism g W P 0 ! P .M / in mod A such that hg D h0 . Proof. Let e1 ; : : : ; en be a set of primitive pairwise orthogonal idempotents of A with 1A D e1 C C en (see Corollary 5.9). Then eN1 D e1 C rad A; : : : ; eNn D en C rad A is a set of primitive pairwise orthogonal idempotents of the semisimple algebra B D A= rad A with 1B D eN1 C C eNn . Moreover, it follows from Proposition 5.16 that, for any i 2 f1; : : : ; ng, rad ej A D ej rad A is a unique maximal right A-submodule of ej A and top.ej A/ Š eNj B is simple. Clearly, then the canonical epimorphism j W ej A ! top.ej A/ is a projective cover in mod A with top.j / W top.ej A/ ! top.ej A/ the identity map. Let M be a module in mod A. Then top.M / D M= rad M is by Corollary 5.15 a semisimple right A-module and a semisimple right B-module. Then, by Proposition 5.16 and Corollary 5.17, there exist isomorphisms of right B-modules ! .eN1 B/r1 ˚ ˚ .eNn B/rn ! top.M /; top.e1 A/r1 ˚ ˚ top.en A/rn

for some r1 0; : : : ; rn 0. We take P .M / D .e1 A/r1 ˚ ˚ .en A/rn . Then P .M / is a projective right A-module and there exists a homomorphism h W P .M / ! M of right A-modules such that the diagram P .M /

h

hP .M /

top.P .M //

/M hM

top.h/

/ top.M / ,

where hP .M / and hM are two canonical minimal epimorphisms, is commutative. Moreover, top.h/ is an isomorphism, by our choice of P .M /. Then hM h D top.h/hP .M / is an epimorphism, and hence h is an epimorphism, because hM is a minimal epimorphism. Moreover, Ker h Ker hM h D Ker top.h/hP .M / D Ker hP .M / , because top.h/ is an isomorphism, and consequently we obtain Ker h .rad e1 A/r1 ˚ ˚ .rad en A/rn D rad P .M /:


77

Since rad P .M / is a superfluous A-submodule of P .M /, Ker h is also a superfluous A-submodule of P .M /. Therefore, the epimorphism h W P .M / ! M is a projective cover of M in mod A. (ii) Let h0 W P 0 ! M be a projective cover of a module M in mod A. Since the map in the projective cover h W P .M / ! M , constructed above, is an epimorphism, the projectivity of P 0 implies that there is a homomorphism g W P 0 ! P .M / in mod A such that hg D h0 . Then, since hg D h0 is surjective, the minimality of h forces that g W P 0 ! P .M / is an epimorphism in mod A. Interchanging the role of P 0 and P .M /, we conclude in a similar way that there exists an epimorphism f W P .M / ! P 0 in mod A such that h0 f D h. Then we get the inequalities dimK P .M / dimK P 0 dimK P .M /, and so dimK P .M / D dimK P 0 . Therefore, g W P 0 ! P .M / is an isomorphism. Corollary 8.5. Let A be a finite dimensional K-algebra. Then the functor top W mod A ! mod A induces a bijection between the isomorphism classes of the nonzero projective modules in mod A and the isomorphism classes of the semisimple modules in mod A. Proof. It follows from Corollary 5.15 that a module M in mod A is semisimple if and only if M Š top.M /. Then, by Theorem 8.4 (i), for a semisimple module M in mod A and its projective cover h W P .M / ! M in mod A, P .M / is a projective module in mod A and top.h/ W top.P .M // ! top.M / is an isomorphism in mod A. Moreover, it follows from Theorem 8.4 (ii), that two semisimple modules M and N in mod A are isomorphic if and only if their projective covers P .M / and P .N / are isomorphic. Finally, observe also that, for a projective module P in mod A, top.P / is a semisimple module in mod A and P Š P .top.P //. As a special case, we get the following bijection, noted by T. Nakayama in [Nak1] (without use of projective modules, introduced later in the book [CE] by H. Cartan and S. Eilenberg (see also the paper [NaNa] by H. Nagao and T. Nakayama)). Corollary 8.6. Let A be a finite dimensional K-algebra. Then the functor top W mod A ! mod A induces a bijection between the isomorphism classes of the indecomposable projective modules in mod A and the isomorphism classes of the simple modules in mod A. Proof. Let e1 ; : : : ; en be a set of primitive pairwise orthogonal idempotents in A such that 1A D e1 C C en . It follows from Proposition 8.2 that a module P in mod A is an indecomposable projective module if and only if P is isomorphic to a module of the form ei A, for some i 2 f1; : : : ; ng. Similarly, by Corollary 5.17, a module S in mod A is a simple module if and only if S is isomorphic to a module of the form ei A=ei rad A D top.ei A/, for some i 2 f1; : : : ; ng. Moreover, by Theorem 8.4, two simple modules ei A=ei rad A and ej A=ej rad A are isomorphic if and only if the indecomposable projective modules ei A and ej A are isomorphic.

78


Let A be a K-algebra, e 2 A an idempotent, and M a module in Mod A. Then the K-vector space eAe D feae j a 2 Ag is a finite dimensional K-algebra with identity e. Observe that eAe is a K-subalgebra of A if and only if e D 1A . The K-vector space M e D fme j m 2 M g is a right eAe-module by .me/.eae/ D meae for m 2 M and a 2 A. Moreover, the K-vector space HomA .eA; M / is a right eAe-module by the action .' eae/.x/ D '.eaex/ for x 2 eA, a 2 A, ' 2 HomA .eA; M /. Lemma 8.7. Let A be a K-algebra, e 2 A an idempotent, and M be a module in Mod A. Then the K-linear map e M W HomA .eA; M / ! M e; e defined by M .'/ D '.e/ D '.e/e for ' 2 HomA .eA; M /, is an isomorphism of right eAe-modules. e Proof. For ' 2 HomA .eA; M / and eae 2 eAe, we have the equalities M .'eae/ D e .'eae/.e/ D '..eae/e/ D '.eae/ D '.e.eae// D '.e/eae D M .'/eae, and e hence M is a homomorphism of right eAe-modules. Consider the K-linear map e e W Me ! HomA .eA; M / defined by M .me/.ea/ D mea for m 2 M and M e a 2 A. Then, for m 2 M , a; b 2 A, we have the equalities D M ..me/.ebe//.ea/ e e e .mebe/.ea/ D mebea D .me/.ebea/ D .me/ebe .ea/, and hence M M M e e e M is a homomorphism of right eAe-modules. We check that M and M are mutue e ally inverse maps. For ' 2 HomA .eA; M / and a 2 A, we have M M .'/.ea/ D e e e M .'.e/e/.ea/ D '.e/ea D '.ea/, and hence M M D id HomA .eA;M / . Sime e e ilarly, for m 2 M , we have M M .me/ D M .me/.e/ D me, and hence e e M M D idMe .

Corollary 8.8. Let A be a K-algebra and e 2 A an idempotent. Then the map e eA W EndA .eA/ ! eAe is an isomorphism of K-algebras. In particular, we obtain the following consequence of the above corollary, Lemma 4.4 and Corollary 5.8. Corollary 8.9. Let A be a finite dimensional K-algebra and e 2 A an idempotent. Then e is a primitive idempotent of A if and only if eAe is a local K-algebra. Similarly, we have the following facts. Lemma 8.10. Let A be a K-algebra, e 2 A an idempotent, and M a module in Mod Aop . Then the K-linear map e ıM W HomAop .Ae; M / ! eM e defined by ıM . /D of left eAe-modules.

.e/ D e .e/ for

2 HomAop .Ae; M / is an isomorphism


79

Corollary 8.11. Let A be a K-algebra and e 2 A an idempotent. Then the map e ıAe W EndAop .Ae/op ! eAe is an isomorphism of K-algebras. The following lemma is also very useful. Lemma 8.12. Let A be a finite dimensional K-algebra and e; f two primitive idempotents of A. The following conditions are equivalent. (i) eA and fA are nonisomorphic as right A-modules. (ii) eAf D e.rad A/f . (iii) fAe D f .rad A/e. (iv) Ae and Af are nonisomorphic as left A-modules. Proof. It follows from Corollary 5.8 and Proposition 5.16 that eA and fA are indecomposable projective right A-modules, and the radicals rad.eA/ D e rad A and rad.fA/ D f rad A are their unique maximal right A-submodules, respectively. Moreover, by Lemma 8.1 (ii), every epimorphism g W eA ! fA or h W fA ! eA is a retraction, and consequently an isomorphism. Therefore, eA and fA are nonisomorphic as right A-modules if and only if HomA .eA; fA/ D HomA .eA; f rad A/, or equivalently, HomA .fA; eA/ D HomA .fA; e rad A/. From Lemma 8.7, this is equivalent to fAe D f .rad A/e, or equivalently, to eAf D e.rad A/f . In a similar way, using Lemma 8.10, we prove that Ae and Af are nonisomorphic as left Amodules if and only if eAf D e.rad A/f , or equivalently, fAe D f .rad A/e. Let A be a finite dimensional K-algebra. A module E in mod A is said to be injective if, for any monomorphism u W M ! N in mod A and w 2 HomA .M; E/, there exists v 2 HomA .N; E/ such that w D vu, that is, making the following diagram commutative u / M N || | | w || v ~|| E (see also Exercise 12.23). Lemma 8.13. Let E be a module in mod A. The following conditions are equivalent. (i) E is an injective module in mod A. (ii) Every monomorphism f W E ! M in mod A is a section.

80


Proof. Assume E is an injective A-module and f W E ! M is a monomorphism in mod A. Then there exists a homomorphism g W M ! E such that gf D idE , and hence f is a section. Thus (i) implies (ii). Conversely, assume that (ii) holds. Let u W M ! N be a monomorphism in mod A and w 2 HomA .M; E/. Consider the factor module W D V =U , where V D E ˚ N and U is the A-submodule of V of the form f.w.m/; u.m// j m 2 M g. Then we obtain a commutative diagram M

u

/N

f

/W ,

g

w

E

where f .x/ D .x; 0/ C U , for x 2 E, and g.y/ D .0; y/ C U , for y 2 N . Observe that indeed f w.m/ gu.m/ D ..w.m/; 0/ C U / ..0; u.m// C U / D .w.m/; u.m// C U D 0 C U , and hence f w.m/ D gu.m/ for any m 2 M . We claim that f is a monomorphism. Let x 2 E and f .x/ D 0. Then .x; 0/ 2 U , and so .x; 0/ D .w.m/; u.m// for some m 2 M . Hence x D w.m/ and u.m/ D 0. Because u is a monomorphism, we obtain m D 0, and consequently x D w.m/ D 0. Thus f is a monomorphism in mod A, and hence a section. Let h W W ! E be a homomorphism in mod A with hf D idE . Then for v D hg W N ! E we have vu D hgu D hf w D w. This shows that E is an injective A-module. Therefore (ii) implies (i). The following injectivity criterion proved by R. Baer [Bae] is very useful. Lemma 8.14. Let A be a finite dimensional K-algebra and E a module in mod A. The following conditions are equivalent. (i) E is an injective module in mod A. (ii) For every right ideal I of A and ' 2 HomA .I; E/, there exists an element 2 HomA .A; E/ such that '.a/ D .a/ for all a 2 I . Proof. Assume E is an injective module in mod A, I a right ideal of A, and u W I ! A the inclusion homomorphism of right A-modules. Then, for every homomorphism ' W I ! E in mod A, there exists a homomorphism W A ! E such that ' D u. Clearly, then '.a/ D .u.a// D .a/ for all a 2 A. Hence (i) implies (ii). Conversely, assume that (ii) holds. Let u W M ! N be a monomorphism in mod A and w 2 HomA .M; E/. Observe that Im u is a right A-submodule of N , u W M ! Im u is an isomorphism in mod A and, for v0 D wu1 2 HomA .Im u; E/, we have w D v0 u. Consider now a maximal right A-submodule X of N containing Im u and with the property that there exists v 2 HomA .X; E/ such that w D vu. We


81

claim that X D N , and consequently E is an injective module in mod A. Assume that X ¤ N , and let n 2 N n X . Consider the set I D fa 2 A j na 2 X g: Observe that I is a right ideal of A, since X is a right A-submodule of N . We define ' 2 HomA .I; E/ by '.a/ D v.na/ for all a 2 I . Then it follows from our assumption (ii) that there exists 2 HomA .A; E/ such that .a/ D '.a/ for all a 2 I . Consider now the right A-submodule Y D X C nA of N and the map f W Y ! E given by f .x C na/ D v.x/ C .a/ for x 2 X and a 2 A. We show that f is well defined. Indeed, let x C na D x 0 C na0 , for some x; x 0 2 X and a; a0 2 A. Then n.a a0 / D x 0 x 2 X , and so a a0 2 I . Hence we obtain v.n.a a0 // D '.a a0 / D .a a0 /. This leads to the equalities .v.x/ C

.a// .v.x 0 / C

.a0 // D v.x x 0 / C .a a0 / D v.x x 0 / C v.n.a a0 // D v..x C na/ .x 0 C na0 // D v.0/ D 0;

as required. Then f 2 HomA .Y; E/, because v and are homomorphisms of right A-modules. Moreover, for m 2 M , we have f u.m/ D f .u.m/Cn0/ D vu.m/ D w.m/, and so f u D w. This contradicts the maximality of X , since X is a proper right A-submodule of Y . Therefore, X D N , and (ii) implies (i). A right A-submodule X of a module M in mod A is said to be essential if X \Y ¤ 0 for any nonzero right A-submodule Y of M . Then a nonzero monomorphism u W L ! M in mod A is said to be a minimal monomorphism if Im u is an essential right A-submodule of M . It follows from Lemma 5.3 that the socle soc.M / of any nonzero module M in mod A is an essential submodule of M , and consequently the canonical monomorphism uM W soc.M / ! M is a minimal monomorphism. The following characterization of minimal monomorphisms in mod A is very useful. Lemma 8.15. Let u W L ! M be a nonzero monomorphism in mod A. Then u is a minimal monomorphism if and only if, for any homomorphism v W M ! N in mod A with vu a monomorphism, v is a monomorphism. Proof. Assume that u is a minimal monomorphism in mod A and v W M ! N is a homomorphism in mod A such that vu W L ! N is a monomorphism. This implies that Im u \ Ker v D 0, and consequently Ker v D 0, because Im u is, by assumption, an essential right A-submodule of M . Thus v is a monomorphism. Conversely, assume that the monomorphism u W L ! M has the stated property. Let X be a right A-submodule of M such that Im u \ X D 0. Consider the factor module N D M=X and the canonical epimorphism v W M ! N , v.m/ D m C X

82


for m 2 M . Then the composed homomorphism vu W L ! N is a monomorphism, because Im u \ Ker v D Im u \ X D 0. Hence, by assumption, v is a monomorphism. This implies that X D 0. Therefore, Im u is an essential right A-submodule of M , and u is a minimal monomorphism. A monomorphism u W M ! E in mod A is said to be an injective envelope of M if E is an injective A-module and u is a minimal monomorphism. We describe now the structure of injective modules in mod A and show that every nonzero module in mod A admits a unique injective envelope. We agreed to identify the category A-mod of finite dimensional left A-modules with the category mod Aop of finite dimensional right mod Aop -modules, where Aop is the opposite K-algebra of A. Moreover, we have the standard duality mod A o

D D

/

mod Aop

with 1mod A Š D B D and 1mod Aop Š D B D, where D D HomK .; K/. Applying the duality D, we will now transfer the main results on the projective modules in mod Aop , described above, to the corresponding dual results on the injective modules in mod A. We start with a general proposition. Proposition 8.16. Let A be a finite dimensional K-algebra and D the standard duality between mod A and mod Aop . (i) A module E in mod A is injective if and only if the module D.E/ in mod Aop is projective. (ii) A module P in mod A is projective if and only if the module D.P / in mod Aop is injective. (iii) A module S in mod A is simple if and only if the module D.S / in mod Aop is simple. (iv) A module M in mod A is semisimple if and only if the module D.M / in mod Aop is semisimple. (v) For every nonzero module M in mod A, we have D.top M / Š soc.D.M // and D.soc M / Š top.D.M //. (vi) u W L ! M is a minimal monomorphism in mod A if and only if D.u/ W D.M / ! D.L/ is a minimal epimorphism in mod Aop .


83

(vii) h W M ! N is a minimal epimorphism in mod A if and only if D.h/ W D.N / ! D.M / is a minimal monomorphism in mod Aop . Proof. This is straightforward and left to the reader.

In particular, the following characterization of injective modules in mod A follows from Lemma 8.1 (for modules in mod Aop ). Lemma 8.17. Let E be a module in mod A. Then E is an injective module if and only if there exists a module E 0 in mod A such that E ˚ E 0 Š D.A A/m for some m 1. Further, Theorem 8.4 for modules in mod Aop is transferred to the following theorem for modules in mod A. Theorem 8.18. Let A be a finite dimensional K-algebra over a field K. (i) For any nonzero module M in mod A there exists an injective envelope u W M ! E.M / such that the induced homomorphism soc.u/ W soc.M / ! soc.E.M // is an isomorphism. (ii) For any injective envelope u0 W M ! E 0 of a nonzero module M in mod A there exists an isomorphism f W E.M / ! E 0 in mod A such that f u D u0 . We will present now a more detailed description of the injective modules in mod A. Let e1 ; : : : ; en be a set of primitive pairwise orthogonal idempotents in A such that 1A D e1 C C en . Obviously then e1 ; : : : ; en is a set of primitive pairwise orthogonal idempotents in Aop with 1Aop D e1 C C en . We know from Proposition 5.16 and Corollary 8.6 that, for any i 2 f1; : : : ; ng, Aei D ei Aop is an indecomposable projective left A-module (right Aop -module) and top.Aei / D Aei =.rad A/ei is a simple left A-module. Further, it follows from Proposition 8.2 that every finite dimensional projective left A-module P is isomorphic to a direct sum of indecomposable projective left A-modules of the form Aei with i 2 f1; : : : ; ng. Similarly, every finite dimensional semisimple left A-module is isomorphic to a direct sum of simple left A-modules of the form Aei =.rad A/ei with i 2 f1; : : : ; ng. Observe also that the right A-module D.A A/ has a decomposition D.A A/ D D.Ae1 ˚ ˚ Aen / D D.Ae1 / ˚ ˚ D.Aen / into a direct sum of indecomposable injective modules in mod A. The following transfer of Proposition 8.2 describes the structure of finite dimensional injective modules over finite dimensional algebras.

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Proposition 8.19. Let A be a finite dimensional K-algebra and e1 ; : : : ; en be a set of primitive pairwise orthogonal idempotents in A with 1A D e1 C C en . Then every nonzero injective module E in mod A is a direct sum E D E1 ˚ ˚ Em , where each module Ej , j 2 f1; : : : ; mg, is isomorphic to a module D.Aei / with i 2 f1; : : : ; ng. We have also the following transfers of Corollaries 8.5 and 8.6. Corollary 8.20. Let A be a finite dimensional K-algebra. Then the functor soc W mod A ! mod A induces a bijection between the isomorphism classes of the nonzero injective modules in mod A and the isomorphism classes of the semisimple modules in mod A. Corollary 8.21. Let A be a finite dimensional K-algebra. Then the functor soc W mod A ! mod A induces a bijection between the isomorphism classes of the indecomposable injective modules in mod A and the isomorphism classes of the simple modules in mod A. A prominent role in our further considerations will be played by the following lemma. Lemma 8.22. Let A be a finite dimensional K-algebra and e1 ; : : : ; en a set of primitive pairwise orthogonal idempotents in A with 1A D e1 C C en . Then soc.D.Aei // Š ei A=ei rad A D top.ei A/ for any i 2 f1; : : : ; ng. Proof. Fix i 2 f1; : : : ; ng. It follows from Proposition 8.16 (v) that soc D.Aei / Š D.top.Aei // D D.Aei =.rad A/ei /. Since soc D.Aei / is a simple right A-module, in order to show that it is isomorphic to top.ei A/, it is enough to prove that there is a nonzero homomorphism ei A ! D.Aei =.rad A/ei / of right A-modules. Applying Lemma 8.7, we obtain isomorphisms of K-vector spaces HomA .ei A; D.Aei =.rad A/ei // Š D.Aei =.rad A/ei /ei Š D.ei Aei =ei .rad A/ei /: Furthermore, ei Aei is a local K-algebra with the Jacobson radical rad ei Aei D ei .rad A/ei , by Corollary 8.9, because the idempotent ei is primitive. It follows that ei Aei =ei .rad A/ei ¤ 0, and consequently D .ei Aei =ei .rad A/ei / ¤ 0. We will illustrate now the results presented above in two special and extreme cases. Lemma 8.23. Let A D Mn .F / be the full n n matrix algebra over a finite dimensional division K-algebra F . Then the nonzero projective, nonzero injective and semisimple modules in mod A coincide. Moreover, there is only one indecomposable module in mod A up to isomorphism.


85

Proof. The elementary diagonal matrices Ei i , i 2 f1; : : : ; ng, form a set of orthogonal primitive idempotents of A with In D E11 C C Enn . Then, as shown in Example 7.9, every indecomposable module in mod A is simple and of the form ˚ Err A D Œaij 2 A j aij D 0 for i ¤ r ; which is the r-th row of A, for some r 2 f1; : : : ; ng. Observe also that for any r; s 2 f1; : : : ; ng the canonical map 'r;s W Err A ! Ess A which shifts the r-th row of A to the s-th row of A is an isomorphism of right A-modules. Clearly, then the projective, injective and semisimple modules in mod A coincide, and there is in mod A exactly one indecomposable module (up to isomorphism). In fact we have another characterization of semisimple algebras completing the Wedderburn structure theorem (Theorem 6.3). Corollary 8.24. Let A be a finite dimensional K-algebra over a field K. The following conditions are equivalent. (i) A is a semisimple algebra. (ii) Every module in mod A is projective. (iii) Every module in mod A is injective. Proof. Let e1 ; : : : ; en be a set of pairwise orthogonal primitive idempotents in A such that 1A D e1 C C en . Assume A is a semisimple algebra. Then we conclude as above, by Proposition 5.12 and Corollary 5.17, that every indecomposable module in mod A is simple and isomorphic to a module of the form ei A, for some i 2 f1; : : : ; ng. Moreover, then P .ei A/ D ei A D E.ei A/ D D.Aei /. Therefore, (i) implies (ii) and (iii). Assume that every module in mod A is projective. Then, by Lemma 8.1, for each i 2 f1; : : : ; ng, the canonical epimorphism ei A ! top.ei A/ D ei A=ei rad A is a retraction, and consequently an isomorphism, because ei A is indecomposable (see Lemma 4.2). Therefore, ei rad A D 0 for any i 2 f1; : : : ; ng, and hence rad A D 0. This shows that (ii) implies (i). Assume that every module in mod A is injective. Then, by Lemma 8.13, for each i 2 f1; : : : ; ng, the canonical monomorphism D.Aei =.rad A/ei / ! D.Aei / is a section, and consequently an isomorphism, because D.Aei / is indecomposable. Therefore .rad A/ei D 0 for any i 2 f1; : : : ; ng, and hence rad A D 0. This proves that (iii) implies (i). Let Q D .Q0 ; Q1 ; s; t / be a finite quiver, I an admissible ideal of the path algebra KQ over a field K, and A D KQ=I the associated bound quiver algebra. It follows from Lemma 1.5, Lemma 3.11 and Corollary 8.9 that A is a finite dimensional K-algebra and the cosets ea D "a C I of the trivial paths "a in Q,

86


a 2 QP 0 , form a set of primitive pairwise orthogonal idempotents of A such that 1A D a2Q0 ea . We proved in Theorem 2.10 that there exists a canonical Klinear equivalence of categories F W mod A ! repK .Q; I /, where repK .Q; I / is the category of finite dimensional representations of the bound quiver .Q; I / over K. We will describe now the simple, semisimple, projective and injective modules in mod A in terms of the representations in repK .Q; I /, and the associated representations in repK .Q; I / will be called the simple, projective, injective representations of .Q; I /, respectively. For a vertex a 2 Q0 , we denote by S.a/ the representation .S.a/b ; '˛ /b2Q0 ;˛2Q1 of .Q; I / defined as ´ S.a/b D

K 0

if b D a, if b ¤ a,

and '˛ D 0 for all arrows ˛ 2 Q1 . Lemma 8.25. Let A D KQ=I be the bound quiver algebra over a field K associated to a bound quiver .Q; I /. The following statements hold. (i) For any a 2 Q0 , S.a/ is a simple representation in repK .Q; I / corresponding to the top of the indecomposable projective module ea A in mod A. (ii) The representations S.a/, a 2 Q0 , form a complete set of pairwise nonisomorphic simple representations in repK .Q; I /. Proof. Clearly, S.a/, a 2 Q0 , are pairwise nonisomorphic simple representations in repK .Q; I /. Moreover, the equivalence functor F W mod A ! repK .Q; I /, defined in the proof of Theorem 2.10, assigns to the simple top ea A=ea rad A of the indecomposable projective right A-module ea A the simple representation S.a/ in repK .Q; I /. This proves the statement (i). The statement (ii) then follows from Theorem 2.10 and Corollary 8.6. We obtain the following immediate corollary. Corollary 8.26. Let A D KQ=I be the bound quiver algebra of a bound quiver .Q; I / over a field K and M D .Ma ; '˛ /a2Q0 ;˛2Q1 a nonzero representation in repK .Q; I /. Then the representation M is semisimple if and only if '˛ D 0 for all arrows ˛ 2 Q1 . Proof. It follows from Lemma 8.25 that M is a semisimple representation of .Q; I / L if and only if M is isomorphic to the direct sum a2Q0 S.a/dimK Ma of simple representations, or equivalently, '˛ D 0 for all ˛ 2 Q1 . The following proposition describes the projective and injective representations in the categories repK .Q; I /.


87

Proposition 8.27. Let A D KQ=I be the bound quiver algebra of a bound quiver .Q; I / over a field K. (i) For each a 2 Q0 , the indecomposable projective representation P .a/ in repK .Q; I / corresponding to the indecomposable projective right A-module ea A is of the form P .a/ D .P .a/b ; '˛ /b2Q0 ;˛2Q1 ; where P .a/b is the K-vector space generated by all cosets wN D w C I , with w the paths in Q from a to b, and for an arrow ˛ W b ! c in Q, the K-linear map '˛ W P .a/b ! P .a/c is given by the right multiplication by ˛N D ˛ C I . (ii) For each a 2 Q0 , the indecomposable injective representation I.a/ in repK .Q; I / corresponding to the indecomposable injective right A-module D.Aea / is of the form I.a/ D .I.a/b ; '˛ /b2Q0 ;˛2Q1 ; where I.a/b is the dual of the K-vector space generated by all cosets wN D w C I , with w the paths in Q from b to a, and for an arrow ˛ W b ! c in Q, the K-linear map '˛ W I.a/b ! I.a/c is given by the dual of the left multiplication by ˛N D ˛ C I . Proof. It follows from the definition of the equivalence functor F W mod A ! repK .Q; I /, given in Theorem 2.10, that, for vertices a; b 2 Q0 , we have P .a/b D .ea A/eb D ea Aeb D ea .KQ=I /eb D "a .KQ/"b ="a I "b : Moreover, if ˛ W b ! c is an arrow in Q, then '˛ W ea Aeb ! ea Aec is given by the right multiplication by the coset ˛N D ˛ C I , that is, if wN is the coset w C I of a path w from a to b in Q, then '˛ .w/ N D wN ˛. N This proves (i). (ii) Similarly as above, for vertices a; b 2 Q0 , we have I.a/b D D.Aea /eb D D.eb Aea / D D."b .KQ/"a ="b I "a /: Moreover, if ˛ W b ! c is an arrow in Q, then '˛ W D.eb Aea / ! D.ec Aea / is N D f .˛N w/ N for f 2 D.eb Aea / D HomK .eb Aea ; K/ and w a defined by '˛ .f /.w/ path from c to a in Q, where ˛N D ˛ C I , wN D w C I . This proves (ii). Corollary 8.28. Let Q be a finite quiver, K a field and I an admissible ideal of KQ. Then, for each a 2 Q0 , we have in repK .Q; I / the equalities top P .a/ D S.a/ D soc I.a/: Proof. This follows from Lemmas 8.22 and 8.25 and Proposition 8.27.

88


Corollary 8.29. Let Q be a finite quiver, K a field, I an admissible ideal of KQ, and A D KQ=I . (i) The modules Pa D ea A (respectively, representations P .a/), a 2 Q0 , form a complete set of pairwise nonisomorphic indecomposable projective modules in mod A (respectively, representations in repK .Q; I /). (ii) The modules Ia D D.Aea / (respectively, representations I.a/), a 2 Q0 , form a complete set of pairwise nonisomorphic indecomposable injective modules in mod A (respectively, representations in repK .Q; I /). Proof. This is a direct consequence of Theorem 2.10, Corollaries 8.6, 8.21 and 8.28, Lemma 8.25 and Proposition 8.27. Let A be a finite dimensional K-algebra over a field K. It follows from Theorems 8.4 and 8.18 that for every module M in mod A there exist a projective cover P .M / ! M and an injective envelope M ! E.M /, and both are uniquely determined (up to isomorphism) by the top top.M / and the socle soc.M / of M , respectively. In order to obtain more precise information on the structure of a module M in mod A, longer approximations of M by projective and injective modules in mod A are needed in general. This leads to projective and injective resolutions of M which we introduce now. hn1 hn A sequence ! Xn1 ! Xn ! XnC1 ! (infinite or finite) of homomorphisms in mod A is called exact if Ker hn D Im hn1 for any n. In particular u r 0 ! L ! M ! N ! 0 is called a short exact sequence if u is a monomorphism, r is an epimorphism and Im u D Ker r. It follows from Lemma 4.2 (see Exercise 12.16) that u is a section if and only if r is a retraction, and if v 2 HomA .M; L/ and s 2 HomA .N; M / are such that vu D idL and rs D idN , then there are direct sum decompositions Im u ˚ Ker v D M D Im s ˚ Ker r. In such a case, we say that the above short exact sequence splits, or is splittable. Following [CE] by a projective resolution of a module M in mod A we mean an exact sequence in mod A of the form dnC1

d1

dn

d0

! PnC1 ! Pn ! Pn1 ! ! P1 ! P0 ! M ! 0; where all modules Pn , n 2 N, are projective. Such a projective resolution is said to d0

be a minimal projective resolution of M in mod A if P0 ! M is a projective cover and dn W Pn ! Im dn is a projective cover for any n 1. Dually, by an injective resolution of a module M in mod A we mean an exact sequence in mod A of the form d0

d1

dn

d nC1

0 ! M ! I0 ! I1 ! ! In1 ! In ! InC1 ! ;


89

where all modules In , n 2 N, are injective. Such an injective resolution of M is d0

said to be a minimal injective resolution of M in mod A if M ! I0 is an injective envelope and Im d n ,! In is an injective envelope for any n 1. Proposition 8.30. Let A be a finite dimensional K-algebra and M be a nonzero module in mod A. Then M admits a minimal projective resolution and a minimal injective resolution in mod A. Proof. We define, applying Theorem 8.4, a minimal projective resolution of M in mod A d2

dn

d1

d0

! Pn ! Pn1 ! ! P2 ! P1 ! P0 ! M ! 0 d0

inductively on n 0 as follows: P0 ! M is a projective cover of M , and, dn

for n 1, Pn ! Pn1 is the composition of a projective cover Pn ! Ker dn1 with the canonical embedding Ker dn1 ,! Pn1 . Similarly, we define, applying Theorem 8.18, a minimal injective resolution of M in mod A d0

d1

d2

dn

0 ! M ! I0 ! I1 ! I2 ! ! In1 ! In ! d0

inductively on n 0 as follows: M ! I0 is an injective envelope of M , and, dn

for n 1, In1 ! In is the composition of the canonical epimorphism In1 ! In1 = Im d n1 D Coker d n1 with an injective envelope Coker d n1 !In . Lemma 8.31. Let A be a finite dimensional K-algebra and M be a nonzero module in mod A. For any two minimal projective resolutions dn

d2

d1

d0

N

N

N

N

! Pn ! Pn1 ! ! P2 ! P1 ! P0 ! M ! 0; d2 d1 d0 dn ! Pxn ! Pxn1 ! ! Px2 ! Px1 ! Px0 ! M ! 0 of M in mod A, there exist isomorphisms fn W Pn ! Pxn , n 2 N, in mod A such that dN0 f0 D d0 and dNnC1 fnC1 D fn dnC1 for n 2 N.

Proof. This is a direct consequence of Theorem 8.4 (ii).

Lemma 8.32. Let A be a finite dimensional K-algebra and M be a nonzero module in mod A. For any two minimal injective resolutions d0

d1

d2

dn

0 ! M ! I0 ! I1 ! I2 ! ! In1 ! In ! ; N0

N1

N2

Nn

d d d d 0 ! M ! IN0 ! IN1 ! IN2 ! ! INn1 ! INn ! of M in mod A, there exist isomorphisms gn W In ! INn , n 2 N, in mod A such that g0 d 0 D dN 0 and dN nC1 gn D gnC1 d nC1 for n 2 N.

90


Proof. This is a direct consequence of Theorem 8.18 (ii).

Let A be a finite dimensional K-algebra over a field K and M be a nonzero module in mod A. The projective dimension pdA M of M is defined as follows: pdA M D n 2 N if there exists a finite minimal projective resolution d2

dn

d1

d0

0 ! Pn ! Pn1 ! ! P2 ! P1 ! P0 ! M ! 0 of M in mod A with Pi ¤ 0 for i 2 f1; : : : ; ng, or pdA M D 1 if M does not admit a finite minimal projective resolution in mod A. Observe that, by Proposition 8.30 and Lemma 8.31, pdA M is well defined. The injective dimension idA M of M is defined as follows: idA M D m 2 N if there is a finite minimal injective resolution d0

d1

d2

dm

0 ! M ! I0 ! I1 ! I2 ! ! Im1 ! Im ! 0 of M in mod A with Ii ¤ 0 for i 2 f1; : : : ; mg, or idA M D 1 if M does not admit a finite minimal injective resolution in mod A. Note that, by Proposition 8.30 and Lemma 8.32, idA M is well defined. For the zero module 0 in mod A, we set pdA 0 D 0 and idA 0 D 0. Observe that for a module M in mod A, we have pdA M D 0 (respectively, idA M D 0) if and only if M is projective (respectively, injective) in mod A. Examples 8.33. (a) Let A D KQ be the path algebra of the quiver QW 1 o

˛

2 :

Then, by Lemma 8.25 and Proposition 8.27, the representations S.1/ W K o

0

and

S.2/ W 0 o

K

form a complete set of pairwise nonisomorphic simple representations in repK .Q/, the representations P .1/ W K o

0

and

P .2/ W K o

1

K

form a complete set of pairwise nonisomorphic indecomposable projective representations in repK .Q/, and the representations I.1/ W K o

1

K

and

I.2/ W 0 o

K

form a complete set of pairwise nonisomorphic indecomposable injective representations in repK .Q/. It follows also from Example 2.9 (b) that S.1/ D P .1/, P .2/ D I.1/ and S.2/ D I.2/ form a complete set of pairwise nonisomorphic


91

indecomposable representations in repK .Q/. Moreover, the canonical exact sequence 0 ! S.1/ ! P .2/ ! S.2/ ! 0 is a minimal projective resolution of S.2/ and a minimal injective resolution of S.1/ in mod A. Then, identifying mod A D repK .Q/, we obtain pdA S.2/ D 1 and idA S.1/ D 1. Obviously, we have pdA S.1/ D 0, pdA P .2/ D 0, idA I.1/ D 0, and idA S.2/ D 0. (b) Let A D KŒx=.x m /, for some m 2. Then A is a unique indecomposable projective module in mod A, S D A= rad A D K is a unique simple module in mod A, and the canonical epimorphism W A ! A= rad A D S is a projective cover of S in mod A. Moreover, we have a projective cover p W A ! rad AD .x/, N m m1 xN D x C .x /, given by p.a/ D xa N for a 2 A, and Ker p D xN Š S. Therefore, S admits an infinite minimal projective resolution in mod A d2

dn

d1

d0

! Pn ! Pn1 ! ! P2 ! P1 ! P0 ! S ! 0 with Pn D A for any n 2 N, d0 D , d2n1 W P2n1 ! P2n2 the composition of p with the inclusion homomorphism u W rad A ,! A, and d2n W P2n ! P2n1 the composition of with the canonical monomorphism w W S ! soc.A/ ,! A, for n 2 N n f0g. In particular, we have pdA S D 1. Similarly, A is a unique indecomposable injective module in mod A and S admits an infinite minimal injective resolution in mod A, d0

d1

d2

dn

0 ! S ! I0 ! I1 ! I2 ! ! In1 ! In ! ; where In D A for any n 2 N, d 0 D w, d 2n1 W I2n1 ! I2n is the composition up, and d 2n W I2n ! I2nC1 is the composition w, for n 2 N n f0g. In particular, we have idA S D 1. (c) Let A D KQ=I , where Q is the quiver ˛

$

o 1

ˇ

2

and I the ideal in KQ generated by ˛ 2 (see Examples 2.9 (d), 2.12 (b) and 5.22 (b)). It follows from Lemma 8.25 and Proposition 8.27 that the representations ' $ o S.1/ W 0 0 0 and S.2/ W K Ko form a complete set of pairwise nonisomorphic simple representations in repK .Q; I /, the representations P .1/ W

01 00

(

K2 o

0

and

P .2/ W

01 00

(

K2 o

0 1

K

92


form a complete set of pairwise nonisomorphic indecomposable projective representations in repK .Q; I /, and the representations I.1/ W

01 00

(

K2 o

10 01

K2

I.2/ W

and

$

0o

K

form a complete set of pairwise nonisomorphic indecomposable injective representations in repK .Q; I /. We identify mod A with repK .Q; I /. Observe that S.1/ D soc.P .1// D rad P .1/, and hence S.1/ admits an infinite minimal projective resolution d2

dn

d1

d0

! Pn ! Pn1 ! ! P2 ! P1 ! P0 ! S.1/ ! 0; where Pn D P .1/ for any n 2 N, d0 D .Œ 0 1 ; 0/ and dn D 00 10 ; 0 for n 2 N n f0g, and consequently pdA S.1/ D 1. On the other hand, S.2/ admits a finite minimal projective resolution in mod A of the form q2

p2

0 ! P .1/ ! P .2/ ! S.2/ ! 0 where p2 D .Œ 0 0 ; 1/ and q2 D 10 01 ; 0 , hence pdA S.2/ D 1. Further, S.2/ D I.2/ is injective, and hence idA S.2/ D 0. We determine a minimal injective resolution of S.1/ in mod A D repK .Q; I /. Observe that the injective envelope u1 W S.1/ ! I.1/ of S.1/ is given by the diagram of K-vector spaces %

0

Ko

0

1 0

01 00

( K2 o

10 01

K2

and hence Coker u1 D I.1/= Im u1 is the direct sum of S.2/ and the representation XW

0

'

Ko

1

K.

Denote by q1 W I.1/ ! S.2/ the composition of the canonical epimorphism I.1/ ! Coker u1 D S.2/˚X with the projection S.2/˚X ! S.2/. Since soc.X / D S.1/, X admits an injective envelope v W X ! I.1/ given by the diagram of K-vector spaces % 1 0 Ko K 1 0

01 00

( K2 o

10 01

1 0

K2 ,

9. Hereditary algebras

93

Then Coker v D I.1/= Im v is isomorphic to X , and the canonical epimorphism w W I.1/ ! X with Ker w D Im v is given by the diagram of K-vector spaces

01 00

(

K2 o

0

01

% Ko

10 01

K2

1

01

K.

Summing up, we conclude that S.1/ admits an infinite minimal injective resolution in mod A d0

d1

d2

dn

0 ! S.1/ ! I0 ! I1 ! I2 ! ! In1 ! In ! where I0 D I.1/, I1 D I2 D I.2/˚I.1/, In D I.1/ for n 2 Nnf0; 1; 2g, d 0 D u1 , 0 q1 , d 2 D 1I.2/ d 1 D Œ vw , d 3 D Œ 0 vw , and d n D vw for n 2 N n f0; 1; 2; 3g, 0 vw and consequently idA S.1/ D 1. (d) Let n 2 be a natural number, K a field and A D K .n/=I.n/, where .n/ is the quiver o 0

˛1

o 1

˛2

o 2

::: o

˛n1

o n2

o n1

˛n

n

and I.n/ is the ideal in K .n/ generated by the paths ˛i ˛i1 for all i 2 f2; : : : ; ng. Then we have in mod A short exact sequences ui

vi

0 ! S.i 1/ ! P .i / ! S.i / ! 0; where vi W P .i / ! S.i / are projective covers and ui W S.i 1/ ! P .i / D I.i 1/ are injective envelopes in mod A, for i 2 f1; : : : ; ng. Moreover, S.0/ is simple projective and S.n/ is simple injective. We conclude that pdA S.i / D i and idA S.i / D n i for any i 2 f0; 1; : : : ; ng.

9 Hereditary algebras In this section we introduce hereditary algebras and describe some of their properties. Let A be a finite dimensional K-algebra over a field K. Then A is said to be right hereditary if any right ideal of A is projective as a right A-module. Similarly, A is said to be left hereditary if any left ideal of A is projective as a left A-module. Finally, A is said to be hereditary if A is left and right hereditary. We note that, by Corollary 8.24, all finite dimensional semisimple K-algebras over a field K are hereditary algebras. We present now characterizations of right hereditary algebras.

94


Theorem 9.1. Let A be a finite dimensional K-algebra over a field K. The following conditions are equivalent. (i) A is right hereditary. (ii) Every right A-submodule of a free module F in mod A is projective. (iii) Every right A-submodule of a projective module P in mod A is projective. (iv) The radical rad P of any indecomposable projective module P in mod A is projective. (v) pdA M 1 for any module M in mod A. Proof. We first prove that (i) implies (ii). Assume F is a free module in mod A. Then F is isomorphic to .AA /m for some positive integer m. Hence F D x1 A˚ ˚xm A, where xi A Š AA , for any i 2 f1; : : : ; mg. Let M be a right A-submodule of F . We claim that M is isomorphic to a direct sum of right L ideals of A, and hence is projective. Consider the right A-submodules Nr D riD1 xi A, r 2 f1; : : : ; mg, of F . Moreover, let N0 D 0. Observe that Nr D Nr1 ˚ xr A as a right A-module for any r 2 f1; : : : ; mg. Hence every element x 2 Nr has a unique presentation of the form x D y C xr ax with y 2 Nr1 and ax 2 A. For each r 2 f1; : : : ; mg, we denote by fr W M \ Nr ! A the homomorphism of right A-modules defined by fr .x/ D ax for any x 2 M \ Nr . Observe that then we have in mod A a short exact sequence ur

fr

0 ! M \ Nr1 ! M \ Nr ! Lr ! 0; where Lr D Im fr and ur is the inclusion homomorphism. Since Lr is a right ideal of A, it follows from the assumption that Lr is a projective right A-module, and consequently fr is a retraction, by Lemma 8.1. Hence, fr gr D idLr for some gr W Lr ! M \ Nr in mod A. Applying Lemma 4.2, we conclude that M \ Nr D Ker fr ˚ Im gr D .M \ Nr1 / ˚ Vr , where Vr D Im gr is isomorphic to Lr as a right A-module. L In order to prove the claim, it is sufficient to show that M D m rD1 Vr . Observe that we have the increasing chain of right A-submodules of M 0 D M \ N0 M \ N1 M \ Nm1 M \ Nm D M: Hence we may assign to every nonzero element Px 2 M the least index sx 2 f1; : : : ; mg such that x 2 M \ Nsx . Let V D m rD1 Vr . Clearly V M . We claim that V D M . Suppose that V ¤ M . Take the least index s 2 f1; : : : ; mg such that s D sx for some element x 2 M n V . Then x 2 M \ Ns D .M \ Ns1 / ˚ Vs , and so x D y C z with y 2 M \ Ns1 and z 2 Vs . Moreover, y ¤ 0, because x … V , and hence z ¤ 0, by the choice of s. Hence we have y D x z 2 M


95

and y … V , since x … V and z 2 V . Therefore, y is an element of M P n V with sy < s, which contradicts the minimality of s. Hence, indeed M D V D m rD1 Vr . Assume now that v1 C C vm D 0 for some elements v1 2 V1 ; : : : ; vm 2 Vm . Then vm D .v1 C Cvm1 / 2 .M \Nm1 /\Vm D 0, so vm D 0. We conclude by the descending induction that vr D 0 forL any r 2 f1; : : : ; mg. Summing up, m we have proved the desired equality M D rD1 Vr , and consequently that (i) implies (ii). We show now that (ii) implies (iii). Let P be a projective module in mod A and M be a right A-submodule of P . It follows from Lemma 8.1 that there exist a free module F and a module P 0 in mod A such that P ˚ P 0 Š F . Then the module M is isomorphic to a right A-submodule N of F . Since, by assumption (ii), N is projective we conclude that M is also a projective right A-module, and so (ii) implies (iii). Observe that the implication (iii) ) (i) is obvious, and consequently the conditions (i), (ii) and (iii) are equivalent. Next we show that the conditions (iii) and (iv) are also equivalent. Clearly, (iii) implies (iv). Assume that (iv) holds. Let P be a projective module in mod A and M be a right A-submodule of P . We prove that M is a projective right A-module, by induction on d D dimK P . For d D 1, P is a simple projective right A-module, and hence M , equal 0 or P , is projective. Assume d 2. We may write P D Q ˚ R for right A-submodules Q and R of P , where Q is indecomposable. We denote by p W P ! Q the canonical projection and by u W M ! P the inclusion homomorphism. Assume first that p.M / D Q. Then pu W M ! Q is an epimorphism, and hence a retraction, because Q is projective (see Lemma 8.1). Thus .pu/v D idQ for some homomorphism v W Q ! M . Applying Lemma 4.2, we conclude that M D Im v ˚ Ker pu, where Im v Š Q. Observe also that Ker pu D M \ R R. Since dimK R D d dimK Q < d , we obtain by the inductive assumption that Ker pu D M \ R is projective. Since Im v, isomorphic to Q, is projective, we conclude that M is a projective right A-module. Assume now that p.M / ¤ Q. Then M .rad Q/ ˚ R, where dimK ..rad Q/ ˚ R/ < d . Moreover, it follows from (iv) that rad Q is projective. Then, by the inductive assumption, we conclude that M is projective. We prove now that (iii) implies (v). Indeed, let M be a module in mod A and h W P .M / ! M a projective cover of M in mod A. Then Ker h is a right Asubmodule of P .M /, and so (iii) implies that Ker h is projective. Hence M admits a minimal projective resolution in mod A of the form d1

d0

0 ! P1 ! P0 ! L ! 0 with P0 D P .M /, P1 D Ker h, d0 D h, and d1 the inclusion homomorphism. This shows that pdA M 1.

96


Finally, observe that (v) implies (iv). Let P be an indecomposable projective module in mod A. Then we have in mod A the canonical short exact sequence u

h

0 ! rad P ! P ! top.P / ! 0; h

where P ! top.P / is the canonical projective cover of the simple module top.P / and u is the inclusion homomorphism. Then pdA top.P / 1 forces rad P to be projective. Hence indeed (v) implies (iv). The following theorem provides characterizations of left hereditary algebras in terms of right modules. Theorem 9.2. Let A be a finite dimensional K-algebra over a field K. The following conditions are equivalent. (i) A is left hereditary. (ii) Every factor module of an injective module E in mod A is injective. (iii) The socle factor E= soc.E/ of any indecomposable injective module E in mod A is injective. (iv) idA M 1 for any module M in mod A. Proof. Consider the standard duality D D HomK .; K/ mod A o

D D

/

mod Aop :

Then a module E in mod A is injective if and only if the associated module D.E/ in mod Aop is projective (see Proposition 8.16). Further, a homomorphism f W M ! N in mod A is an epimorphism (respectively, minimal epimorphism) if and only if D.f / W D.N / ! D.M / is a monomorphism (respectively, minimal monomorphism) in mod Aop . Moreover, for an indecomposable injective module E in mod A, the canonical short exact sequence 0 ! soc.E/ ! E ! E= soc.E/ ! 0 in mod A leads to the short exact sequence 0 ! D.E= soc.E// ! D.E/ ! D.soc.E// ! 0 in mod Aop , where D.E/ is an indecomposable projective module and D.soc.E// is a simple module, and consequently D.E= soc.E// Š rad D.E/ in mod Aop . Since A is left hereditary, Aop is right hereditary and hence Theorem 9.1 holds for Aop . Then, applying the duality D, we conclude the required equivalences of (i), (ii), (iii) and (iv) for A.


97

The following third theorem clarifies the relationship between the left hereditary and right hereditary algebras. Theorem 9.3. Let A be a finite dimensional K-algebra over a field K. Then A is left hereditary if and only if A is right hereditary. Proof. We will show that if A is right hereditary then A is left hereditary. Observe that the converse implication follows then from the fact that Aop right hereditary implies Aop left hereditary. Assume that A is a right hereditary algebra. Let v W M ! N be an epimorphism in mod A with M an injective right A-module. We claim that then N is an injective module. Then, applying Theorem 9.2, we obtain that A is a left hereditary algebra. In order to show that N is an injective module in mod A, we apply Baer’s Lemma 8.14. Let I be a right ideal of A and u W I ! A be the inclusion homomorphism. Take a homomorphism ' W I ! N in mod A. Since A is right hereditary, by assumption, we conclude that the right ideal I of A is a projective module in mod A, and there exists a homomorphism f W I ! M such that vf D '. Further, there exists a homomorphism g W A ! M such that gu D f , because M is injective. Summing up, for the composed homomorphism D vg W A ! N in mod A, we obtain the equalities u D .vg/u D v.gu/ D vf D ', and so N is injective, by Lemma 8.14. Corollary 9.4. Let A be a finite dimensional hereditary K-algebra over a field K. The following statements hold. (i) Every nonzero homomorphism between indecomposable projective modules in mod A is a monomorphism. (ii) For every indecomposable projective module P in mod A, EndA .P / is a division K-algebra. (iii) Every nonzero homomorphism between indecomposable injective modules in mod A is an epimorphism. (iv) For every indecomposable injective module E in mod A, EndA .E/ is a division K-algebra. Proof. (i) Let f W P ! Q be a nonzero homomorphism in mod A, with P and Q indecomposable projective modules. Then we have the canonical short exact sequence 0 ! Ker f ! P ! Im f ! 0 in mod A. Since Im f is a right A-submodule of Q, it follows from Theorem 9.1 that Im f is projective, and consequently P Š Im f ˚ Ker f , by Lemmas 4.2 and 8.1. Since Im f ¤ 0 and P is indecomposable, we obtain Ker f D 0, or equivalently, f is a monomorphism.

98


(ii) Let P be an indecomposable projective module in mod A. Then it follows from (i) that every 0 ¤ f 2 EndA .P / is a monomorphism, and hence an isomorphism. This shows that EndA .P / is a division K-algebra. The statements (iii) and (iv) follow in a similar way, or from (i) and (ii) by Proposition 8.16. The following theorem provides a wide class of finite dimensional hereditary algebras. Theorem 9.5. Let K be a field and Q be a finite acyclic quiver. Then the path algebra KQ is a hereditary algebra. Proof. We note that, by Lemma 1.3, A is a finite dimensional K-algebra and the set "a , a 2 Q0 , of all trivial paths of Q P is a complete set of pairwise orthogonal primitive idempotents of A with 1A D a2Q0 "a . Moreover, Aop D KQop , where clearly Qop is a finite acyclic quiver. Hence, by Theorem 9.3, in order to show that A is a hereditary algebra, it is enough to prove that A is a right hereditary algebra. We will show that the radical of any indecomposable projective module in mod A is also projective. This will imply, by Theorem 9.1, that A is right hereditary. We identify mod A with repK .Q/. It follows from Proposition 8.27 that, for each a 2 Q0 , the indecomposable projective right A-module "a A is the representation P .a/ D .P .a/b ; '˛ /b2Q0 ;˛2Q1 , where P .a/b is the K-vector space whose basis is formed by all paths in Q from a to b, and, for an arrow ˛ in Q with b D s.˛/ and c D t .˛/, the K-linear map '˛ W P .a/b ! P .a/c is given by the right multiplication by ˛. Then rad P .a/ D .P .a/b ; '˛ /b2Q0 ;˛2Q1 where P .a/a D 0, P .a/b D P .a/b for all b 2 Q0 n fag, and '˛ D '˛ for any arrow in Q with s.˛/ ¤ a. Let ˇ1 ; : : : ; ˇr be all arrows of Q with a D s.ˇi / for i 2 f1; : : : ; rg. Observe now that every path w in Q with a D s.w/ is of the form w D ˇi u for a path u in Q with t .ˇi / D s.u/, uniquely determined by w. Then it follows that r M rad P .a/ D P .t .ˇi //; iD1

and consequently rad P .a/ is projective.

In fact, we have also the following converse theorem. Theorem 9.6. Let K be a field, Q a finite quiver, I an admissible ideal of the path algebra KQ, and A D KQ=I the associated bound quiver algebra. The following conditions are equivalent. (i) A is a hereditary algebra. (ii) I D 0 and the quiver Q is acyclic.


99

Proof. Observe that the implication (ii) ) (i) follows from Theorem 9.5, because the zero ideal in KQ is admissible if and only if the quiver Q is acyclic. Hence we have to show that (i) implies (ii). Assume that A is a hereditary algebra. We identify mod A with repK .Q; I /. Then it follows from Proposition 8.27 that the indecomposable projective module P .a/ in mod A associated to the vertex a 2 Q0 is of the form P .a/ D .P .a/b ; '˛ /b2Q0 ;˛2Q1 , where P .a/b is the K-vector space generated by all cosets wN D w C I , with w the paths in Q from a to b, and for an arrow ˛ from b to c in Q, the K-linear map '˛ W P .a/b ! P .a/c is given by the right multiplication by ˛N D ˛ C I . In particular, we have dimK P .a/b D dimK "a .KQ/"b dimK "a I "b for any b 2 Q0 . Further, for each arrow in Q, we have a nonzero homomorphism of right A-modules f W P .t . // ! P .s.// which assigns to the coset uN D u C I of a path u in Q from t . / to a vertex b, the coset N uN D u C I of the path u from s./ to b. Further, f is not an isomorphism, because N D C I belongs to rad.KQ=I / (see Lemma 3.6). We claim now that the quiver Q is acyclic. Indeed, suppose Q contains a cycle ˛1

˛2

˛m

a D a0 ! a1 ! a2 ! ! am1 ! am D a: Then we obtain a cycle f˛2

f˛m

f˛1

P .a/ D P .am / ! P .am1 / ! ! P .a2 / ! P .a1 / ! P .a0 / D P .a/ of nonzero nonisomorphisms in mod A between the indecomposable projective Amodules. Further, by Corollary 9.4, we obtain that f˛m ; : : : ; f˛2 ; f˛1 are proper monomorphisms, which gives the proper monomorphism f˛1 f˛2 : : : f˛m W P .a/ ! P .a/, a contradiction. We prove now that I D 0. Assume I ¤ 0. Since the quiver Q is acyclic and finite, we may number the vertices of Q such that Q0 D f1; : : : ; ng and for any path from a to b in Q we have a > b. Then there is a least a in Q0 such that "a I "b ¤ 0 for some b 2 Q0 . In particular, there is in Q a path from a to b, and consequently rad P .a/ ¤ 0. Let ˇ1 ; : : : ; ˇr be all arrows of Q with a D s.ˇi / for each i 2 f1; : : : ; rg. Since A is a right hereditary algebra, we know from Theorem 9.1 that rad P .a/ is a projective right A-module. Then we conclude that rad P .a/ D

r M

P .t .ˇi //:

iD1

It follows from our choice of a that " t.ˇi / I "c D 0, and consequently we have dimK P .t.ˇi //ec D dimK " t.ˇi / .KQ/"c , for any i 2 f1; : : : ; rg and c 2 Q0 . We

100


then conclude that dimK P .a/ D 1 C dimK rad P .a/ D 1 C

r X

dimK P .t .ˇi //

iD1

D1C

r X

dimK " t.ˇi / .KQ/ D 1 C dimK rad "a .KQ/

iD1

D dimK "a .KQ/: This leads to a contradiction, because X X dimK P .a/ec ; dimK "a .KQ/ D dimK "a .KQ/"c ; dimK P .a/ D c2Q0

dimK P .a/ec D dimK "a .KQ/"c dimK "a I "c

c2Q0

for any c 2 Q0 ;

and "a I "b ¤ 0.

10 Nakayama algebras In this section we describe the structure of module categories of Nakayama algebras, for which every indecomposable module has a unique composition series. Let A be a finite dimensional K-algebra over a field K. A nonzero module M in mod A is said to be uniserial if its submodule lattice is a chain. Observe that every uniserial module M in mod A has a unique composition series, and hence has a simple socle and a simple top. In particular, a uniserial module M in mod A is necessarily indecomposable. The following proposition gives a characterization of uniserial modules. Proposition 10.1. Let A be a finite dimensional K-algebra over a field K and M be a nonzero module in mod A. The following conditions are equivalent. (i) M is a uniserial module. (ii) The radical series of M is a composition series of M . (iii) ``.M / D `.M /. Proof. Let M rad M rad2 M radn1 M radn M D 0 be the radical series of M , and so n D ``.M /. We prove that (i) implies (ii) by induction on `.M /. Assume M is a uniserial module and `.M / D m. If m D 1 then M is a simple module, rad M D 0, and the statement (ii) is obvious. Assume m 2 and (ii) holds for every uniserial module N in mod A with `.N / < m. Observe that rad M is a uniserial submodule of M with `.rad M / < `.M /, and hence rad M rad2 M radn M D 0

10. Nakayama algebras

101

is a composition series of rad M . Moreover, rad M is a unique maximal right Asubmodule of M , because M is uniserial. Hence M rad M rad2 M radn1 M radn M D 0 is a composition series of M . Therefore, (i) implies (ii). Obviously (ii) implies (iii). We show that (iii) implies (i). Assume ``.M / D `.M /. Then it follows from Lemma 7.3 that, for each i 2 f0; 1; : : : ; n 1g, radi M= radiC1 M is a simple right A-module, and consequently radiC1 M is a maximal right A-submodule of radi M . Since radiC1 M D rad.radi M / is the intersection of all maximal right A-submodules of radi M , we conclude that radiC1 M is a unique maximal right A-submodule of radi M . In particular, rad M is a unique maximal right A-submodule of M . Let N be a right A-submodule of M different from M and 0. Then N rad M . Let i be a maximal element of f1; : : : ; n 1g such that N radi M . Because radiC1 M is a unique maximal right A-submodule of radi M and N is not contained in radiC1 M , we infer that N D radi M . Therefore, the radical series of M is formed by all right A-submodules of M , and so M is a uniserial module. Hence (iii) implies (i). A finite dimensional K-algebra A over a field K is called a Nakayama algebra if every indecomposable projective module in mod A and in mod Aop is uniserial. It follows from Proposition 8.16 that a finite dimensional K-algebra A is a Nakayama algebra if and only if all indecomposable projective modules and all indecomposable injective modules in mod A are uniserial modules. Clearly, A is a Nakayama algebra if and only if Aop is a Nakayama algebra. We also note that the semisimple algebras, described in Section 6, are trivially Nakayama algebras, because their indecomposable finite dimensional modules are simple. Lemma 10.2. Let A be a finite dimensional Nakayama K-algebra and J be a proper two-sided ideal of A. Then A=J is also a Nakayama algebra. Proof. Let e1 ; : : : ; en be a set of pairwise orthogonal primitive idempotents of A such that 1A D e1 C C en . We may assume (without loss of generality) that e1 ; : : : ; em , for some m n, are all idempotents from fe1 ; : : : ; en g which do not belong to J . Hence the cosets eN1 D e1 C J; : : : ; eNm D em C J form a set of pairwise orthogonal idempotents of AN D A=J such that 1AN D eN1 C C eNm . It follows from Proposition 8.2 that a module P in mod A (respectively, mod Aop D A-mod) is an indecomposable projective module if and only if P is isomorphic to a module ei A (respectively, Aei ) for some i 2 f1; : : : ; ng. Then a module N is an indecomposable projective Q in mod AN (respectively, mod ANop D A-mod) module if and only if Q is isomorphic to a module eNi AN Š ei A=ei J (respectively, ANeNi Š Aei =Jei ) for some i 2 f1; : : : ; mg. Since A is a Nakayama algebra, the modules e1 A; : : : ; en A and Ae1 ; : : : ; Aen are uniserial A-modules. Clearly, then N : : : ; eNm AN and ANeN1 ; : : : ; ANeNm are uniserial A-modules N (see Lemma 3.17), and eN1 A; N consequently A D A=J is a Nakayama algebra.

102


We exhibit now a wide class of Nakayama algebras. Theorem 10.3. Let Q be a finite connected quiver, K a field, I an admissible ideal of the path algebra KQ, and A D KQ=I the associated bound quiver algebra. Then the following conditions are equivalent. (i) A is a Nakayama algebra. (ii) Q is one of the following two quivers o 1

˛1

o 2

˛2

o 3

: : : o ˛i 1 o i

˛3

1 1

˛n

˛i

: : : o ˛n2 o ˛n1 ; n n1

˛1

" 2

n B ˛n1

˛2

3

n 1 Q ˛n2

d ˛i

q i

˛3

˛i 1

with n 1 vertices. Proof. Let Q D .Q0 ; Q1 ; s; t /. We first show that (i) implies (ii). Assume A is a Nakayama algebra. Observe that Q is of one of the forms presented in (ii) if and only if, for each vertex a 2 Q0 , there exists at most one arrow ˛ 2 Q with a D s.˛/ and at most one arrow with a D t . /. Suppose Q admits a vertex i and two different arrows ˛ and ˇ with s.˛/ D i D s.ˇ/. Let J be the ideal of A D KQ=I generated by the cosets N D C I of all arrows 2 Q1 different from ˛ and ˇ and the idempotents ea D "a C I for a 2 Q0 different from i , j D t.˛/ and k D t .ˇ/. Consider the quotient algebra B D A=J . Then B is the path algebra K of the quiver of one of the forms j o

˛

i

ˇ

/k

or

˛

i

ˇ

// j D k :

It follows from Lemmas 3.6 and 10.2 that B is Nakayama K-algebra with .rad B/2 D 0. On the other hand, applying Proposition 8.27, we conclude that for the indecomposable projective representation P .i / in repK . / we have rad P .i / D S.j / ˚ S.k/ D K˛ ˚ Kˇ, where S.j / and S.k/ are the simple representations in repK . / at the vertices j and k, respectively. Therefore, P .i / is not a

10. Nakayama algebras

103

uniserial representation, and hence the indecomposable projective B-module eNi B, for eNi D ei C J , is not a uniserial right B-module, a contradiction. We have proved that every vertex a 2 Q0 is the source of at most one arrow of Q. Observe also that Aop D KQop =I op is also a Nakayama algebra, and hence every vertex a of Qop is the source of at most one arrow of Qop . This implies that every vertex a 2 Q0 is the target of at most one arrow of Q. Consequently, (i) implies (ii). Assume now that Q is one of the quivers presented in (ii). Then applying Lemma 3.6 and Proposition 8.27, we conclude that, for each vertex a 2 Q0 , the radical series of the indecomposable projective representation P .a/ in repK .Q; I / at the vertex a is a composition series of P .a/. Invoking now the equivalence mod A ! repK .Q; I /, established in Theorem 2.10, and Proposition 10.1, we obtain that every indecomposable projective module in mod A is uniserial. Observe also that Aop Š A because Q coincides with Qop , and hence every indecomposable projective module in mod Aop is uniserial. Therefore, A is a Nakayama algebra. This shows that (ii) implies (i). Lemma 10.4. Let A be a finite dimensional Nakayama K-algebra and P be an indecomposable projective module in mod A with ``.P / D ``.AA /. Then P is an injective module in mod A. Proof. Let u W P ! E.P / be an injective envelope of P in mod A. Since A is a Nakayama algebra, P is a uniserial module. In particular, the socle soc.P / of P is simple. Then the socle soc.E.P // of E.P / is simple, because u is an essential monomorphism inducing the isomorphism soc.u/ W soc.P / ! soc.E.P //, and consequently E.P / is an indecomposable injective module in mod A. Therefore, E.P / is a uniserial module and we obtain the inequalities ``.AA / D ``.P / D `.P / `.E.P // D ``.E.P // ``.AA /; by Corollary 5.19 and Proposition 10.1. Hence, `.P / D `.E.P //, and this forces u to be an isomorphism. Theorem 10.5. Let A be a finite dimensional Nakayama K-algebra over a field K. Then every indecomposable module M in mod A is isomorphic to a module of the form P = radm P for some indecomposable projective module P in mod A and an integer m with 1 m ``.P /. Proof. Observe first that, if P is an indecomposable projective module in mod A and m 2 f1; : : : ; ``.P /g, then P = radm P is a uniserial module, and hence indecomposable. Let M be an indecomposable module in mod A, and m D ``.M /. Then M radm A D radm M D 0, and hence M has a natural structure of a right .A= radm A/-module via m.a C radm A/ D ma for m 2 M and a 2 A. Moreover, by Lemma 10.2, A= radm A is a Nakayama algebra with ``.A= radm A/ D m. Let

104


h W Px ! M be a projective cover of M in mod.A= radm A/, and Px D Px1 ˚ ˚ Pxn a decomposition of Px into a direct sum of indecomposable projective modules in mod.A= radm A/. Then we obtain the inequalities ˚ m D ``.M / max ``.Px1 /; : : : ; ``.Pxn / ``.A= radm A/ D m: For each i 2 f1; : : : ; ng, denote by hi W Pxi ! M the restriction of h to Pxi . Since A= radm A is a Nakayama algebra, Px1 ; : : : ; Pxn are uniserial modules. Hence, if hi W Pxi ! M is not aP monomorphism, then ``.Im hi / D ``.Pxi = Ker hi / ``.Pxi / 1. Moreover, M D niD1 Im hi , and so ``.M / maxf``.Im h1 /; : : : ; ``.Im hn /g: Hence, there exists r 2 f1; : : : ; ng such that ``.Pxr / D m D ``.Im hr /, and then hr is a monomorphism. It follows then from Lemma 10.4 that Pxr is an injective module in mod.A= radm A/, and consequently hr W Pxr ! M is a section (see Lemma 8.13). Then M D Im hr ˚ M 0 for an A-submodule M 0 of M , by Lemma 4.2. This forces M D Im hr because M is indecomposable and Im hr ¤ 0. Therefore, hr W Pxr ! M is an isomorphism in mod.A= radm A/. We also note that Pxr D eA=e radm A for a primitive idempotent e of A. Hence P D eA is an indecomposable projective module in mod A and P = radm P D eA=e radm A is isomorphic to M in mod A= radm A, and so in mod A. The following characterization of Nakayama algebras, proved by T. Nakayama in [Nak3], is an immediate consequence of Theorem 10.5. Corollary 10.6. Let A be a finite dimensional K-algebra over a field K. Then A is a Nakayama algebra if and only if every indecomposable module in mod A is uniserial. A finite dimensional K-algebra A over a field K is said to be of finite representation type if the number of isomorphism classes of indecomposable modules in mod A is finite. Theorem 10.7. Let A be a finite dimensional Nakayama K-algebra over a field K. Then A is of finite representation type and the number of isomorphism classes of indecomposable modules in mod A is equal to n X

``.Pi /;

iD1

where P1 ; : : : ; Pn is a complete set of pairwise nonisomorphic indecomposable projective right A-modules.

11. The Grothendieck group and the Cartan matrix

105

Proof. It follows from Theorem 10.5 that every indecomposable module M in mod A is isomorphic to a module P = radm P for some indecomposable projective module P in mod A and an integer m with 1 m ``.P /. We know also from Theorem 8.4 and Corollary 8.6 that, for an indecomposable projective module P in mod A and an integer r 2 f1; : : : ; ``.P /g, the canonical epimorphism P ! P = radr P is a projective cover. Since A is a Nakayama algebra, the radical series of P is its unique composition series, and hence P = radr P is uniserial with r D ``.P = radr P / D `.P = radr P / for any r 2 f1; : : : ; ``.P /g. Then, for two indecomposable projective modules P and Q in mod A and r 2 f1; : : : ; ``.P /g, s 2 f1; : : : ; ``.Q/g, we have P = radr P Š Q= rads Q in mod A if and only if r D s and P Š Q in mod A. It follows from Proposition 8.2 that the number of isomorphism classes of indecomposable projective modules in mod A is finite. Therefore, A is of finite representation type and the number of isomorphism classes of indecomposable modules in mod A equals to ``.P1 / C C ``.Pn /, for a complete set P1 ; : : : ; Pn of pairwise nonisomorphic indecomposable projective right A-modules.

11 The Grothendieck group and the Cartan matrix Let A be a finite dimensional K-algebra over a field K. The Grothendieck group of A is the abelian group K0 .A/ D F=F0 , where F is the free abelian group having as Z-basis the set of isomorphism classes fM g of modules M in mod A and F0 is the subgroup of F generated by the elements fM g fLg fN g given by all short exact sequences 0 ! L ! M ! N ! 0 in mod A. We denote by ŒM the image of the isomorphism class fM g of a module M in mod A via the canonical group epimorphism F ! F=F0 D K0 .A/. We also note that F is a set because for each module M in mod A there exists an epimorphism Am ! M , for some positive integer m. Theorem 11.1. Let A be a finite dimensional K-algebra over a field K and let S1 ; : : : ; Sn be a complete set of pairwise nonisomorphic simple modules in mod A. Then the Grothendieck group K0 .A/ of A is a free abelian group and ŒS1 ; : : : ; ŒSn form a Z-basis of K0 .A/. Proof. Let M be a module in mod A and 0 D M0 M1 Mm D M be a composition series of M . For each i 2 f1; : : : ; ng, we abbreviate by ci .M / the composition multiplicity cSi .M / of the simple module Si in M . Observe that, for each j 2 f1; : : : ; mg, we have the short exact sequence 0 ! Mj 1 ! Mj ! Mj =Mj 1 ! 0

106


with Mj =Mj 1 a simple module in mod A. Then it follows from the definition of K0 .A/ that m n X X ŒMj =Mj 1 D ci .M /ŒSi : ŒM D j D1

iD1

Therefore, ŒS1 ; : : : ; ŒSn generate the abelian group K0 .A/. Observe that we may assign to M the vector c.M / D .c1 .M /; : : : ; cn .M // in Nn Zn . Clearly, if M Š N in mod A, then c.M / D c.N /. Therefore, c induces a unique group homomorphism c W K0 .A/ ! Zn and e1 D c.S1 /; : : : ;P en D c.Sn / form a canonical Z-basis ofP Zn . Hence, if r1 ; : : : ; rn 2 Z and niD1 ri ŒSi D 0, n then .r1 ; : : : ; rn / D c iD1 ri ŒSi D 0, and consequently ŒS1 ; : : : ; ŒSn are Z-linearly independent in K0 .A/. It follows that K0 .A/ is a free abelian group with Z-basis ŒS1 ; : : : ; ŒSn , and c W K0 .A/ ! Zn is an isomorphism of abelian groups. Let A be a finite dimensional K-algebra over a field K. Let P1 ; : : : ; Pn be a complete set of pairwise nonisomorphic indecomposable projective modules in mod A. It follows from Corollary 8.6 that S1 D P1 = rad P1 ; : : : ; Sn D Pn = rad Pn form a complete set of pairwise nonisomorphic simple modules in mod A. For a module M in mod A, the vector c.M / D cS1 .M /; : : : ; cSn .M / is called the composition vector of M . The Cartan matrix of A is the n n matrix CA D Œcij 2 Mn .Z/, where cij D cSi .Pj / is the composition multiplicity of Si in Pj , for all i; j 2 f1; : : : ; ng. Therefore, the j -th column of CA is given by the transpose c.Pj /t of the composition vector c.Pj / D cS1 .Pj /; : : : ; cSn .Pj / of Pj . We give also an alternative description of the coefficients of the Cartan matrix CA of A. Observe first that, by Lemma 5.1, F1 D EndA .S1 /; : : : ; Fn D EndA .Sn / are finite dimensional division K-algebras. We note that then A= rad A Š Mm1 .F1 / Mmn .Fn / for some positive integers m1 ; : : : ; mn , by Theorem 6.3. For each i 2 f1; : : : ; ng, we set ƒi D EndA .Pi /. Observe that ƒi is a local finite dimensional K-algebra, because Pi is indecomposable (see Lemma 4.4). Moreover, we have the canonical embedding HomA .Pi ; rad Pi / ! ƒi of K-vector spaces given by the inclusion map rad Pi ,! Pi . Lemma 11.2. For each i 2 f1; : : : ; ng, we have (i) rad ƒi D HomA .Pi ; rad Pi /; (ii) ƒi = rad ƒi Š Fi . Proof. Since Pi is indecomposable projective, rad Pi is a unique maximal right Asubmodule of Pi , by Propositions 5.16 and 8.2. Further, by Lemmas 4.2 and 8.1, an endomorphism f 2 EndA .Pi / D ƒi is noninvertible if and only if Im f rad Pi .

11. The Grothendieck group and the Cartan matrix

107

Then the equality rad ƒi D Homƒ .Pi ; rad Pi / follows from Lemma 3.8. Observe that for any f 2 EndA .Pi / D ƒi we have a commutative diagram in mod A, 0

/ rad Pi

0

/ rad Pi

f0

/ Pi

i

/0

fN

f

/ Pi

/ Si

i

/ Si

/ 0,

where f 0 is the restriction of f to rad Pi and fN is given by fN.x Crad Pi / D f .x/C rad Pi , for any x 2 Pi . Observe also that fN D 0 if and only if Im f rad Pi . Moreover, since Pi is projective, for any g 2 EndA .Si / there exists f 2 EndA .Pi / such that g D fN. Therefore, the map EndA .Pi / ! EndA .Si / defined above is ! Fi of a K-algebra epimorphism which induces an isomorphism ƒi = rad ƒi K-algebras. For i 2 f1; : : : ; ng and a module M in mod A, the vector space HomA .Pi ; M / has the natural structure of a right ƒi -module given by the composition gf , for g 2 HomA .Pi ; M / and f 2 ƒi D EndA .Pi /. Lemma 11.3. Let M be a module in mod A, and i 2 f1; : : : ; ng. Then the composition multiplicity cSi .M / of Si in M is the length `ƒi .HomA .Pi ; M // of the module HomA .Pi ; M / in mod ƒi . Proof. Let 0 D M0 M1 Mm D M be a composition series of M in mod A. Then we obtain a chain 0 D HomA .Pi ; M0 / HomA .Pi ; M1 / HomA .Pi ; Mm / D HomA .Pi ; M / of right ƒi -submodules of HomA .Pi ; M /. Moreover, for each j 2 f1; : : : ; mg, the short exact sequence 0 ! Mj 1 ,! Mj ! Mj =Mj 1 ! 0 in mod A induces the short exact sequence 0 ! HomA .Pi ; Mj 1 / ! HomA .Pi ; Mj / ! HomA .Pi ; Mj =Mj 1 / ! 0 in mod ƒi . Furthermore, since Mj =Mj 1 is a simple right A-module, we obtain that HomA .Pi ; Mj =Mj 1 / Š HomA .Pi ; Si / in mod ƒi if Mj =Mj 1 Š Si , and HomA .Pi ; Mj =Mj 1 / D 0 if Mj =Mj 1 © Si . Observe also that HomA .Pi ; Si / is a simple right ƒi -module generated by the canonical epimorphism i W Pi ! Pi = rad Pi D Si in mod A. Indeed, for any nonzero homomorphism g 2 HomA .Pi ; Si /, there exists f 2 EndA .Pi / such that g D i f , because Pi is projective and g is an epimorphism. Therefore, the required equality `ƒi .HomA .Pi ; M // D cSi .M / holds.

108


Corollary 11.4. For i; j 2 f1; : : : ; ng, we have cij D `ƒi .HomA .Pi ; Pj //: The following proposition shows that the Cartan matrix of a bound quiver algebra is easy to determine. Proposition 11.5. Let K be a field, Q D .Q0 ; Q1 / a finite quiver, I an admissible ideal of KQ, and A D KQ=I . Then CA D Œcab a;b2Q0 ; where cab D dimK eb Aea D dimK ."b .KQ/"a ="b I "a / for all a; b 2 Q0 . Proof. The simple right A-modules Sa D ea A=ea rad A corresponding to the simple representations S.a/ in repK .Q; I /, via the K-linear equivalence F W mod A ! repK .Q; I / established in Theorem 2.10, are one-dimensional K-vector spaces. Further, by Corollary 8.29, the right A-modules ea A, corresponding to the representations P .a/ of .Q; I / over K (via the functor F ), form a complete set of pairwise nonisomorphic indecomposable projective modules in mod A. Then the required description of CA follows from Proposition 8.27 and the fact that the Klinear equivalence functor F W mod A ! repK .Q; I / carries the composition series in mod A into the composition series in repK .Q; I /. Corollary 11.6. Let K be a field, Q D .Q0 ; Q1 / a finite acyclic quiver and A D KQ. Then CA D Œcab a;b2Q0 ; where cab is the number of pairwise different paths in Q with source b and target a, for all a; b 2 Q0 . Proof. It follows from Proposition 11.5, since, for I D 0, dimK "b .KQ/"a is the number of pairwise different paths in Q with source b and target a, for all vertices a; b 2 Q0 . We end this section with a distinguished property of the Cartan matrices of the bound quiver algebras of acyclic bound quivers. Proposition 11.7. Let K be a field, Q a finite acyclic quiver, I an admissible ideal of KQ, and A D KQ=I . Then det CA D 1. Proof. Let Q D .Q0 ; Q1 ; s; t / and jQ0 j D n. Since the quiver Q is acyclic, there exists a bijection W Q0 ! f1; : : : ; ng such that, for each arrow ˛ 2 Q1 , we have .s.˛// > .t .˛//. Observe that such a bijection may be constructed as follows. Take a vertex a 2 Q0 which is a sink in Q, that is, there is no arrow ˛ 2 Q1 with a D s.˛/. Define .a/ D 1. Next consider the full subquiver

12. Exercises

109

Q.a/ of Q having Q0 n fag as the set of vertices. If n 2, then Q.a/ contains a sink b, and we set .b/ D 2. We continue this process and the claim follows by induction on n D jQ0 j. Therefore, without loss of generality, we may assume j > i . From that Q0 D f1; : : : ; ng and, for every arrow j ! i in Q, we have Proposition 11.5 we have CA D Œcij i;j 2Q0 , where cij D dimK "j .KQ/"i ="j I "i for all vertices i; j 2 Q0 . It follows from our assumption that (1) ci i D 1 for any i 2 Q0 ; (2) cij D 0 for any i; j 2 Q0 with i > j . Obviously then det CA D 1.

12 Exercises 1. Let A be a finite dimensional K-algebra over a field K, a1 ; : : : ; an and b1 ; : : : ; bn two bases of the K-vector space, and ˛ij k ; ˇij k 2 K, i; j; k 2 f1; : : : ; ng, the associated structure constants such that aj ak D

n X

and bj bk D

˛ij k ai

iD1

n X

ˇij k bi

iD1

x (respectively, Rt , R xt ) the first (refor all j; k 2 f1; : : : ; ng. Denote by L, L spectively, second) regular representations of A over K associated to the bases a1 ; : : : ; an and b1 ; : : : ; bn , respectively. Prove that: x are equivalent. (a) The representations L and L xt are equivalent. (b) The representations Rt and R 2. Let Q be the quiver o

1 o

˛ ˇ

2

and KQ the path algebra of Q over a field K. For each element a 2 K consider the elements e1;a D "1 C ˛ C aˇ and e2;a D "1 ˛ aˇ, where "1 and "2 are the trivial paths of Q at the vertices 1 and 2, respectively. Prove that e1;a and e2;a are orthogonal primitive elements of KQ such that 1KQ D e1;a C e2;a . 3. Let K be a field, Q the quiver o 1

˛

o 2

ˇ

; 3

I the ideal in KQ generated by ˇ˛, and A D KQ=I the associated bound quiver algebra. Consider the elements of A f1 D ."1 C ˛/ C I;

f2 D ."2 ˛ ˇ/ C I;

f3 D ."3 C ˇ/ C I;

110


where "1 ; "2 ; "3 are the trivial paths of Q at the vertices 1; 2; 3, respectively. Prove that f1 ; f2 ; f3 are pairwise orthogonal primitive idempotents of A with 1A D f1 C f2 C f3 . 4. Let K be a field, Q the quiver ˛ ttt

2 eKK ˇ KK KK

t ytt o o ; 1 3 4 I1 the ideal in KQ generated by , I2 the ideal in KQ generated by ˇ˛, and A1 D KQ=I1 , A2 D KQ=I2 the associated bound quiver algebras. Prove that the K-algebras A1 and A2 are isomorphic.

5. Let K be a field, Q the quiver oo 1

˛ ˇ

o 2

: 3

For each element a 2 K, consider the ideal Ia in KQ generated by ˛ C aˇ, and the associated bound quiver algebra ƒa D KQ=Ia . Show that the K-algebras ƒa , a 2 K, are pairwise isomorphic. 6. Let K be a field, Q the quiver 2 eKK ˇ KK KK t t yt o 5 ; 1 eKK t KK t K tt 4 K ytt 3 I1 the ideal in KQ generated by ˇ˛, I2 the ideal in KQ generated by ˇ˛ C , and A1 D KQ=I1 , A2 D KQ=I2 the associated bound quiver algebras. Prove that dimK A1 D dimK A2 but the K-algebras A1 and A2 are nonisomorphic. ˛ ttt

7. Let Q be the quiver

3 4 2 @@ @@ ~~ ~ @ˇ ~~ ˛ @@ ~~ 1 and KQ the path algebra of Q over a field K. Prove the following: (a) KQ is isomorphic to the matrix K-algebra 9 2 3 82 3 K 0 0 0 a 0 0 0 ˆ > ˆ > = 6K K 0 0 7 ˆ > : ; K 0 0 K z 0 0 d

12. Exercises

111

(b) The representations 0? 0 0 ?? ?? K

0 K? 0 ?? ?? 0

0 0 KA AA AA 1 K

0 0? K ?? ??1 K

0 KA K AA AA1 1 K

0 K KA AA } AA }} } ~} 1 1 K

0 0 ?? K ?? ?? 0 0 K 0? ?? }} ?? } ~}} 1 K K 0? K ?? } ??1 }}} ~} 1 K

K 0= 0 == == 0 K K B K BB 0 || | BB1 1 ! }|| 1 0 1 K2 K KA K AA } AA1 }}} ~} 1 1 K

form a complete family of pairwise nonisomorphic indecomposable representations in repK .Q/. (c) Describe the K-vector spaces HomQ .M; N / for all representations M and N listed in (b). 8. Let K be a field, Q the quiver ˛

$

o 1

ˇ

2

z

and I the ideal in KQ generated by ˛ 4 , 4 , and ˇ˛ ˇ. Prove that the bound quiver algebra KQ=I is isomorphic to the matrix K-algebra ³ ² 0 a 0 ˇˇ KŒx=.x 4 / 4 D a; b; c 2 KŒx=.x / : KŒx=.x 4 / KŒx=.x 4 / b c 9. Let K be a field, Q the quiver 1 O

˛

4 o

/ 2 ˇ

3

and I the ideal in KQ generated by ˛ˇ. (a) Describe the isomorphism classes of indecomposable representations in repK .Q; I /.

112


(b) Let M be the representation in repK .Q; I / of the form

KO 3 3

2

110 000

/ K2

00 41 05 01

K2 o

01

K.

1 0

Decompose M into a direct sum of indecomposable representations in repK .Q; I /. 10. Let Q be the quiver 6

/ / o o ı 1 2 3 4 5 and A D KQ the path algebra of Q over a field K. Consider the representation M in repK .Q/ of the form ˇ

˛

K 22 1 0 3 1 0

K

/ K2

3 10 40 15 00

41 15 01 2

2

/ K3 o

3 00 41 05 01

K o 2

0 1

K.

(a) Prove that EndQ .M / ! K, and hence M is an indecomposable representation in repK .Q/. (b) Find a composition series and simple composition factors of the right Amodule G.M / associated to M via the equivalence functor G W repK .Q/ ! mod A, described in the proof of Theorem 2.10. (c) Find the radical series of the A-module G.M /. 11. Let K be a field, Q the quiver 2 1 @_@@ @@@@ˇ1 ˛2 ~~~~~? @@@@ ~~~~ ˛1 @@@ ~~~~ ˇ2 ~ 0 O ˇ3

˛3

3,

I the ideal in KQ generated by ˛1 ˇ2 , ˛1 ˇ3 , ˛2 ˇ1 , ˛2 ˇ3 , ˛3 ˇ1 , ˛3 ˇ2 , ˇ1 ˛1 ˇ2 ˛2 , ˇ2 ˛2 ˇ3 ˛3 , and A D KQ=I the associated bound quiver algebra.

12. Exercises

113

(a) Show that in mod A the projective modules coincide with the injective modules. (b) Determine the Cartan matrix CA of A and show that CA is nonsingular. 12. Let K be a field, Q the quiver 1 _@@@ @@@@˛1 @@@@ ˇ1 @@@ ~? o ˛2 ~~~~ 3 ~~~~ ~~~~~ ˇ2 2

~? 5 ~~~~~ ~ ~~~~˛5 / ~~~ @ 4 _@@@@@@@@ˇ6 @@ ˛6 @@@ 6, ˇ5

I the ideal in KQ generated by ˇ1 ˛1 , ˇ2 ˛2 , ˇ5 ˛5 , ˇ6 ˛6 , ˛1 ˇ2 , ˛2 ˇ1 , ˛1 , ˛2 , ˇ1 , ˇ2 , ˛5 ˇ6 , ˛6 ˇ5 , ˛5 , ˛6 , ˇ5 , ˇ6 , and A D KQ=I the associated bound quiver algebra. (a) Prove that in mod A the projective modules coincide with the injective modules. (b) Determine the Cartan matrix CA of A. (c) Show that CA is singular. 13. Let K be a field, Q the quiver ˛

$

1

z

ˇ

and I the ideal of KQ generated by ˛ 2 , ˇ 2 , ˛ˇ ˇ˛. For each positive integer m, consider the representation in repK .Q; I / of the form M.m/ W

.m/

'˛

&

K m˚K m

x

.m/

'ˇ

;

where '˛.m/ and 'ˇ.m/ are the 2m2m-matrices '˛.m/ D

0 Im

0 ; 0

0 0 ; Jm .0/ 0

'ˇ.m/ D

where Im is the m m identity matrix and Jm .0/ is the m m Jordan block with 0 on the diagonal. Prove that M.m/, m 1, form a family of pairwise nonisomorphic indecomposable representations in repK .Q; I /.

114


14. Let G D Z2 Z2 be the Klein four group and KG the group algebra of G over a field K. Prove that KG is of finite representation type if and only if K is of characteristic different from 2. 15. Let Q D .Q0 ; Q1 ; s; t / be a finite quiver, I an admissible ideal of the path algebra KQ over a field K, and M D .Ma ; '˛ /a2Q0 ;˛2Q1 a representation in repK .Q; I /. (a) Determine the radical rad M of M . (b) Determine the top top.M / of M . (c) Determine the socle soc.M / of M . 16. Let A be a finite dimensional K-algebra over a field K and f

g

0 ! L ! M ! N ! 0 a short exact sequence in mod A. Prove that f is a section in mod A if and only if g is a retraction in mod A. 17. Let A be a finite dimensional K-algebra over a field K and 0

/L

0

/X

f

/M

u

/Y

p

g

v

/N

/0

w

q

/Z

/0

a commutative diagram in mod A with exact rows. Prove that, if any two of u; v; w are isomorphisms, then the third one is also an isomorphism. f

g

18. Let A be a finite dimensional K-algebra over a field K, and X !Z Y homomorphisms in mod A. Consider the fibered product (pull-back) X Z Y D f.x; y/ 2 X ˚ Y j f .x/ D g.y/g f0

g0

of X and Y over Z, via f and g, and the maps X X Z Y ! Y given by f 0 .x; y/ D x and g 0 .x; y/ D y for any .x; y/ 2 X Z Y . Prove the following: (a) X Z Y is a module in mod A and f 0 , g 0 are homomorphisms in mod A. u

v

(b) For every homomorphism X M ! Y in mod A with f u D gv there exists exactly one homomorphism h W M ! X Z Y in mod A such that u D f 0 h and v D g 0 h (the universal property of the fibered product).

115

12. Exercises

19. Let A be a finite dimensional K-algebra over a field K and let f

g

0!L !M !N !0 be a short exact sequence in mod A. (a) Let v W V ! N be a homomorphism in mod A and M N V the fibered product of M and V over N , via g and v. Show that there is a commutative diagram in mod A 0

/L

0

/L

v0

/ M N V

i

/V

g0

idL

v

/M

f

/0

/N

g

/0

with exact rows. (b) Let 0

j

/L

˛

/V

g

/N

f

/M

/0 v

ˇ

idL

/L

/W

/0 0 be a commutative diagram in mod A with exact rows. Show that there is an isomorphism h W W ! M N V in mod A such that i D hj , ˛ D v 0 h, ˇ D g 0 h. u

v

20. Let A be a finite dimensional K-algebra over a field K, and X Z ! Y homomorphisms in mod A. Consider the fibered sum (push-out) X ˚Z Y D X ˚ Y = f.u.z/; v.z// j z 2 Zg u0

v0

! X ˚Z Y Y given by of X and Y over Z, via u and v, and the maps X u0 .x/ D .x; 0/ and v 0 .y/ D .0; y/, where .x; y/ is the image of .x; y/ under the canonical epimorphism X ˚ Y ! X ˚Z Y . Prove the following statements. (a) X ˚Z Y is a module in mod A and u0 , v 0 are homomorphisms in mod A. f

g

(b) For every homomorphism X ! N Y in mod A with f u D gv there exists exactly one homomorphism h W X ˚Z Y ! N in mod A such that f D hu0 and g D hv 0 (the universal property of the fibered sum). 21. Let A be a finite dimensional K-algebra over a field K and let f

g

0!L !M !N !0 be a short exact sequence in mod A.

116


(a) Let u W L ! U be a homomorphism in mod A and U ˚L M the fibered sum of U and M over L, via u and f . Show that there is a commutative diagram in mod A f g /L /M /N /0 0 /U

0

f0

u u0

idN

/ U ˚L M

/N

p

/0

with exact rows. (b) Let /L

0

/U

f

u

/M

g

/W

/N

/0

idN

q

/N

/0 0 be a commutative diagram in mod A with exact rows. Show that there is an isomorphism h W U ˚V M ! W in mod A such that D hu0 , D hf 0 , p D qh.

22. Let A be a finite dimensional K-algebra over a field K and P a projective module in mod A. Prove that for any epimorphism h W M ! N in Mod A and f 2 HomA .P; N / there exists g 2 HomA .P; M / such that hg D f . 23. Let A be a finite dimensional K-algebra over a field K and E an injective module in mod A. Prove that for any monomorphism u W M ! N in Mod A and w 2 HomA .M; E/ there exists v 2 HomA .N; E/ such that w D vu. 24. Let A be a finite dimensional K-algebra over a field K and M a module in mod A. (a) Assume d2

dm

d1

d0

0 ! Pm ! Pm1 ! ! P2 ! P1 ! P0 ! M ! 0 is a finite projective resolution of M in mod A. Prove that m pdA M . (b) Assume pdA M D 1. Prove that every projective resolution of M in mod A is infinite. 25. Let A be a finite dimensional K-algebra over a field K and M a module in mod A. (a) Assume d0

d1

d2

dm

0 ! M ! I0 ! I1 ! I2 ! ! Im1 ! Im ! 0 is a finite injective resolution of M in mod A. Prove that m idA M .

12. Exercises

117

(b) Assume idA M D 1. Prove that every injective resolution of M in mod A is infinite. 26. Let K be a field, Q the quiver 2 _@ @@ ˇ ~ ˛ ~~ @@ ~ @@ ~ ~ ~ 4 1 _@ @@ ~~ @@ ~ ~~ @@ ~~ 3, I the ideal in KQ generated by ˇ˛ , and A D KQ=I (see Example 1.7 (c)). Prove the following statements: (a) There are exactly 11 isomorphism classes of indecomposable representations in repK .Q; I /, and hence A is of finite representation type. (b) pdA M 2 and idA M 2 for any module M in mod A. (c) There exist indecomposable modules L and N in mod A such that pdA N D 2 and idA L D 2. 27. Let K be a field, Q the quiver 2 _@ @@ ˇ ~ ˛ ~~ @@ ~ @@ ~ ~ ~ 4 1 _@ @@ ~~ @@ ~ ~~ @@ ~~ 3 and A D KQ. Prove the following statements: (a) A is of infinite representation type. (b) pdA M 1 and idA M 1 for any module M in mod A. 28. Let K be a field, Q be a finite acyclic quiver, I an admissible ideal of KQ, and A D KQ=I the associated bound quiver algebra. Denote by d the length of the longest path in Q. Prove that pdA M d and idA M d for every module M in mod A.

118


29. Let A be a finite dimensional K-algebra over a field K. Prove that A is an indecomposable K-algebra if and only if for every indecomposable projective modules P and Q in mod A there is a sequence of indecomposable projective modules P D P1 ; P2 ; : : : ; Pm1 ; Pm D Q, m 2, in mod A such that HomA .Pi ; PiC1 / ¤ 0 or HomA .PiC1 ; Pi / ¤ 0 for any i 2 f1; : : : ; m 1g. 30. Let K be a field, Q the quiver o 1

˛1

o 2

˛2

o 3

˛3

o 4

˛4

o 5

˛5

o 6

˛6

; 7

I the ideal in KQ generated by the paths ˛4 ˛3 ˛2 ˛1 and ˛5 ˛4 ˛3 and A D KQ=I . (a) Describe the isomorphism classes of indecomposable modules in mod A (respectively, indecomposable representations in repK .Q; I /). (b) Describe the pdA M and idA M for any indecomposable module M in mod A. 31. Let K be a field, Q the quiver 1 ?~ @@ @@˛1 ˛3 ~~ @@ ~~ @ ~ ~ 2, 3 o ˛2

I the ideal in KQ generated by ˛1 ˛2 ˛3 ˛1 ˛2 , ˛2 ˛3 ˛1 ˛2 ˛3 and ˛3 ˛1 ˛2 ˛3 ˛1 , and A D KQ=I . (a) Describe the isomorphism classes of indecomposable modules in mod A (respectively, indecomposable representations in repK .Q; I /). (b) Prove that the projective modules and the injective modules in mod A coincide. (c) Prove that pdA M D 1 and idA M D 1 for any indecomposable nonprojective module M in mod A. 32. Let K be a field, Q the quiver 1 ?~ @@ @@˛1 ˛3 ~~ @@ ~~ @ ~ ~ 2, 3 o ˛2

I the ideal in KQ generated by ˛1 ˛2 ˛3 ˛1 and ˛3 ˛1 ˛2 , and A D KQ=I .

12. Exercises

119

(a) Describe the isomorphism classes of indecomposable modules in mod A (respectively, indecomposable representations in repK .Q; I /). (b) Prove that there exist indecomposable modules X and Y in mod A such that X is projective but noninjective and Y is injective but nonprojective. (c) Describe the pdA M and idA M for any indecomposable module M in mod A. 33. Let

² ³ ˇ R 0 a 0 D 2 M2 .C/ ˇ a 2 R; b; c 2 C ; C C c b ² ³ ˇ C 0 a 0 ˇ BD D 2 M2 .C/ a; c 2 C; b 2 R : C R c b

AD

(a) Show that A and B are nonisomorphic R-subalgebras of M2 .C/. (b) Show that B is isomorphic to the opposite algebra Aop of A. (c) Determine the Cartan matrices CA and CB . (d) Prove that neither A nor B is isomorphic to a bound quiver R-algebra RQ=I of a bound quiver .Q; I /. (e) Prove that A and B are hereditary R-algebras. 34. Let

² ³ ˇ R 0 a 0 ˇ ƒD D 2 M2 .H/ a; b 2 R; c 2 H H R c b

be the R-subalgebra of M2 .H/. Prove that ƒ is isomorphic to the path algebra RQ of the quiver Q of the form

35. Let

o

˛1

o

˛2

1 o

˛3

o

˛4

2.

² ³ ˇ R 0 a 0 ˇ AD D 2 M2 .H/ a 2 R; b; c 2 H ; H H c b ² ³ ˇ H 0 a 0 BD D 2 M2 .H/ ˇ a; c 2 H; b 2 R : H R c b

120


(a) Show that A and B are nonisomorphic R-subalgebras of M2 .H/. (b) Show that B is not isomorphic to the opposite algebra Aop of A. (c) Determine the Cartan matrices CA and CB . (d) Prove that neither A nor B is isomorphic to a bound quiver algebra RQ=I of a bound quiver .Q; I /. (e) Prove that A and B are hereditary R-algebras. 36. Let

² ³ ˇ R 0 a 0 ˇ AD D 2 M2 .H/ a 2 R; b 2 C; c 2 H ; H C c b ² ³ ˇ C 0 a 0 ˇ BD D 2 M2 .H/ a 2 C; b 2 R; c 2 H : H R c b

(a) Show that A and B are nonisomorphic R-subalgebras of M2 .C/. (b) Show that B is not isomorphic to the opposite algebra Aop of A. (c) Determine the Cartan matrices CA and CB . (d) Prove that neither A nor B is isomorphic to a bound quiver algebra RQ=I of a bound quiver .Q; I /. (e) Prove that A and B are hereditary R-algebras. 37. Let F , G, L be finite field extensions of a field K such that F L and G L. Let ² ³ ˇ a 0 F 0 ˇ AD D 2 M2 .L/ a 2 F; b 2 G; c 2 L : L G c b (a) Prove that A is a hereditary K-algebra. (b) Determine the K-dimensions and the lengths of the indecomposable projective modules in mod A. (c) Determine the K-dimensions and the lengths of the indecomposable injective modules in mod A. (d) Determine the Cartan matrix CA . 38. Let F be a finite dimensional division K-algebra over an algebraically closed field K. Prove that the canonical K-algebra homomorphism K ! F is an isomorphism.

12. Exercises

121

39. Prove that a finite dimensional commutative K-algebra over a field K is a product of finitely many finite dimensional commutative local K-algebras. 40. Let K be a field, n a positive integer, and A D K K the product of n copies of the field K. Prove that there are exactly n pairwise different K-algebra homomorphisms from A to K. 41. Let K be a field, f .x/ a polynomial in KŒx of degree 1, and .f .x// the principal ideal of KŒx generated by f .x/. Prove that the K-algebra KŒx=.f .x// is semisimple if and only if f .x/ has no multiple irreducible factors. 42. Let K be a field of characteristic ¤ 2, and a; b 2 K n f0g. Consider the 4-dimensional vector space V with basis 1, i, j , k. Moreover, define on V the K-bilinear multiplication with 1 being its identity and the multiplications of the remaining basis elements as follows: ij D j i D k;

j k D kj D bi;

i D a; 2

j D b; 2

ki D i k D aj;

k D ab: 2

Then V together with this multiplication is denoted by a;b and called a generalized K quaternion algebra over K. Prove that (a) a;b is a simple K-algebra whose center is K; K (b) a;b is a division K-algebra if and only if 1 D 2 D 3 D 0 is the unique K solution of the equation 21 D a22 C b23 in K. 43. Let K be a field of characteristic ¤ 2, and a; b 2 K nf0g. Consider the quotient algebra H D KhX; Y i=.X 2 a; Y 2 b; X Y C YX / of polynomial K -algebra KhX; Y i in two noncommuting variables X , Y by the generated by X 2 a, a;bideal 2 Y b, XY C YX . Show that H is isomorphic to K . 44. Let K be a field of characteristic ¤ 2, and a; b; c 2 K n f0g. Prove that there exist isomorphisms of K-algebras ! ! ! ac 2 ; b a; b a; bc 2 Š Š : K K K 45. Let a; b 2 R n f0g. Prove that (a) a;b is isomorphic as an R-algebra to one of the R-algebras R ! ! 1; 1 1; 1 and H D I R R

122


1;1

Š M2 .R/ Š

1;1

as R-algebras. R 1;2 is a division Q-algebra and L D Q ˚ QŒi and 46. Prove that F D Q M D Q ˚ QŒj are nonisomorphic maximal subfields of Q. (b)

R

47. Prove that there exist isomorphisms of Q-algebras ! ! 1; 2 1; 2 Š M2 .Q/ Š : Q Q 48. Let H D R ˚ Ri ˚ Rj ˚ Rk be the quaternion R-algebra, G D f1; 1; i; i; j; j; k; kg the associated quaternion group of order 8, and RG the group algebra of G over R. Prove that there exists an isomorphism of R-algebras RG Š H R R R R.

Chapter II

Morita theory

The aim of this chapter is to introduce the Morita theory of equivalences and dualities between some categories. The revolutionary paper [Mor] in 1958 by K. Morita was the first to successfully apply the notion of category to modules. In the middle of the 1960s, Morita’s work was exhibited by H. Bass in [Bas1], [Bas2], and had become popular. The equivalence theory is now one of the fundamental theories in many branches of mathematics, and the duality theory includes many known dualities, for example, as shown in [Mor], the Pontrjagin duality for locally compact abelian groups. We show in this chapter an essential part of the Morita theory related to module categories over finite dimensional algebras over a field. We refer to the original paper [Mor] of K. Morita for more details.

1 Categories and functors A category C is a triple C D .Ob C; HomC ; B/, where Ob C is called the class of objects of C, HomC is called the class of morphisms of C, and B is a partial binary operation on morphisms of C satisfying the following conditions: (1) for each pair of objects X; Y of C, a set HomC .X; Y /, called the set of morphisms from X to Y , is associated, and if .X; Y / ¤ .Z; U / then the intersection of the sets HomC .X; Y / and HomC .Z; U / is empty; (2) for each triple of objects X; Y; Z of C, the operation B W HomC .Y; Z/ HomC .X; Y / ! HomC .X; Z/ is defined, g B f is called the composition of f 2 HomC .X; Y / and g 2 HomC .Y; Z/, and the following two properties are satisfied: (a) h B .g B f / D .h B g/ B f for all f 2 HomC .X; Y /, g 2 HomC .Y; Z/, h 2 HomC .Z; U /, (b) for each object X of C, there is an element idX 2 HomC .X; X /, called the identity morphism of X , such that f B idX D f and idX Bg D g for all f 2 HomC .X; Y / and g 2 HomC .Z; X /. We will frequently abbreviate the composition g B f of morphisms f and g in C to gf . Moreover, we will call an element f 2 HomC .X; Y / a morphism from X to Y , and usually write it as f W X ! Y .

124

Chapter II. Morita theory

For a category C, we define the opposite category C op of C to be the category with the same objects as C, with HomCop .X; Y / D HomC .Y; X / for all objects X and Y of Cop , and the composition Bop W HomCop .Y; Z/ HomCop .X; Y / ! HomCop .X; Z/ of morphisms in Cop is defined by g Bop f D f B g for f 2 HomCop .X; Y / D HomC .Y; X/ and g 2 HomCop .Y; Z/ D HomC .Z; Y /. For a morphism f W X ! Y in C, we denote by f op W Y ! X the morphism in Cop associated to f . Observe also that .Cop /op D C. A category C0 is called a subcategory of a category C if the following conditions are satisfied: (1) the class Ob C0 of objects of C0 is a subclass of the class Ob C of objects of C; (2) for each pair of objects X; Y of C0 , HomC0 .X; Y / is a subset of HomC .X; Y /; (3) the composition of morphisms in C0 is the same as in C; 0 (4) for each object X of C0 , the identity morphism idX of X in C0 coincides with the identity morphism idX of X in C.

A subcategory C0 of a category C is said to be a full subcategory of C if HomC0 .X; Y / D HomC .X; Y / for all objects X; Y of C0 . For a field K, a category C is said to be a K-category if, for each pair of objects X; Y of C, the set HomC .X; Y / is equipped with a K-vector space structure such that the composition B of morphisms in C is K-bilinear. Let C and D be two categories. A covariant functor T W C ! D from C to D is defined by assigning to each object X of C an object T.X / of D, and to each morphism h W X ! Y in C a morphism T.h/ W T.X / ! T.Y / in D such that the following conditions are satisfied: (1) T.idX / D idT.X/ for each object X of C; (2) for each pair of morphisms f W X ! Y and g W Y ! Z in C, the equality T.g B f / D T.g/ B T.f / holds in D. A contravariant functor T W C ! D from C to D is defined by assigning to each object X of C an object T.X / of D, and to each morphism h W X ! Y in C a morphism T.h/ W T.Y / ! T.X / in D such that the following conditions are satisfied: (1) T.idX / D idT.X/ for each object X of C; (2) for each pair of morphisms f W X ! Y and g W Y ! Z in C, the equality T.g B f / D T.f / B T.g/ holds in D. Throughout this chapter, K will denote a field and by a K-category we mean a full subcategory of Mod A or Mod Aop over a K-algebra A.

2. Bimodules

125

2 Bimodules Let A, B and C be K-algebras. An .A; B/-bimodule M , denoted by A MB when we stress the sides of the operation of rings, is a K-vector space which is a left A-module and a right B-module and satisfies the associativity condition a.xb/ D .ax/b for x 2 M; a 2 A; b 2 B. In this case, a.xb/ is also denoted by axb. In particular, an .A; A/-bimodule is called an A-bimodule. Every right A-module M is a .K; A/-bimodule, where the structure of left K-module on M is given by m D m D m.1A / for 2 K and m 2 M . Similarly, every left A-module N is an .A; K/-bimodule, where the structure of right K-module on N is given by n D n D .1A /n for 2 K and n 2 N . Let M be a right A-module. For an endomorphism u of MA , we write ux D u.x/; .u/x D u.x/; .u/x D u.x/ for x 2 M and 2 K. Then, from the definition of homomorphisms of A-modules it holds that u.x C y/ D ux C uy; .u C v/x D ux C vx; .vu/x D v.ux/; idM x D x; .u/x D .u/.x/ D u.x/ D u.x/; for all u; v 2 EndA .M /; x 2 M , 2 K, where idM denotes the identity endomorphism of M . This shows that the action of u 2 EndA .M / on the left of the right A-module M makes M into a left EndA .M /-module. Moreover, the associativity condition u.xa/ D .ux/a holds, because u.xa/ D u.x/a for all x 2 M and a 2 A. Thus M may be regarded as an .EndA .M /; A/-bimodule. For a left A-module N , an endomorphism u 2 EndAop .N / satisfies, by definition, u.ax/ D au.x/ for x 2 M and a 2 A. We define u and u for 2 K by .u/.x/ D u.x/ and .u/.x/ D u.x/ for x 2 N , respectively. In order to visualize the associativity, we write the endomorphism u on the right of the left A-module N , and then the fact that u is an endomorphism of A N is expressed by .ax/u D a.xu/. Note that x.vu/ D .vu/.x/ D v.u.x// D .xu/v for all u; v 2 EndA .N /, and hence, for the opposite algebra EndA .N /op D .EndA .N /; / (see Example I.1.1(g)), the associativity condition x.u v/ D .xu/v holds. Thus, as in the case of the right A-module M , it is easy to see that N becomes an .A; EndA .N /op /-bimodule. There is an important relation between a direct sum decomposition of a module and a set of pairwise orthogonal idempotents of the endomorphism algebra of the module. The following lemma is a generalization of Corollaries I.5.9 and I.5.10 on an algebra to the endomorphism algebra of a module.

126


Lemma 2.1. Let A be a K-algebra, M be a right A-module and B D EndA .M /. The following statements hold for the (B; A)-bimodule M . (i) For a set of pairwise orthogonal idempotents e1 ; : : : ; en of B with 1B D e1 C C en ; M is the direct sum M D e1 M ˚ ˚ en M of A-submodules e1 M; : : : ; en M of M . (ii) For a decomposition of M into a direct sum of A-submodules M D M1 ˚ ˚ Mn ; there is a set of pairwise orthogonal idempotents e1 ; : : : ; en of B such that 1B D e1 C C en and ei M D Mi for i 2 f1; : : : ; ng. Moreover, in this case, it holds that Mi Š Mj as right A-modules if and only if Bei Š Bej as left B-modules. Proof. (i) Assume that 1B D e1 C C en is a sum of pairwise orthogonal idempotents of B. Then M D e1 M C C en M , because, for m 2 M , we have m D 1B m D .e1 C C en /m D e1 m C C en m 2 e1 M C C en M . To show that the sum is direct, take x1 2 e1 M; : : : ; xn 2 en M such that x1 C C xn D 0. Then 0 D ei .x1 C C xn / D ei e1 x1 C C ei en xn D xi for all i 2 f1; : : : ; ng, because xj D ej xj for each j 2 f1; : : : ; ng and Pnthe idempotents e1 ; : : : ; en are pairwise orthogonal. It therefore follows that iD1 ei M is indeed a direct sum Ln iD1 ei M . (ii) Assume that M is a direct sum of A-submodules M1 ; : : : ; Mn , and let ui W Mi ! M and pi W M ! Mi be the canonical injection and canonical projection, respectively, for i 2 f1; : : : ; ng. Letting ei D ui pi , it is easy to see that these mappings satisfy the equalities pi ui D idMi ; 1B D u1 p1 C C un pn ; ei2 D .ui pi /.ui pi / D ui .pi ui /pi D ui idMi pi D ei ; ei ej D .ui pi /.uj pj / D ui .pi uj /pj D 0 for i ¤ j; and ei M D ui pi .M / D ui .Mi / D Mi , for all i; j 2 f1; : : : ; ng. Assume now that there is an isomorphism ˇ W Bei ! Bej of left B-modules, for some i; j 2 f1; : : : ; ng, and ˇ 1 W Bej ! Bei is the inverse of ˇ. Let b D ˇ.ei / and b 0 D ˇ 1 .ej /. Then it holds that b D ei bej ; b 0 D ej b 0 ei ; bb 0 D ei and b 0 b D ej , because bb 0 D bˇ 1 .ej / D ˇ 1 .bej / D ˇ 1 .b/ D ˇ 1 .ˇ.ei // D ei and b 0 b D b 0 ˇ.ei / D ˇ.b 0 ei / D ˇ.b 0 / D ˇˇ 1 .ej / D ej . Now consider the Klinear maps ˛ W ei M ! ej M and ˛ 0 W ej M ! ei M being the left multiplications bL0

2. Bimodules

127

and bL by b 0 and b, respectively, that is, bL0 .ei m/ D b 0 ei m D b 0 m and bL .ej n/ D bej n D bn for m; n 2 M . Observe that in fact ˛ and ˛ 0 are homomorphisms of right A-modules. Moreover, ˛ 0 ˛ D idei M and ˛˛ 0 D idej M . Indeed, since bb 0 D ei and b 0 b D ej , we have ˛ 0 ˛ D bL bL0 D .bb 0 /L D ei L which is the identity on ei M , and similarly ˛˛ 0 D bL0 bL D .b 0 b/L D ej L is the identity on ej M . Thus ˛ is an isomorphism of right A-modules with the inverse ˛ 0 . Conversely, assume that there is an isomorphism ˛ W ei M ! ej M of right Amodules, for some i; j 2 f1; : : : ; ng, and ˛ 1 W ej M ! ei M is the inverse of ˛. Let b 0 D uj ˛pi and b D ui ˛ 1 pj . Then b and b 0 belong to B and b D ei bej ; b 0 D ej b 0 ei ; bb 0 D ei ; b 0 b D ej : Indeed, we have the equalities b 0 D uj idMj ˛ idMi pi D uj .pj uj /˛.pi ui /pi D .uj pj /.uj ˛pi /.ui pi / D ej b 0 ei ; and bb 0 D .ui ˛ 1 pj /.uj ˛pi / D ui ˛ 1 .pj uj /˛pi D ui ˛ 1 ˛pi D ui pi D ei ; and similarly b D ei bej and b 0 b D ej . Consider the K-linear maps ˇ W Bei ! 0 by b and b 0 , Bej and ˇ 0 W Bej ! Bei being the right multiplications bR and bR 0 0 respectively, that is, bR .xei / D xei b D xb and bR .yej / D yej b D yb 0 for x; y 2 B. Then ˇ and ˇ 0 are homomorphisms of left B-modules and, as above, we have ˇ 0 ˇ D idBei and ˇˇ 0 D idBej . This shows that ˇ is an isomorphism of left B-modules. Lemma 2.1 is particularly important for a direct sum decomposition of a module into indecomposable submodules, in which case the statement is as follows. Corollary 2.2. Let A be a K-algebra and M be a right A-module. Then M is a direct sum M D M1 ˚ ˚ Mn of indecomposable A-submodules if and only if there is a set of pairwise orthogonal primitive idempotents e1 ; : : : ; en in EndA .M / with 1EndA .M / D e1 C C en . In this case, we may take the decomposition and the set of idempotents such that Mi D ei M for all i 2 f1; : : : ; ng. For two bimodules A MB and A NC , the K-vector space HomAop .M; N / of all homomorphisms from A M to A N has a .B; C /-bimodule structure defined by .bf c/.x/ D f .xb/c for all f 2 HomA .M; N /; b 2 B; c 2 C and x 2 M . Similarly, two bimodules 0 0 0 0 B MA ; C NA induce a (C; B)-bimodule structure on HomA .M ; N / defined by .cgb/.x/ D cg.bx/

128


for all g 2 HomA .M 0 ; N 0 /; b 2 B; c 2 C , and x 2 M 0 . For an idempotent e 2 A and a .B; A/-bimodule M , eA is an .eAe; A/-bimodule and HomA .eA; M / is a .B; eAe/-bimodule. By Lemma I.8.7 there is a K-linear isomorphism e M W HomA .eA; M / ! M e; e given by M .'/ D '.e/ for ' 2 HomA .eA; M /, which is in fact an isomorphism e e .'/ of .B; eAe/-bimodules. Indeed, we have M .b'/ D .b'/.e/ D b'.e/ D b M e e and M .'a/ D .'a/.e/ D '.a/ D '.e/a D M .'/a, for all b 2 B, a D eae 2 eAe and ' 2 HomA .eA; M /. Therefore we have proved the following bimodule version of Lemma I.8.7.

Lemma 2.3. Let A; B be K-algebras and M be a .B; A/-bimodule. Then, for an idempotent e 2 A, the K-linear map e M W HomA .eA; M / ! M e; e .'/ D '.e/ for ' 2 HomA .eA; M /, is an isomorphism of .B; eAe/given by M bimodules. 1 W HomA .A; M / ! M , for 1 D 1A , is an isomorphism In particular, M D M of .B; A/-bimodules.

The next lemma says that, for a finite set of right A-modules X1 ; : : : ; Xn , there is a bijective correspondence between the families Ln fi 2 Hom A .Xi ; M /; i 2 f1; : : : ; ng, and the homomorphisms f 2 HomA X ; M , under the relai iD1 L tion f ui D fi for all i 2 f1; : : : ; ng, where ui W Xi ! jnD1 Xj is the canonical injection. Lemma 2.4. Let A and B be K-algebras. Let X1 ; : : : ; Xn and Y1 ; : : : ; Yn be right B-modules, and let M be an (A; B)-bimodule. Then there exist canonical isomorphisms n n M M HomB Xi ; M Š HomB .Xi ; M / iD1

iD1

of left A-modules, and n n M M Yi Š HomB .M; Yi / HomB M; iD1

iD1

of right A-modules. Proof. We shall the first isomorphism. Lexhibit n Let X D X iD1 i and let ui W Xi ! X and pi W X ! Xi , i 2 f1; : : : ; ng, be the canonical injections and projections, respectively. For a homomorphism

2. Bimodules

129

f 2 HomB .X; M / let .f / D .f u1 ; : : : ; f un / 2

n M

HomB .Xi ; M /;

iD1

and, for fi 2 HomB .Xi ; M /; i 2 f1; : : : ; ng, let %.f1 ; : : : ; fn / D f1 p1 C C fn pn 2 HomB .X; M /: Ln Then iD1 HomB .Xi ; M / and Litn is easy to see that both W HomB .X; M / ! %W iD1 HomB .Xi ; M / ! HomB .X; M / are homomorphisms of left A-modules, and % and % are the identity homomorphisms. This implies that is an isomorphism of left A-modules. The second isomorphism is defined in a similar way. Now we start to study some functors induced from a bimodule. Let A MB be an .A; B/-bimodule. The functor HomB .M; / W Mod B ! Mod A assigns to a right B-module X the right A-module HomB .M; X / and carries a homomorphism u 2 HomB .X; Y / onto the homomorphism HomB .M; u/ W HomB .M; X / ! HomB .M; Y / in Mod A, defined by HomB .M; u/.f / D uf for all f 2 HomB .M; X /. For a right B-module X , the right A-module structure on HomB .M; X / is given by .f a/.m/ D f .am/ for a 2 A, m 2 M and f 2 HomB .M; X /. Observe that then .f a/.mb/ D f .a.mb// D f ..am/b/ D f .am/b D .f a/.m/b for b 2 B, and HomB .M; u/.f a/ D u.f a/ D .uf /a D HomB .M; u/.f /a, since .u.f a//.m/ D u..f a/.m// D u.f .am// D .uf /.am/ D ..uf /a/.m/, which shows that f a is a homomorphism of right B-modules and HomB .M; u/ is a homomorphism of right A-modules. Moreover, it holds that HomB .M; idX / D idHomB .M;X/ and HomB .M; vu/ D HomB .M; v/ HomB .M; u/; for two consecutive homomorphisms u W X ! Y and v W Y ! Z in Mod B. This shows that HomB .M; / is a covariant functor from Mod B to Mod A. Similarly, we have the covariant functor HomAop .M; / W Mod Aop ! Mod B op

130


which assigns to each left A-module X the left B-module HomAop .M; X / and carries each homomorphism u 2 HomAop .X; Y / in Mod Aop onto the homomorphism HomAop .M; u/ W HomAop .M; X / ! HomAop .M; Y / in Mod B op , defined by HomAop .M; u/.f / D uf for all f 2 HomAop .M; X /. For a left A-module X, the left B-module structure on HomAop .M; X / is given by .bf /.m/ D f .mb/ for b 2 B, m 2 M and f 2 HomAop .M; X /. Observe that then .bf /.am/ D f ..am/b/ D f .a.mb// D af .mb/ D a.bf /.m/ for a 2 A, and HomAop .M; u/.bf / D u.bf / D b.uf / D b HomAop .M; u/.f / for u 2 HomAop .X; Y /, since .u.bf //.m/ D u..bf /.m// D u.f .mb// D .uf /.mb/ D .b.uf //.m/ for m 2 M , which shows that bf is a homomorphism of left A-modules and HomAop .M; u/ is a homomorphism of left B-modules. Moreover, HomAop .M; idX / D idHomAop .M;X/ and HomAop .M; uv/ D HomAop .M; u/ HomAop .M; v/; for two consecutive homomorphisms u W X ! Y and v W Y ! Z in Mod Aop . Hence, HomAop .M; / is a covariant functor from Mod Aop to Mod B op . Further, we have the contravariant functor HomB .; M / W Mod B ! Mod Aop ; which assigns to a right B-module X the left A-module HomB .X; M / and carries a homomorphism u 2 HomB .X; Y / in Mod B onto the homomorphism HomB .u; M / W HomB .Y; M / ! HomB .X; M / in Mod Aop , defined by HomB .u; M /.g/ D gu for all g 2 HomB .Y; M /. For a right B-module X, the left A-module structure on HomB .X; M / is given by .af /.x/ D af .x/ for a 2 A, x 2 X and f 2 HomB .X; M /. Observe that then .af /.xb/ D af .xb/ D a.f .x/b/ D .af .x//b D .af /.x/b for b 2 B, and hence af is a homomorphism of right B-modules. Moreover, for a 2 A, u 2 HomB .X; Y / and g 2 HomB .Y; M /, we have HomB .u; M /.ag/ D .ag/u D a.gu/ D a HomB .u; M /.g/, since ..ag/u/.x/ D .ag/.u.x// D a.g.u.x/// D a.gu/.x/ for x 2 X , which shows that HomB .u; M / is a homomorphism of left A-modules. Obviously, it holds that HomB .idX ; M / D idHomB .X;M / and HomB .vu; M / D HomB .u; M / HomB .v; M /; for two consecutive homomorphisms u W X ! Y and v W Y ! Z in Mod B. Therefore, HomB .; M / is a contravariant functor from Mod B to Mod Aop .

2. Bimodules

131

Finally, we have the contravariant functor HomAop .; M / W Mod Aop ! Mod B which assigns to a left A-module X the right B-module HomAop .X; M / and carries a homomorphism u 2 HomAop .X; Y / in Mod Aop onto the homomorphism HomAop .u; M / W HomAop .Y; M / ! HomAop .X; M / in Mod B op , defined by HomAop .u; M /.g/ D gu for all g 2 HomAop .Y; M /. For a left A-module X , the right B-module structure on HomAop .X; M / is given by .f b/.x/ D f .x/b for b 2 B, x 2 X and f 2 HomAop .X; M /. Observe that then .f b/.ax/ D f .ax/b D .af .x//b D a.f .x/b/ D a.f b/.x/ for a 2 A, and so f b is a homomorphism of left A-modules. Further, for b 2 B, u 2 HomAop .X; Y / and g 2 HomAop .Y; M /, we have HomAop .u; M /.gb/ D .gb/u D .gu/b D HomAop .u; M /.g/b, since ..gb/u/.x/ D .gb/.u.x// D g.u.x//b D .gu/.x/b D ..gu/b/.x/ for x 2 X , which shows that HomAop .u; M / is a homomorphism of right B-modules. Hence HomAop .; M / is a contravariant functor from Mod Aop to Mod B. In case A MB is finite dimensional as a K-vector space, HomB .M; X / and HomB .X; M / are also finite dimensional for all finite dimensional right B-modules X. This ensures that HomB .M; / and HomB .; M / induce functors from mod B to mod A and from mod B to mod Aop , respectively. Similarly, HomAop .M; / and HomAop .; M / define functors from mod Aop to mod B op and from mod Aop to mod B, respectively. The covariant and contravariant functors HomB .M; /, HomAop .M; /, HomB .; M / and HomAop .; M /, defined above, are called hom functors. Let F W A ! B be a covariant functor between K-categories. The functor F is said to be K-linear if, for any modules X; Y 2 A, the induced mapping HomA .X; Y / ! HomB .F .X /; F .Y // is K-linear. For example, as seen above, the hom functors on the categories of modules over K-algebras defined by bimodules are K-linear. Given a short exact sequence in A f

g

0 ! X ! Y ! Z ! 0; any K-linear covariant functor F W A ! B transforms it into a sequence F .f /

F .g/

0 ! F .X / ! F .Y / ! F .Z/ ! 0 in B with F .g/F .f / D F .gf / D F .0Z / D 0. The functor F is said to be left exact if the above transformed sequence is exact at both F .X / and F .Y /, while

132


F is said to be right exact if the transformed sequence is exact at both F .Y / and F .Z/. An exact functor is by definition left and right exact. A contravariant functor G W A ! B is said to be left exact (respectively, right exact) if the covariant functor G 0 W Aop ! B is left exact (respectively, right exact), where G 0 is the functor such that G 0 .X/ D G.X / and G 0 .f op / D G.f / for all objects X and morphisms f op in Aop . Lemma 2.5. Let A and B be K-algebras. For an .A; B/-bimodule M , the hom functors HomB .; M / W Mod B ! Mod Aop ; HomB .M; / W Mod B ! Mod A; HomAop .M; / W Mod Aop ! Mod B op ; HomAop .; M / W Mod Aop ! Mod B are left exact. Proof. We shall show the left exactness of HomB .M; / only, and the proofs of the remaining statements are left to the reader (Exercise 8.7). f

g

Let 0 ! X ! Z ! 0 be an exact sequence of right B-modules. ! Y Since f is a monomorphism, f u D 0 implies u D 0 for any homomorphism u 2 HomB .M; X /. This shows that HomB .M; f / is a monomorphism in Mod A. Next, to show the exactness of the sequence HomB .M;f /

HomB .M;g/

HomB .M; X / ! HomB .M; Y / ! HomB .M; Z/ in Mod A, we have to show that the image of HomB .M; f / coincides with the kernel of HomB .M; g/. It is clear that Im HomB .M; f / Ker HomB .M; g/ because gf D 0 and hence HomB .M; g/ HomB .M; f / D HomB .M; gf / D 0. For each u 2 Ker HomB .M; g/, we have gu D 0, and hence there is a homomorphism v 2 HomB .M; X/ such that u D f v, because Im f D Ker g and f is a monomorphism M }} } } u }} ~ }f } /X /Y v

0

g

/Z

/ 0.

This implies that u D HomB .M; f /.v/, and so the inclusion Ker HomB .M; g/ Im HomB .M; f / holds. Hence we obtain Im HomB .M; f / D Ker HomB .M; g/. Therefore, the functor HomB .M; / W Mod B ! Mod A is left exact. Let A be a finite dimensional K-algebra, M a finite dimensional right A-module and f

g

0 ! X ! Y ! Z ! 0

3. Tensor products of modules

133

a short exact sequence in mod A. Then M is a .K; A/-bimodule. It follows from Lemma 2.5 that the induced sequence of K-vector spaces HomA .M;f /

HomA .M;g/

0 ! HomA .M; X / ! HomA .M; Y / ! HomA .M; Z/ ! 0 is exact in mod A if and only if it is exact at HomA .M; Z/, which is equivalent to saying that any homomorphism from M to Z in mod A can be lifted to Y along g. Similarly, the sequence HomA .g;M /

HomA .f;M /

0 ! HomA .Z; M / ! HomA .Y; M / ! HomA .X; M / ! 0 of K-vector spaces is exact if and only if it is exact at HomA .X; M /, or equivalently, any homomorphism from X to M in mod A can be extended to Y along f . Thus we have the following functorial characterization of projective and injective modules in mod A. Proposition 2.6. Let A be a finite dimensional K-algebra and M be a finite dimensional right A-module. The following statements hold. (i) M is projective in mod A if and only if the induced covariant functor HomA .M; / W mod A ! mod K is exact. (ii) M is injective in mod A if and only if the induced contravariant functor HomA .; M / W mod A ! mod K is exact.

3 Tensor products of modules Let A be a K-algebra, X a right A-module and Y a left A-module. We shall construct a K-vector space X ˝A Y . Consider the K-vector space F D KfX Y g with the basis consisting of all ordered pairs .x; y/ from the product X Y , and let F0 be the K-vector subspace of F generated by all elements of the form .x C x 0 ; y/ .x; y/ .x 0 ; y/; .x; y C y 0 / .x; y/ .x; y 0 /; .xa; y/ .x; ay/; .x; y/ .x; y/; for all x; x 0 2 X; y; y 0 2 Y; a 2 A and 2 K. We consider the factor space X ˝A Y D F=F0 of F by F0 , and denote by x ˝ y the element .x; y/ C F0 of X ˝A Y given by .x; y/ 2 X Y . Then, the following equalities hold:

134


.x C x 0 / ˝ y D x ˝ y C x 0 ˝ y; x ˝ .y C y 0 / D x ˝ y C x ˝ y 0 ; .xa/ ˝ y D x ˝ .ay/; .x ˝ y/ D .x/ ˝ y; Y , a 2 A and 2 K. The elements of X ˝A Y are for x; x 0 2 X , y; y 0 2 P expressions of the form niD1 xi ˝ yi , with xi 2 X; yi 2 Y . The K-vector space X ˝A Y is called the tensor product of X and Y over A, and is also denoted by X ˝ Y (without the ring A) if there is no ambiguity. For a K-vector space V , a map W X Y ! V is said to be A-bilinear provided, for any x 2 X and y 2 Y , the induced maps .x; / W Y ! V and

.; y/ W X ! V are K-linear, and .xa; y/ D .x; ay/ for any a 2 A. Clearly the canonical map W X Y ! X ˝A Y , defined by .x; y/ D x ˝ y, for x 2 X and y 2 Y , is A-bilinear. Observe that we may extend a K-bilinear map

W X Y ! V to a K-linear map N from F D KfX Y g to V naturally by

N

m X

m X i .xi ; yi / D i .xi ; yi /;

iD1

iD1

Pm

for all iD1 i .xi ; yi / 2 F , which is uniquely determined by . Then the AN 0 / D 0. bilinearity of is nothing else than .F Proposition 3.1. Let X be a right A-module, Y a left A-module, V a K-vector space and W X Y ! V be an A-bilinear map. Then there exists a unique K-linear map ' W X ˝A Y ! V such that '.x ˝ y/ D .x; y/ for all x 2 X; y 2 Y . Pm Pm Pm Proof. We put ' iD1 xi ˝yi D iD1 .xi ; yi /, for an element PmiD1 xi ˝yi 2 X Pm For this, assume Pnthat 0 iD10 xi ˝ yi D Pn˝A Y0, and0 verify that ' is well defined. j D1 xj ˝ yj in X ˝A Y . Since iD1 .xi ; yi / D j D1 .xj ; yj / , we have m X

.xi ; yi /

iD1

n X

.xj0 ; yj0 / 2 F0 ;

j D1

which is contained in the kernel of N as noticed above. It follows that m X

.xi ; yi / D

.xj0 ; yj0 /;

j D1

iD1

0 0 and hence ' iD1 xi ˝ yi D ' j D1 xj ˝ yj , which ensures that ' is well defined. Observe that two K-linear maps ' W X ˝A Y ! V and W X ˝A Y ! V coincide if and only if '.x ˝ y/ D .x ˝ y/ for all elements x ˝ y 2 X ˝A Y . This shows the uniqueness of '. Pm

n X

Pn


135

The property of the tensor product presented in Proposition 3.1 is often called the universal property of tensor product, expressed by the property of the canonical map W X Y ! X ˝A Y : for any A-bilinear map W X Y ! V there exists a unique K-linear map ' W X ˝A Y ! V with D ', that is, the following diagram is commutative: / X ˝A Y X YL LLL LLL ' LLL L% V.

Let A MB be an (A; B)-bimodule, X be a right A-module and Y be a left Bmodule. The tensor products X ˝A M and M ˝B Y then have a right B-module and a left A-module structure, respectively, with the operations of elements of A and B such that .x ˝ m/b D x ˝ .mb/;

a.m ˝ y/ D .am/ ˝ y

for all x 2 X, y 2 Y , m 2 M , a 2 A, b 2 B. Moreover, an (A; C )-bimodule A XC and a (C; B)-bimodule C YB make X ˝C Y an (A; B)-bimodule by the formula a.x ˝ y/b D .ax/ ˝ .yb/ for all x 2 X; y 2 Y; a 2 A; b 2 B. Lemma 3.2. For a right A-module XA , an (A; B)-bimodule A YB and a left Bmodule B Z, the canonical mapping 'X;Y;Z W .X ˝A Y / ˝B Z ! X ˝A .Y ˝B Z/; given by 'X;Y;Z ..x ˝ y/ ˝ z/ D x ˝ .y ˝ z/, for all x 2 X , y 2 Y , z 2 Z, is a K-linear isomorphism. Proof. For a fixed z 2 Z, let z W X Y ! X ˝A .Y ˝B Z/ be the map defined by

z .x; y/ D x ˝ .y ˝ z/ for all x 2 X; y 2 Y . The A-bilinearity of z is an immediate consequence of the property of tensor product. Hence, by Proposition 3.1, there is a K-linear map 'z W X ˝A Y ! X ˝A .Y ˝B Z/ with 'z .x ˝ y/ D x ˝ .y ˝ z/ for x 2 X; y 2 Y . Next let .w; z/ D 'z .w/ for all w 2 X ˝A Y and z 2 Z. As easily seen, this is a B-bilinear map from .X ˝A Y / Z to X ˝A .Y ˝B Z/. Applying Proposition 3.1 again, we conclude that there is a K-linear map 'X;Y;Z W .X ˝A Y / ˝B Z ! X ˝A .Y ˝B Z/

136


such that 'X;Y;Z ..x ˝ y/ ˝ z/ D x ˝ .y ˝ z/ for all x 2 X; y 2 Y; z 2 Z. Similarly, there is a K-linear map X;Y;Z W

X ˝A .Y ˝B Z/ ! .X ˝A Y / ˝B Z

such that X;Y;Z .x ˝ .y ˝ z// D .x ˝ y/ ˝ z for all x 2 X, y 2 Y , z 2 Z. Clearly, 'X;Y;Z and X;Y;Z are mutually inverse. We turn now our attention to the functors induced by tensor products. Let A MB be an (A; B)-bimodule. For a homomorphism u W X ! Y of right A-modules, we define a homomorphism of right B-modules u ˝A M W X ˝A M ! Y ˝A M by .u˝A M /.x ˝m/ D u.x/˝m for all x 2 X; m 2 M , which is also well defined by Proposition 3.1. Then, for the identity endomorphism idX on X , idX ˝A M is the identity endomorphism on X ˝A M , and it holds that .vu/ ˝ M D .v ˝ M /.u ˝ M / for any consecutive homomorphisms u W X ! Y and v W Y ! Z in Mod A. Thus we have the covariant functor ˝A M W Mod A ! Mod B which assigns to a right A-module X the right B-module X ˝A MB , and carries a homomorphism u in Mod A to the homomorphism u ˝A M in Mod B. Similarly, we have the functor M ˝B W Mod B op ! Mod Aop such that .M ˝B /.Y / D M ˝B Y and .M ˝B /.v/ D M ˝B v for a left B-module Y and a homomorphism v in Mod B op . The covariant functors ˝A M and M ˝B are called tensor functors. We note that, if A and B are finite dimensional K-algebras and A MB an (A; B)bimodule of finite dimension over K, then we have also the tensor functors ˝A M W mod A ! mod B

and

M ˝B W mod B op ! mod Aop :

Lemma 3.3. Let A and B be K-algebras. For an (A; B)-bimodule M , the induced tensor functors ˝A M W Mod A ! Mod B; are right exact.

M ˝B W Mod B op ! Mod Aop


137

Proof. We shall show that ˝A M is right exact. For a short exact sequence f

g

0!X !Y ! Z ! 0 in Mod A, we have to show that fN

gN

X ˝A M ! Y ˝A M ! Z ˝A M ! 0 is an exact sequence in Mod B, where fN D f ˝A M , gN D g ˝A M . The exactness at Z ˝A M is easy. Indeed, any element z ˝ m of Z ˝A M is the image of some y ˝ m under g, N where y 2 Y is an element with g.y/ D z. Note that such a choice of y is possible because g is surjective by assumption. Then the claim follows by the K-linearity of gN and general form of elements of Y ˝A M and Z ˝A M . The exactness at Y ˝A M means Im fN D Ker g. N Observe that Im fN Ker gN follows from the fact that gf D 0 and gN fN D .gf / ˝A M D 0. In order to show the converse inclusion Ker gN Im fN, we shall show that for the canonical surjective homomorphism ˛ W Y ˝A M ! .Y ˝A M /= Im fN; given by ˛.y ˝m/ D y ˝mCIm fN for y 2 Y and m 2 M , there is a K-linear map ˇ W Z ˝A M ! .Y ˝A M /= Im fN such that ˛ D ˇ g. N The inclusion Ker gN Im fN then follows, because ˛.Ker g/ N D ˇ g.Ker N g/ N D 0 and so Ker gN Ker ˛ D Im fN. To define the map ˇ we use the universal property for the tensor product. Consider the map W Z M ! .Y ˝A M /= Im fN; given by .z; m/ D ˛.y ˝ m/, for z 2 Z; m 2 M , and y 2 Y with g.y/ D z. Observe that is well defined. Indeed, for any y 0 2 Y with g.y 0 / D z, we have y y 0 2 Ker g and hence y y 0 2 Im f , because Im f D Ker g. Hence, we can write y y 0 D f .x/, for some x 2 X . Then we obtain ˛..y y 0 / ˝ m/ D ˛.f .x/ ˝ m/ D ˛.fN.x ˝ m// 2 ˛.Im fN/ D 0: Thus ˛..y y 0 /˝m/ D 0 and so ˛.y ˝m/ D ˛.y 0 ˝m/. This shows that ˛.y ˝m/ does not depend on a choice of y with g.y/ D z, and so is well defined. Next we show that is A-bilinear. Indeed, for any z; z 0 2 Z, take y; y 0 2 Y with g.y/ D z; g.y 0 / D z 0 , and observe that the equalities .z C z 0 ; m/ D ˛..y C y 0 / ˝ m/ D ˛.y ˝ m/ C ˛.y 0 ˝ m/ D .z; m/ C .z 0 ; m/; .za; m/ D ˛.ya ˝ m/ D ˛.y ˝ am/ D .z; am/; hold for all m 2 M; a 2 A. Furthermore, we have .z; m C m0 / D ˛.y ˝ .m C m0 // D ˛.y ˝ m/ C ˛.y ˝ m0 / D .z; m/ C .z; m0 / for all m; m0 2 M . Therefore, by Proposition 3.1, there exists a K-linear map ˇ W Z ˝A M ! .Y ˝A M /= Im fN such that ˛ D ˇ g. N Hence, Im fN D Ker g. N

138


Corollary 3.4. Let B be a K-algebra and A a K-subalgebra of B. Then B is an A-bimodule and we have the right exact functors ˝A B W Mod A ! Mod B and B ˝A W Mod Aop ! Mod B op : Moreover, if the K algebras A and B are finite dimensional, then we have also the right exact functors ˝A B W mod A ! mod B and B ˝A W mod B op ! mod Aop : Let A be a K-algebra, X a right A-module and Y a left A-module. Since A is an A-bimodule, we conclude that X ˝A A is a right A-module and A ˝A Y is a left A-module with .x ˝ a/b D x ˝ ab and b.a ˝ y/ D ba ˝ y for all a; b 2 A; x 2 X and y 2 Y . In fact, the following facts hold. Lemma 3.5. Let A be a K-algebra. The following statements hold. (i) For any right A-module X, there is a canonical isomorphism of right Amodules 'X W X ˝A A ! X . (ii) For any left A-module Y , there is a canonical isomorphism of left A-modules ' Y W A ˝A Y ! Y . Proof. (i) Let X be a right A-module. Then the K-linear map 'NX W X A ! X given by 'NX .x; a/ D xa, for x 2 A and a 2 A, is A-bilinear. Hence, by Proposition 3.1, there is a K-linear map 'X W X ˝A A ! X such that 'X .x ˝ a/ D xa for x 2 X and a 2 A. Observe also that 'X ..x ˝ a/b/ D 'X .x ˝ ab/ D x.ab/ D .xa/b D 'X .x ˝ a/b for x 2 X and a; b 2 A, and so 'X is a homomorphism of right A-modules. Consider now the K-linear homomorphism 'X0 W X ! X ˝A A given by 'X0 .x/ D x ˝ 1 for x 2 X . Then 'X0 is a homomorphism of right A-modules, because 'X0 .xa/ D xa ˝ 1 D x ˝ a D .x ˝ 1/a D 'X0 .x/a for x 2 X and a 2 A. Finally, observe that 'X0 'X D idX˝A A and 'X 'X0 D idX . Therefore 'X is an isomorphism of right A-modules. The proof of (ii) is similar. The tensor products of modules allow us also to consider important classes of algebras. Example 3.6. (a) Let K be a field and A1 ; A2 ; : : : ; An , n 2, be K-algebras. Then the tensor product A1 ˝K A2 ˝K ˝K An is a K-algebra with the multiplication given by .a1 ˝ a2 ˝ ˝ an /.b1 ˝ b2 ˝ ˝ bn / D .a1 b1 ˝ a2 b2 ˝ ˝ an bn /


139

for a1 ; b1 2 A1 , a2 ; b2 2 A2 , : : : , an ; bn 2 An , and the identity 1A1 ˝K A2 ˝K ˝K An D 1A1 ˝K 1A2 ˝K ˝K 1An : Moreover, the K-algebra A1 ˝K A2 ˝K ˝K An is finite dimensional if and only if the K-algebras A1 ; A2 ; : : : ; An are finite dimensional. Further, if this is the case, then dimK .A1 ˝K A2 ˝K ˝K An / D .dimK A1 /.dimK A2 / : : : .dimK An / (see Exercise 8.3). In the special case, A1 D A2 D D An D A, we set A˝n D A ˝K A ˝K ˝K A for the tensor product of n-copies of the K-algebra A over K, with n 2, and call the n-th tensor algebra of A. (b) Let A be a K-algebra over a field K and M be an A-bimodule. We may consider the family of A-bimodules TAn .M /, n 0, defined as TA0 .M / D A, TA1 .M / D M and TAn .M / D M ˝A M ˝A ˝A M the tensor product of n copies of the A-bimodule M for n 2. Then the K-vector space (in fact also an A-bimodule) 1 M TAn .M / TA .M / D nD0

is a K-algebra, called the tensor algebra of M over A. The multiplication in TA .M / is given by .x1 ˝ ˝ xm /.y1 ˝ ˝ yn / D x1 ˝ ˝ xm ˝ y1 ˝ ˝ yn for x1 ˝ ˝ xm 2 TAm .M /, y1 ˝ ˝ yn 2 TAn .M /, with m; n 1, and a.x1 ˝ ˝ xm / D .ax1 / ˝ ˝ xm ;

.x1 ˝ ˝ xm /a D x1 ˝ ˝ .xm a/;

for a 2 A D TA0 .M /, x1 ˝ ˝xm 2 TAm .M /, m 1, and the K-algebra structure of A. Clearly then the identity 1A of A is the identity 1TA .M / of TA .M /. Moreover, consider also the quotient algebra SA .M / D TA .M /=IA .M / of TA .M / by the two-sided ideal IA .M / generated by all elements x ˝ y y ˝ x with x; y 2 M . Then SA .M / is a commutative K-algebra, called the symmetric algebra of M over A. We note also that as K-vector space SA .M / D

1 M nD0

SAn .M /;

140


where SAn .M / D TAn .M / C IA .M / =IA .M /, for any n 0, SA0 .M / Š A and SA1 .M / Š M . In particular, for a K-vector space V over a field K, we have the tensor algebra TK .V / of V over K and the symmetric algebra SK .V / of V over K (see Exercises 8.4 and 8.5 for universal properties of TK .V / and SK .V /). Let A and B be K-algebras. We denote by Bimod.A; B/ the category of all .A; B/-bimodules, that is, the category whose objects are the .A; B/-bimodules, the morphisms are homomorphisms of .A; B/-bimodules, and the composition of morphisms is the usual composition of maps. By a homomorphism f W M ! N of .A; B/-bimodules we mean a K-linear homomorphism f W M ! N such that f .amb/ D af .m/b for all a 2 A, b 2 B and m 2 M . Moreover, we denote by bimod.A; B/ the full subcategory of Bimod.A; B/ whose objects are the finite dimensional (over K) .A; B/-bimodules. For A D B, we will write Bimod A and bimod A instead of Bimod.A; A/ and bimod.A; A/, respectively. Proposition 3.7. Let A and B be K-algebras. Then there exists a K-linear equivalence of categories F W Mod.Aop ˝K B/ ! Bimod.A; B/ which restricts to a K-linear equivalence of categories F W mod.Aop ˝K B/ ! bimod.A; B/: Proof. Let M be a right .Aop ˝K B/-module. Then F .M / is the .A; B/-bimodule whose underlying K-vector space is M and the .A; B/-bimodule structure is given by amb D m.a ˝ b/ for any a 2 A, b 2 B and m 2 M . Observe that, for a; c 2 A, b; d 2 B and m 2 M , we have the equalities a.cmd /b D .cmd /.a ˝ b/ D .m.c ˝ d //.a ˝ b/ D m..c ˝ d /.a ˝ b// D m.ac ˝ db/ D .ac/m.db/: Clearly, every .A; B/-bimodule is of the form F .M / for some right .Aop ˝K B/module M . Further, for any homomorphism f W M ! N of right .Aop ˝K B/modules, the K-linear homomorphism F .f / D f W F .M / ! F .N / is a homomorphism of .A; B/-bimodules, because f .amb/ D f .m.a ˝ b// D f .m/.a ˝ b/ D af .m/b; for a 2 A, b 2 B and m 2 M .

4 Adjunctions and natural isomorphisms Let A and B be K-categories and let F and G be covariant functors from A to B. A natural transformation of functors ' W F ! G assigns to each object X of A a

4. Adjunctions and natural isomorphisms

141

morphism 'X W F .X / ! G.X / in B such that, for every morphism f W X ! Y in A, the following diagram in B is commutative: F .X /

'X

G.f /

F .f /

F .Y /

/ G.X /

'Y

/ G.Y / .

In case the morphism 'X is an isomorphism for all X of A, ' is called a natural isomorphism of functors. The identity functor 1A of A satisfies 1A .X / D X and 1A .f / D f for all objects X and morphisms f of A. The composite GF of two covariant functors F W A ! B and G W B ! C is defined by .GF /.X / D G.F .X // and

.GF /.f / D G.F .f //

for all objects X and morphisms f of A. Now let F W A ! B and G W B ! A be covariant K-linear functors. A triple hF; G; 'i is said to be an adjunction from A to B if, for any objects X of A and Y of B, there is a K-linear isomorphism 'X;Y W HomB .F .X /; Y / ! HomA .X; G.Y // which is natural in both arguments X and Y , that is, for any morphisms u W X 0 ! X in A and v W Y ! Y 0 in B, the following diagram in Mod K is commutative: HomB .F .X /; Y / Hom.F .u/;v/

HomB .F .X 0 /; Y 0 /

'X;Y

/ HomA .X; G.Y // Hom.u;G.v//

'X 0 ;Y 0

/ HomA .X 0 ; G.Y 0 // .

An adjunction hF; G; 'i is often abbreviated to hF; Gi, and called an adjoint pair, and ' is called an adjunction between F and G, while F is called a left adjoint for G and G is called a right adjoint for F . We shall briefly recall some basic facts on the adjunctions. For more details we refer to the book [ML2]. Let hF; G; 'i be an adjunction from A to B. For any object X of A, the image of the identity idF .X/ 2 HomB .F .X /; F .X // under 'X;F .X/ is denoted by X , that is,

X D 'X;F .X/ .idF .X/ / W X ! GF .X /: The morphisms X , for all objects X of A, yield a natural transformation W 1A ! GF of functors called a unit, which determines the adjunction ' by the following relation 'X;Y .u/ D G.u/ X

142


for all morphisms u 2 HomB .F .X /; Y /. In fact, from the commutative square HomB .F .X /; F .X //

'X;F .X/

HomB .F .X/;u/

HomB .F .X /; Y /

/ HomA .X; GF .X // HomA .X;G.u//

'X;Y

/ HomA .X; G.Y // ,

we have 'X;Y .u/ D 'X;Y HomB .F .X /; u/ .idF .X/ / D HomA .X; G.u//'X;F .X/ .idF .X/ / D G.u/ X : Dually, for an object Y of B, let "Y be the inverse image of idG.Y / under 'G.Y /;Y , that is, 1 "Y D 'G.Y /;Y .idG.Y / / W F G.Y / ! Y: As before, the morphisms "Y , for all objects Y of B, yield a natural transformation " W F G ! 1B of functors, called a counit, and 1 'X;Y .v/ D "Y F .v/

for all v 2 HomA .X; G.Y //. Observe also that the following equalities hold: idG.Y / D 'G.Y /;Y ."Y / D 'G.Y /;Y HomB .F G.Y /; "Y /.idF G.Y / / D HomA .G.Y /; G."Y //'G.Y /;F G.Y / .idF G.Y / / D G."Y / G.Y / : Hence, the composed natural transformation of functors G

G."/

G."/ G W G ! .GF /G D G.F G/ ! G is the identity transformation of G. Lemma 4.1. Any two left adjoints for a functor are naturally isomorphic. Proof. Let hF; G; 'i and hF 0 ; G; ' 0 i be adjunctions from A to B and W 1A ! GF and 0 W 1A ! GF 0 their units, respectively. For an object X of A, consider the following K-linear maps: 0 'X;F 0 .X/

'X;F 0 .X /

HomB .F 0 .X /; F 0 .X// ! HomA .X; GF 0 .X // HomB .F .X /; F 0 .X // 'X;F .X/

0 'X;F .X/

HomB .F .X /; F .X// ! HomA .X; GF .X // HomB .F 0 .X /; F .X //:

For 0 ˛X D .'X;F 0 .X/ /1 . X / W F .X / ! F 0 .X /


143

and ˇX D .' 0X;F .X/ /1 . X / W F 0 .X / ! F .X /; we have 0 0 G.ˇX / X D 'X;F .X/ .ˇX / D X ;

0 G.˛X / X D 'X;F 0 .X/ .˛X / D X ;

and hence

X D G.ˇX / .G.˛X / X / D .G.ˇX /G.˛X // X D G.ˇX ˛X / X D 'X;F .X/ .ˇX ˛X /: Therefore, we have 'X;F .X/ idF .X/ D 'X;F .X/ .ˇX ˛X /: Since 'X;F .X/ is an isomorphism, it follows that idF .X/ D ˇX ˛X . Similarly, 0 'X;F 0 .X/ idF 0 .X/ D X D 'X;F 0 .X/ .˛X ˇX / and hence idF 0 .X/ D ˛X ˇX . Thus ˛X W F .X / ! F 0 .X / is an isomorphism in B and ˇX is its inverse, for any object X of A. Moreover, ˛X is natural in X , and hence ˛ W F ! F 0 is a natural isomorphism of functors. Indeed, since 0 W 1A ! GF 0 is a natural transformation of functors, for any morphism f W X ! Y in A, we have 0

0Y f D GF 0 .f / X . Then we have the equalities 0 'X;F 0 .Y / .F 0 .f /˛X / D GF 0 .f /'X;F 0 .Y / .˛X / D GF 0 .f / X 0 D Y f D 'Y;F 0 .Y / .˛Y /f D 'X;F 0 .Y / .˛Y F .f //;

and so F 0 .f /˛X D ˛Y F .f /, because 'X;F 0 .Y / is an isomorphism.

Similarly, using counits, one proves that the following lemma holds. Lemma 4.2. Any two right adjoints for a functor are naturally isomorphic. An important example of an adjunction is provided by ˝ and Hom. Let A and B be K-algebras, and A MB be an .A; B/-bimodule. Consider the functors ˝A M W Mod A ! Mod B;

HomB .M; / W Mod B ! Mod A:

For a right A-module X and a right B-module Y , define the K-linear homomorphism 'M .X; Y / W HomB .X ˝A M; Y / ! HomA .X; HomB .M; Y // by the formula

'M .X; Y /.f / .x/ .m/ D f .x ˝ m/

144


for f 2 HomB .X ˝A M; Y /; x 2 X; m 2 M . We claim that 'M .X; Y / is natural in X and Y . Indeed, for any homomorphisms u 2 HomA .X 0 ; X / and v 2 HomB .Y; Y 0 /, consider the diagram HomB .X ˝A M; Y /

'M .X;Y /

HomB .u˝A M;v/

HomB .X 0 ˝A M; Y 0 /

/ HomA .X; HomB .M; Y // HomA .u;HomB .M;v//

'M .X 0 ;Y 0 /

/ HomA .X 0 ; HomB .M; Y 0 // .

For all f 2 HomB .X ˝A M; Y /; x 0 2 X 0 ; m 2 M , we have the equalities .HomA .u; HomB .M; v//'M .X; Y //.f / .x 0 / .m/ D v 'M .X; Y /.f /.u.x 0 //.m/ D v.f .u.x 0 / ˝ m// D .vf .u ˝A M //.x 0 ˝ m/ D ..'M .X 0 ; Y 0 / HomB .u ˝A M; v//.f //.x 0 / .m/: Therefore, we obtain that HomA .u; HomB .M; v//'M .X; Y / D 'M .X 0 ; Y 0 / HomB .u ˝A M; v/: Hence we know that 'M .; / W HomB . ˝A M; / ! HomA .; HomB .M; // is a natural transformation of functors, where both HomB . ˝A M; / and HomA .; HomB .M; // are functors from Mod A Mod B to Mod K. Observe that, in the case that A MB is finite dimensional over K, both HomB . ˝A M; / and HomA .; HomB .M; // induce functors from mod A mod B to mod K. The following theorem, called the adjoint theorem, asserts that the functors ˝A M and HomB .M; / form an adjoint pair. Theorem 4.3. Let A and B be K-algebras and M be an .A; B/-bimodule. For any right A-module X and any right B-module Y , the K-linear map 'M .X; Y / W HomB .X ˝A M; Y / ! HomA .X; HomB .M; Y // is an isomorphism and natural in X and Y . Proof. In order to show that 'M .X; Y / is an isomorphism, consider the K-linear map 0 'M .X; Y / W HomA .X; HomB .M; Y // ! HomB .X ˝A M; Y / defined by

0 'M .X; Y /.g/ .x ˝ m/ D g.x/.m/


145

0 .X; Y / for g 2 HomA .X; HomB .M; Y //; x 2 X and m 2 M . This map 'M is well defined by the universal property of the tensor product (Proposition 3.1). 0 Let ' D 'M .X; Y / and ' 0 D 'M .X; Y /, for simplicity. We show that ' 0 ' and 0 ' ' are the identity maps on HomB .X ˝A M; Y / and HomA .X; HomB .M; Y //, respectively. By definition of ' and ' 0 , we have 0 .' '/.f / .x ˝ m/ D ' 0 '.f / .x ˝ m/ D '.f /.x/ .m/ D f .x ˝ m/

for all f 2 HomB .X ˝A M; Y /; x 2 X and m 2 M . Hence .' 0 '/.f / D f for all f 2 HomB .X ˝A M; Y /, which implies that ' 0 ' D idHomB .X ˝A M;Y / . Similarly, .'' 0 /.g//.x/ .m/ D '.' 0 .g//.x/ .m/ D ' 0 .g/.x ˝ m/ D .g.x//.m/ for all x 2 X; m 2 M and g 2 HomA .X; HomB .M; Y //, which implies that .'' 0 /.g/ D g for all g 2 HomA .X; HomB .M; Y //, and hence we obtain ' ' 0 D idHomA .X;HomB .M;Y // . This shows that ' D 'M .X; Y / is an isomorphism, and natural in X and Y , as we have proved above. Lemma 4.4. Let A and B be K-algebras, and let hF; Gi be an adjoint pair between Mod A and Mod B. Then F .A/ is an .A; B/-bimodule and there exist natural isomorphisms of functors F Š ˝A F .A/;

G Š HomB .F .A/; /:

Proof. Let ' be the adjunction defining the adjoint pair hF; Gi. Hence, for a right A-module X and right B-module Y , we have the K-linear isomorphism 'X;Y W HomB .F .X /; Y / ! HomA .X; G.Y //; which is natural in X and Y . Taking the right A-module AA as X , we obtain then the composite K-linear isomorphism 'A;Y

G.Y /

HomB .F .A/; Y / ! HomA .A; G.Y // ! G.Y /; where G.Y / W HomA .A; G.Y // ! G.Y / is the isomorphism of right A-modules defined by G.Y / .f / D f .1/ for all f 2 HomA .A; G.Y // (see Lemma 2.3). Moreover, G.Y / is natural in Y . Hence, to conclude that G Š HomB .F .A/; / as functors from Mod B to Mod A, it suffices to show that F .A/ has a left A-module structure and 'A;Y W HomB .F .A/; Y / ! HomA .A; G.Y // is a homomorphism of right A-modules, where the right A-module structures on HomB .F .A/; Y / and HomA .A; G.Y // are induced from the left A-module structure of F .A/ and the left A-module structure of A, respectively. Observe also that every element of HomA .AA ; AA / is of the form aL for some a 2 A, where aL W AA ! AA is the left multiplication by a, that is, aL .x/ D ax for all x 2 A.

146


Now, the algebra homomorphism F W HomA .A; A/ ! HomB .F .A/; F .A// makes F .A/ a left A-module by the action ax D F .aL /.x/ 0 for all a 2 A and x 2 F .A/. Note that .aa0 /x D F ..aa0 /L /.x/ D F .aL aL /.x/ D 0 0 0 0 .F .aL /F .aL //.x/ D F .aL /.F .aL /.x// D a.a x/ for all a; a 2 A and x 2 F .A/. Then F .A/ becomes an .A; B/-bimodule, because .ax/b D F .aL /.x/b D F .aL /.xb/ D a.xb/, for all a 2 A; b 2 B and x 2 F .A/. The following commutative diagram implies that 'A;Y is a homomorphism of right A-modules

HomB .F .A/; Y /

'A;Y

HomB .F .aL /;Y /

HomB .F .A/; Y /

/ HomA .A; G.Y // HomA .aL ;G.Y //

'A;Y

/ HomA .A; G.Y // .

Indeed, for f 2 HomB .F .A/; Y / and x 2 F .A/, we have .f a/.x/ D f .ax/ D f .F .aL /.x// D .HomB .F .aL /; Y /.f //.x/; and hence f a D HomB .F .aL /; Y /.f /. Therefore, from the commutativity of the above diagram, we conclude that 'A;Y .f a/ D 'A;Y HomB .F .aL /; Y / .f / D .HomA .aL ; G.Y //'A;Y /.f / D HomA .aL ; G.Y //.'A;Y .f //: On the other hand, .HomA .aL ; G.Y //'A;Y /.f / .x/ D .'A;Y .f //.aL .x// D .'A;Y .f //.ax/ D .'A;Y .f /a/.x/ for all x 2 A, which implies that .HomA .aL ; G.Y //'A;Y /.f / D 'A;Y .f /a. Consequently we have 'A;Y .f a/ D 'A;Y .f /a. Thus 'A;Y is a homomorphism of right A-modules as desired. Hence, the functors HomB .F .A/; / and G from Mod B to Mod A are naturally isomorphic. We know from Theorem 4.3 that the functor ˝A F .A/ W Mod A ! Mod B is left adjoint to the functor HomB .F .A/; / W Mod B ! Mod A, and so is left adjoint to the functor G. Since by assumption the functor F W Mod A ! Mod B is left adjoint to G, applying Lemma 4.1, we conclude that the functors F and ˝A F .A/ from Mod A to Mod B are naturally isomorphic.


147

Other important natural K-linear isomorphisms, invoking the hom and tensor functors, are given by projective modules. Let A and B be K-algebras and P be a left A-module. For an (A; B)-bimodule X and a right B-module Y , we consider the K-linear map P .X; Y / W

HomB .X; Y / ˝A P ! HomB .HomAop .P; X /; Y /

given by the formula

P .X; Y /.f

˝ p/ .u/ D f .u.p//

for all f 2 HomB .X; Y /; u 2 HomAop .P; X / and p 2 P . Here, HomB .X; Y / is a right A-module induced from the left A-module X and HomAop .P; X / is a right B-module induced from the right B-module X , and the map P .X; Y / is well defined by the universal property of the tensor product (Proposition 3.1). Similarly, for a (B; A)-bimodule X and a left B-module Y , we consider the K-linear map 0 P .Y; X / W

HomB op .Y; X / ˝A P ! HomB op .Y; X ˝A P /

given by the formula

0 P .Y; X /.g

˝ p/ .y/ D g.y/ ˝ p

for all g 2 HomB op .Y; X /; p 2 P and y 2 Y . Here, HomB op .Y; X / is a right A-module induced from the right A-module X and X ˝A P is a left B-module induced from the left B-module X , and the map P0 .Y; X / is well defined by the universal property of the tensor product. Theorem 4.5. Let A, B and C be K-algebras and P a finite dimensional .A; C /bimodule such that P is projective in mod Aop . Then the following statements hold. (i) For an (A; B)-bimodule X and a right B-module Y , the K-linear map P .X; Y / W

HomB .X; Y / ˝A P ! HomB .HomAop .P; X /; Y /

is an isomorphism of right C -modules and natural in X and Y . (ii) For a (B; A)-bimodule X and a left B-module Y , the K-linear map 0 P .Y; X / W

HomB op .Y; X / ˝A P ! HomB op .Y; X ˝A P /

is an isomorphism of right C -modules and natural in X and Y .

148


Proof. We shall give the proof only for (i), and leave to the reader to verify the statement (ii) (see Exercise 8.39). It is clear that P .X; Y / is a homomorphism of right C -modules, and it is straightforward to check the naturality of P .X; Y / in X and Y . The proof that P .X; Y / is an isomorphism will be divided into three steps. Before starting the proof, observe that the mapping P .X; Y / is defined for an arbitrary left A-module P (without projectivity), and in that case, for fixed modules X and Y , P .X; Y / is a natural transformation in P . First, consider the case when P D A. It is easy to check that the composition HomB .ˇ; Y / A .X; Y /˛ of the homomorphisms A .X;Y /

˛

HomB .X; Y / ˝A A ! HomB .HomAop .A; X /; Y / HomB .X; Y / ! HomB .ˇ;Y /

! HomB .X; Y / is the identity homomorphism idHomB .X;Y / , where ˛ and ˇ W X ! HomAop .A; X / are canonical K-linear isomorphisms such that ˛.f / D f ˝ 1A ;

ˇ.x/.a/ D ax;

for all f 2 HomB .X; Y /; x 2 X; a 2 A. Indeed, for f 2 HomB .X; Y / and x 2 X, we have ..HomB .ˇ; Y /

A .X; Y /˛/.f

//.x/ D .HomB .ˇ; Y /. A .X; Y /.f ˝ 1A ///.x/ D . A .X; Y /.f ˝ 1A //.ˇ.x// D f .ˇ.x/.1A // D f .1A x/ D f .x/:

Since ˛ and HomB .ˇ; Y / are isomorphisms, it follows that the homomorphism 1 1 is an isomorphism. A .X; Y / D HomB .ˇ; Y / ˛ Next, consider the case when P is a free A-module, that is, P is isomorphic to a direct sum of finitely many copies of A A. The fact that P .X; Y / is an isomorphism then follows by the naturality of P .X; Y / in P and the fact shown above that A .X; Y / is an isomorphism. Finally, let F be a finite dimensional free left A-module and u W P ! F and v W F ! P homomorphisms in mod Aop with vu D idP the identity on P (see Lemma I.8.1). Then we have the diagram of K-vector spaces P .X;Y /

HomB .X; ?xY / ˝A P ! HomB .Hom?Axop .P; X /; Y / ?? ?? u1y?v1 u2y?v2 HomB .X; Y / ˝A F ! HomB .HomAop .F; X /; Y / F .X;Y /

such that u2

P .X; Y /

D

F .X; Y /u1 ;

v2

F .X; Y /

D

P .X; Y /v1 ;


149

where v1 D HomB .X; Y / ˝A v; u1 D HomB .X; Y / ˝A u; u2 D HomB .HomAop .u; X /; Y /; v2 D HomB .HomAop .v; X /; Y /: Let N P .X; Y / D v1 N P .X; Y /

1 F .X; Y / u2 . P .X; Y /

D .v1 D .v1 .

Then, we obtain the equalities 1 F .X; Y / /.u2 1

F .X; Y /

/.

P .X; Y // F .X; Y /u1 /

D v1 . F .X; Y /1 F .X; Y //u1 D v1 u1 D HomB .X; Y / ˝ vu D HomB .X; Y / ˝ idP D idHomB .X;Y /˝A P : Similarly, we show that P .X; Y / N P .X; Y / D v2 u2 D idHomB .HomAop .P;X/;Y / . Therefore, P .X; Y / is an isomorphism. Corollary 4.6. Let A and B be K-algebras and P a finite dimensional (A; B)bimodule such that P is projective in mod Aop . (i) For a right A-module X , the K-linear map X ˝A P ! HomA .HomAop .P; A/; X /; defined by P .X /.x ˝ p/ .u/ D xu.p/ for x 2 X; p 2 P and u 2 HomAop .P; A/, is an isomorphism of right B-modules and natural in X .

P .X / W

(ii) The K-linear map 0 P

W HomAop .P; A/ ˝A P ! EndAop .P /; defined by P0 .u ˝ p/ .p 0 / D u.p 0 /p for u 2 HomAop .P; A/ and p; p 0 2 P , is an isomorphism of B-bimodules.

Proof. Observe that

P .X / is the composition of isomorphisms of right B-modules

ˇ ˝A P

P .A;X/

X ˝A P ! HomA .A; X / ˝A P ! HomA .HomAop .P; A/; X /; and

0 P

is the composition of isomorphisms of B-bimodules 0 P .P;A/

HomAop .P;/

HomAop .P; A/ ˝A P ! HomAop .P; A ˝A P / ! HomAop .P; P /; where ˇ W X ! HomA .A; X / is the canonical isomorphism of right A-modules (as in the proof of Theorem 4.5 (i)), W A ˝A P ! P is the canonical isomorphism of left A-modules from Lemma 3.5 (ii), and P .A; X /, P0 .P; A/ are the isomorphisms defined in Theorem 4.5. The observation shows that P .X / is an isomorphism of right B-modules and P0 is an isomorphism of B-bimodules.

150


Similarly, we conclude that the following fact holds. Corollary 4.7. Let A and B be K-algebras and P be a finite dimensional (B; A)bimodule such that P is projective in mod A. Then, for a left A-module X , the K-linear map 'P .X / W P ˝A X ! HomAop .HomA .P; A/; X / defined by 'P .X /.p ˝ x/ .u/ D u.p/x for p 2 P; x 2 X and u 2 HomA .P; A/, is an isomorphism of left B-modules and natural in X . Proposition 4.8. Let A be a K-algebra and P a finite dimensional projective right A-module. Then the following statements hold. (i) HomA .P; A/ is a projective left A-module. (ii) The K-linear map P W P ! HomAop .HomA .P; A/; A/; defined by P .p/ .u/ D u.p/ for all p 2 P; u 2 HomA .P; A/, is an isomorphism of right A-modules.

Proof. (i) It follows from Lemma I.8.1 that there exists a free module F D .AA /n , for some positive integer n, and homomorphisms u W P ! F and v W F ! P of right A-modules with vu D idP . Then the composition of the induced K-linear homomorphisms HomA .v;A/

HomA .u;A/

HomA .P; A/ ! HomA .F; A/ ! HomA .P; A/ is the identity on HomA .P; A/ and hence, by Lemma I.4.2, the left A-module HomA .P; A/ is isomorphic to a direct summand of the left A-module HomA .F; A/. Hence, by Lemma I.8.1, HomA .P; A/ is a projective left A-module, because the left A-module HomA .F; A/ is isomorphic to the free left A-module .A A/n . (ii) Since P is a finite dimensional .K; A/-bimodule, by Corollary 4.7 we have that 'P .A/ W P ˝A A ! HomAop .HomA .P; A/; A/ is an isomorphism of left K-modules, and hence of K-vector spaces. Let ˛ W P ! P ˝A A be the canonical isomorphism of right A-modules, hence of K-vector spaces, defined by ˛.p/ D p ˝ 1A for p 2 P . Observe now that P is the composition 'P .A/˛, because .'P .A/˛/.p/ .u/ D 'P .A/.p ˝ 1A / .u/ D u.p/1A D P .p/.u/, for p 2 P and u 2 HomA .P; A/. This shows that P is an isomorphism of K-vector spaces. Further, we have, for p 2 P , u 2 HomA .P; A/ and a 2 A, the equalities P .pa/.u/

and so

P .pa/

D

D u.pa/ D .au/.p/ D P .p/a.

Hence,

P

P .p/.au/

D.

P .p/a/ .u/;

is an isomorphism of right A-modules.

5. Progenerators

151

Corollary 4.9. Let A and B be K-algebras and P a finite dimensional (B; A)bimodule such that P is projective in mod A. Then, for any right A-module X , the K-linear map 0 P .X / W

X ˝A HomA .P; A/ ! HomA .P; X /; defined by P0 .X /.x ˝u/ .p/ D xu.p/ for x 2 X; u 2 HomA .P; A/ and p 2 P , is an isomorphism of right B-modules and natural in X .

Proof. Let P D HomA .P; A/, which is an (A; B)-bimodule. By Proposition 4.8, the left A-module P is projective and P W P ! HomAop .P ; A/ is an isomorphism of right A-modules. On the other hand, by Corollary 4.6, P .X / W X ˝A P ! HomA .HomAop .P ; A/; X / is an isomorphism of right B-modules. Thus we have the composed isomorphism of right B-modules P .X/

‰P .X / W X ˝A HomA .P; A/ ! HomA .HomAop .P ; A/; X / HomA .

P ;X/

! HomA .P; X /; such that

‰P .X /.x ˝ u/ .p/ D xu.p/

for all x 2 X; u 2 HomA .P; A/; p 2 P . Hence P0 .X / D ‰P .X / is an isomorphism of right B-modules. The naturality of P0 .X / in X follows by the naturality of ‰P .X/ in X .

5 Progenerators We shall now introduce generators of a K-category and characterize them, aiming to develop the Morita theory. Let C be a K-category. An object M of C is called a generator of C if, for any two different morphisms f; g 2 HomC .X; Y /, there exists a morphism h 2 HomC .M; X/ such that f h ¤ gh, or equivalently, for any nonzero morphism f 2 HomC .X; Y /, there is a morphism h 2 HomC .M; X / with f h ¤ 0. We note that this is equivalent to saying that the functor HomC .M; / W C ! Mod K is faithful (see Section 6). Obviously, an object M of a full K-subcategory D of C is a generator of D if M is a generator of C. The following lemma is an immediate consequence of the definition of generator. Lemma 5.1. Let C be a K-category and M an object of C. The following statements hold. (i) If M is a generator of C and X an object of C , then the direct sum M ˚ X is a generator of C.

152


(ii) M is a generator of C if M m is a generator of C for some positive integer m. Lemma 5.2. A K-algebra A is a generator of Mod A. In particular, if A is finite dimensional, then A is also a generator of mod A. Proof. For modules X , Y in Mod A and 0 ¤ f 2 HomA .X; Y /, take an element ˛ 2 X with f .˛/ ¤ 0. Then, for the left multiplication ˛L W A ! X by ˛, ˛L .a/ D ˛a, for a 2 A, we have f ˛L ¤ 0, because .f ˛L /.1A / D f .˛L .1A // D f .˛/ ¤ 0. Hence A is a generator of Mod A. In case A is finite dimensional, A belongs to the full subcategory mod A of Mod A and hence obviously A is a generator of mod A. Proposition 5.3. Let A be a finite dimensional K-algebra and M be a finite dimensional right A-module. The following conditions are equivalent. (i) M is a generator of Mod A. (ii) M is a generator of mod A. (iii) For any module X in mod A, there is an epimorphism from M n to X , for some positive integer n. (iv) There is an epimorphism from M n to the right A-module AA , for some positive integer n, or equivalently, M n Š AA ˚ N in mod A, for some positive integer n and a right A-module N . (v) M has a projective direct summand P D P1 ˚ ˚ Pm , where P1 ; : : : ; Pm is a complete set of pairwise nonisomorphic indecomposable projective modules in mod A. Proof. (i) ) (ii) This is trivial, because M is in mod A and mod A is a full subcategory of Mod A. (ii) ) (iii) Let X be a module in mod A and N be the A-submodule of X generated by the images of all homomorphisms f 2 HomA .M; X /, that is, X N D f .M /: f 2HomA .M;X/

Since N is a finite dimensional, hence finitely generated A-module, there are hoPn momorphisms P f1 ; : : : ; fn 2 HomA .M; X / such that N D iD1 fi .M /. Indeed, let N D jr D1 xj A for some x1 ; : : : ; xr 2 X . Then each xj is contained in a Prj submodule P f .M /, where rj is a positive integer and fjk 2 HomA .M; X /. kD1 jk Hence N D j;k fjk .M /. It is then enough to take as f1 ; : : : ; fn the finite set ffjk gj;k .

5. Progenerators

153

Ln be copies of MA and ' W a hoNow let Mi , i 2 f1; : : : ; ng, P iD1 Mi ! X Lbe n n / for m D .m ; : : : ; m / 2 momorphism such that '.m/ D f .m i i 1 n iD1 Mi . iD1 L Clearly, '. niD1 Mi / D N . We claim that ' is surjective. For this, to the contrary, suppose that ' is not surjective, and let g W X ! X=N be the canonical homomorphism with g.x/ D x C N for x 2 X. Notice that g.N / D 0. Since g is nonzero, by the definition of a generator, there is a homomorphism f W M ! X in mod A such that gf ¤ 0. It follows that 0 ¤ gf .M / D g.f .M // g.N /, which implies that g.N / ¤ 0, a contradiction. (iii) ) (iv) The first part of (iv) follows trivially from (iii), and the second part is a consequence of Lemma I.8.1. (iv) ) (v) We may take direct summands ej A of A, j 2 f1; : : : ; mg, which form a complete set of pairwise nonisomorphic indecomposable projective right Amodules. Assume that there is an isomorphism M n Š A ˚ N of right A-modules, for some positive integer n and a right A-module N . Then, by the Krull–Schmidt theorem (Theorem I.4.6), for each j 2 f1; : : : ; mg, M has a direct summand Pr.j / isomorphic to ej A. Clearly, the set Pr.1/ ; : : : ; Pr.m/ is a complete set of pairwise nonisomorphic indecomposable projective modules. Therefore, it follows from P Proposition I.4.7 that the sum P D jmD1 Pr.j / is a direct sum and a direct summand of M . L (v) ) (i) Let P D jmD1 Pj be a direct summand of M , where P1 ; : : : ; Pm is a complete set of pairwise nonisomorphic indecomposable projective modules in mod A. Then, by Proposition I.8.2, AA is isomorphic to a direct summand of P r for some positive integer r. Therefore, by Lemmas 5.1 and 5.2, P r and so P are generators of Mod A. Since P is a direct summand of M , applying Lemma 5.1 again, we conclude that M is a generator of Mod A. Lemma 5.4. Let A be a finite dimensional K-algebra. Let P1 ; : : : ; Pm be a complete set of pairwise nonisomorphic indecomposable projective modules in mod A, and P D P1 ˚ ˚ Pm . Then P is a projective generator of mod A, and P is isomorphic to a module eA for some idempotent e of A. Proof. Each Pi is isomorphic to a direct summand ei A of AA , where ei is a primitive idempotent of A. Then the set e1 A; : : : ; em A is a complete set of pairwise nonisomorphic projective modules. Hence, by Proposition I.4.7, the submodule Pm e A of A A is a direct sum of e1 A; : : : ; em A and a direct summand of A. iD1 i Moreover, for e D e1 C C em , we have P Š e1 A ˚ ˚ em A D e1 A C C em A D eA: However, AA is clearly isomorphic to a direct summand of P r , for some positive integer r, which implies, by Proposition 5.3, that P is a generator of mod A. A finite dimensional right A-module M is called a progenerator of mod A if M is both a generator and a projective A-module. The progenerator P of mod A

154


defined in Lemma 5.4 does not depend, up to isomorphism, on a choice of a complete set of pairwise nonisomorphic indecomposable projective modules, and we call it a minimal progenerator of mod A. Indeed, let P10 ; : : : ; Pm0 be another complete set of pairwise nonisomorphic indecomposable projective modules of mod A, and let P 0 D P10 ˚ ˚ Pm0 . Here, from the definition of the “complete set”, it should be noted that the number of the Pj0 ’s is the same as the number of the Pi ’s, and, moreover, each Pj0 is isomorphic to exactly one Pi . This implies that P 0 is isomorphic to P , and we conclude that a minimal progenerator of mod A is uniquely determined up to isomorphism. A right module M over a finite dimensional K-algebra A is said to be faithful if M a ¤ 0 for any nonzero element a 2 A. Obviously, the right A-module AA is faithful. The following lemma shows that every faithful module in mod A involves the right A-module A. Lemma 5.5. Let A be a finite dimensional K-algebra. A finite dimensional right A-module M is faithful if and only if AA is isomorphic to an A-submodule of M r , for some positive integer r, that is, there is an exact sequence in mod A of the form 0 ! A ! M r : Pr Proof. Since M is finite dimensional over K, we may write M D iD1 mi K for some m1 ; : : : ; mr 2 M . Let f W A ! M r be the homomorphism of right A-modules defined by f .a/ D .m1 a; : : : ; mr a/ for a 2 A. Now suppose that M is a faithful module. We claim that f is a monomorphism. For this, let Pf .a/ D 0 for some a 2 A. Then, mi a D 0, for all i 2 f1; : : : ; rg, so that M a D riD1 mi aK D 0. This holds only for a D 0, because by the assumption M is a faithful right A-module. Conversely, assume that there is a monomorphism f W A ! M r in mod A, for some r, and take a 2 A with M a D 0. Then f .A/a .M r /a D .M a/r D 0, and hence f .Aa/ D f .A/a D 0. Since f is injective, it follows that Aa D 0, and so a D 0. This shows that M is faithful. Corollary 5.6. Let A be a finite dimensional K-algebra and u W AA ! E.AA / an injective envelope of AA in mod A. Then E.AA / is a faithful right A-module. Applying Lemma 5.5 and the standard duality D W mod A ! mod Aop , for a finite dimensional K-algebra A, we obtain the following fact. Corollary 5.7. Let A be a finite dimensional K-algebra. Then a finite dimensional right A-module M is faithful if and only if there is an epimorphism from D.M /r to D.A/ of left A-modules, for some positive integer r, where D D HomK .; K/. As a direct consequence of Proposition 5.3 and Lemma 5.5 we obtain also the following property of generators.

5. Progenerators

155

Corollary 5.8. Let A be a finite dimensional K-algebra. Then every generator M of mod A is faithful. As is seen in the next example, the converse of Corollary 5.8 is not true in general. See Exercise 8.46 for a finite dimensional K-algebra A whose all faithful finite dimensional right A-modules are generators of mod A. Example 5.9. Let AD

K K

0 K

be the algebra of the lower triangular 2 2 matrices over a field K, and let 1 0 0 0 0 0 e1 D ; e2 D ; aD : 0 0 0 1 1 0 Then it is easy to see that e2 A is a faithful right A-module. Namely, there is a monomorphism A D e1 A ˚ e2 A ! e2 A ˚ e2 A from the right A-module A to the direct sum e2 A ˚ e2 A of 2 copies of e2 A, which assigns to .x; y/ 2 e1 A ˚ e2 A the element .ax; y/ 2 e2 A ˚ e2 A. On the other hand, e2 A is not a generator of mod A. Indeed, if e2 A is a generator of mod A, then, by Proposition 5.3, a direct sum .e2 A/r of finitely many copies of e2 A has a direct summand isomorphic to AA . Then, by the Krull–Schmidt theorem (Theorem I.4.6), e2 A has to be isomorphic to the direct summand e1 A of A, which is not possible. Let A and B be K-algebras and B MA be a (B; A)-bimodule. Then we have two canonical homomorphisms of K-algebras ./L W B ! EndA .MA /

and

./R W A ! EndB op .B M /op

such that bL .x/ D bx and aR .x/ D xa for b 2 B, a 2 A, x 2 M . We say that B MA has the double centralizer property if ./L and ./R are isomorphisms of K-algebras. Moreover, we say that a right A-module MA has the double centralizer property if M regarded as an .EndA .M /; A/-bimodule has the double centralizer property, or equivalently, the correspondence A ! EndEndA .M /op .M /op ; a 7! aR , is an isomorphism of K-algebras. We may also introduce the double centralizer property for left A-modules. Namely, a left A-module A N has the double centralizer property when the right Aop -module NAop has the double centralizer property. We note that, if a right A-module M has the double centralizer property, then the left EndA .M /-module M has the double centralizer property (see Exercise 8.29). K. Morita [Mor] proved the following fundamental theorem which plays an important role for the Morita equivalence theorem, in which the terminology “generator” was first introduced by H. Bass [Bas1] from the category theoretical view point.

156


Theorem 5.10. Let A be a finite dimensional K-algebra, M be a module in mod A, and let B be the endomorphism algebra EndA .M /. Then the following conditions hold. (i) If MA is projective, then B M is a generator of mod B op . (ii) MA is a generator of mod A if and only if B M is projective and MA has the double centralizer property. Proof. (i) Assume that MA is projective. Then, by Proposition I.8.2, there is in mod A an isomorphism of the form An Š M ˚ X; for a positive integer n and a right A-module X . Applying the functor HomA .; M / to this isomorphism, we obtain isomorphisms of left B-modules .B M /n Š B HomA .An ; M / Š B HomA .M; M / ˚ B HomA .X; M / D B B ˚B HomA .X; M /: Hence it follows from Proposition 5.3 that B M is a generator of mod B op . (ii) Assume that MA is a generator of mod A. Then it follows from Proposition 5.3 that there is a positive integer n and an isomorphism M n Š A ˚ X in mod A for an A-module X . Applying HomA .; M / to this isomorphism, we obtain isomorphisms in mod B op B n D HomA .M n ; M / Š HomA .A; M / ˚ HomA .X; M / Š B M ˚ HomA .X; M /: This implies that B M is a projective left B-module. Next we shall show that the canonical homomorphism of K-algebras D ./R W A ! EndB op .M /op is an isomorphism. Notice that is a monomorphism, because MA is a faithful right A-module, by Corollary 5.8. Hence, it suffices to show that is an epimorphism. Take homomorphisms ˛i 2 HomA .M; A/, i 2 f1; : : : ; ng, such that Œ˛1 : : : ˛n W M1 ˚ ˚ Mn ! A is surjective, where M1 ;P : : : ; Mn are copies of MA . Moreover, let mi 2 Mi , i 2 f1; : : : ; ng, be such that niD1 ˛i .mi / D 1A . For any x 2 M and i 2 f1; : : : ; ng, let ˇi .x/ be the endomorphism of MA P such that ˇi .x/.y/P D x˛i .y/ for y 2 M . Then ˇ1 .x/; : : : ; ˇn .x/ 2 B and x D P niD1 x˛i .mi / D niD1 ˇi .x/.mi /: Now, for any f 2 EndB op .M /op , taking a D niD1 ˛i .f .mi // 2 A, we obtain f .x/ D D

n X iD1 n X iD1

f .ˇi .x/.mi // D x˛i .f .mi // D x

n X iD1 n X iD1

f .ˇi .x/mi / D

n X iD1

˛i .f .mi // D xa;

ˇi .x/f .mi /

6. Morita equivalence

157

where ˇi .x/f .mi / D ˇi .x/.f .mi // by the definition of the left B-module structure on M . Thus f .x/ D .a/.x/ for all x 2 M , and hence f D .a/, which shows that D ./R is surjective. Conversely, assume that B M is projective and W A ! EndB op .M /op is an isomorphism of K-algebras. Then, by (i), M is a generator of mod EndB op .M /op , and hence of mod A, because of the isomorphism W A ! EndB op .M /op . The following proposition is a direct consequence of Theorem 5.10. Proposition 5.11. Let A and B be finite dimensional K-algebras and M be a finite dimensional (B; A)-bimodule. Then the following conditions are equivalent. (i) MA is a progenerator of mod A and the canonical map ./L W B ! EndA .M / is an isomorphism of K-algebras. (ii)

BM

is a progenerator of mod B op and the canonical map ./R W A ! EndB op .M /op

is an isomorphism of K-algebras.

6 Morita equivalence Let A and B be K-categories and let F and G be K-linear covariant functors from A to B. A natural transformation of functors W F ! G is called a natural equivalence or natural isomorphism when the morphism X W F .X / ! G.X / is an isomorphism for all objects X of A and then the inverse morphism X1 is a component of a natural isomorphism 1 W G ! F of functors. In this case, we write F Š G. Two K-categories A and B are said to be equivalent provided that there exist functors F W A ! B and G W B ! A such that the composite GF is naturally isomorphic to the identity functor 1A on A and the composite F G is naturally isomorphic to the identity functor 1B on B. In that case, F and G are called (a pair of ) equivalences or mutually inverse equivalences between A and B. Moreover, we will also say that F W A ! B is an equivalence of categories and G is a quasi-inverse functor for F . Any covariant K-linear functor F W A ! B defines a K-linear homomorphism FX;Y W HomA .X; Y / ! HomB .F .X /; F .Y //; for any pair of objects X; Y of A, which assigns to a morphism u W X ! Y in A the morphism F .u/ W F .X / ! F .Y / in B. If this homomorphism is injective or surjective, the functor F is said to be faithful or full, respectively. Moreover, the functor F is said to be dense if any object of B is isomorphic to one of the form F .X/ for some object X of A.

158


Proposition 6.1. Let A and B be K-categories and F W A ! B a covariant Klinear functor. Then the following conditions are equivalent. (i) F is an equivalence of categories. (ii) F is faithful, full, and dense. Proof. Assume that F is an equivalence of categories, G W B ! A is a quasi-inverse functor for F , and W GF ! 1A and W F G ! 1B are natural equivalences of functors. We will show that the functors F and G are faithful, full, and dense. The density of F and G is obvious because we have the natural isomorphisms Y W F G.Y / ! Y , for all objects Y of B, and X W GF .X / ! X , for all objects X of A. We will show now that the functor F is faithful. Let u; v be morphisms from X to Y in A satisfying F .u/ D F .v/. Then, by the natural isomorphism W GF ! 1A , there is the commutative diagram in A for t 2 fu; vg, GF .X /

X

/X

GF .t/

GF .Y /

t

Y

/Y.

Hence we obtain u D Y GF .u/X1 D Y GF .v/X1 D v. Similarly, we show that the functor G is also faithful. Next, we show that the functor F is full. Let w W F .X / ! F .Y / be any morphism in B and let u be the composite Y G.w/X1 , so we have in A the commutative diagram GF .X /

X

u

G.w/

GF .Y /

/X

Y

/ Y.

On the other hand, since u D Y GF .u/X1 , it follows that Y GF .u/X1 D Y G.w/X1 ; and therefore G.w/ D G.F .u//. This implies that w D F .u/, because the functor G is faithful. Hence the functor F is full. Using the natural isomorphism W F G ! 1B and the fact that the functor F is faithful, we show similarly that the functor G is also full. In particular, we conclude that (i) implies (ii). Conversely, assume that the functor F W A ! B is faithful, full and dense. We will define a quasi-inverse functor G W B ! A for F . Since the functor F is


159

dense, we may fix for any object Y of B an object XY of A and an isomorphism Y W F .XY / ! Y in B. We define G.Y / D XY . Moreover, since the functor F is faithful and full, we may choose for any morphism g W Y ! Y 0 in B a unique morphism f W XY ! XY 0 such that F .f / D 1 Y 0 gY , that is, the following diagram in B, Y /Y F .XY / g

F .f /

F .XY 0 /

Y 0

/ Y0,

is commutative. Then we define G.g/ D f . Obviously this defines a covariant K-linear functor G W B ! A, and the family Y W F G.Y / ! Y of isomorphisms in B, for all objects Y of B, defines a natural equivalence W F G ! 1B of functors. We define now a natural equivalence of functors W GF ! 1A as follows. For any object X of A, we define YX D F .X /. Observe that F .X/ D YX is the morphism YX W F .GF .X // D F G.YX / ! YX D F .X /: Since the functor F is faithful and full, there is a unique isomorphism X W GF .X / ! X in A such that F .X / D F .X/ D YX . Let f W X ! X 0 be a morphism in A. We claim that the following diagram in A, GF .X /

X

GF .f /

GF .X 0 /

X 0

/X f

/ X0 ,

is commutative. Since W F G ! 1B is a natural isomorphism of functors, we have in B the commutative diagram F G.F .X //

F .X /

F G.F .f //

F G.F .X 0 //

F .X 0 /

/ F .X / F .f /

/ F .X 0 / .

Moreover, we have F .X / D F .X/ and F .X 0 / D F .X 0 / . Hence we obtain the equalities F .f X / D F .f /F .X / D F .f /F .X/ D F .X 0 / F G.F .f // D F .X 0 /F .GF .f // D F .X 0 GF .f //;

160


and then the required equality f X D X 0 GF .f /, because the functor F is faithful. Therefore, the family of isomorphisms X W GF .X / ! X , for all objects X of A, defines a natural isomorphism W GF ! 1A of functors. Summing up, G is a quasi-inverse functor for F . This shows that (ii) implies (i). We also note the following fact. Lemma 6.2. Let F W A ! B and G W B ! A be covariant K-linear functors between K-categories such that GF Š 1A and F G Š 1B . Then F is a left adjoint for G, and G is a left adjoint for F . Proof. Let W GF ! 1A and W F G ! 1B be natural isomorphisms of functors. Observe that, for any objects X of A and Y of B, we have natural K-linear isomorphisms HomB .F .X /; Y / W HomB .F .X /; F G.Y // ! HomB .F .X /; Y /; FX;G.Y / W HomA .X; G.Y // ! HomB .F .X /; F G.Y //; because the functor F is faithful and full. Then the composed isomorphisms 1 1 W HomB .F .X /; Y / ! HomA .X; G.Y // FX;G.Y / Hom B .F .X /; Y /

are natural in X and Y , and consequently F is a left adjoint for G. Similarly, we prove that G is a left adjoint for F . Two K-algebras A and B are said to be Morita equivalent if the categories Mod A and Mod B are equivalent, and a functor F W Mod A ! Mod B defining such an equivalence is called a Morita equivalence. As is shown later, in the case when A; B are finite dimensional K-algebras, the existence of an equivalence between Mod A and Mod B is equivalent to the existence of an equivalence between mod A and mod B. Thus we also call an equivalence between mod A and mod B a Morita equivalence. The following simple lemma shows that an isomorphism of K-algebras induces a Morita equivalence of the associated module categories. Lemma 6.3. Let f W A ! B be an isomorphism of K-algebras. Then we have a canonical equivalence Ff W Mod B ! Mod A of categories. Proof. For a module M in Mod B, we define the module Mf D Ff .M / in Mod A as follows: Mf D M as K-vector space and m a D mf .a/ for any m 2 M and a 2 A. Observe that, for any homomorphism u W M ! N in Mod B, m 2 M and a 2 A, we have u.m/ a D u.m/f .a/ D u.mf .a// D u.m a/, and

161


so u D Ff .u/ W Ff .M / ! Ff .N / is a homomorphism in Mod A. Obviously, Ff .vu/ D Ff .v/Ff .u/ for all homomorphisms u W M ! N and v W N ! L in Mod B, and Ff .idM / D idFf .M / for any module M in Mod B. Hence, we have a covariant K-linear functor Ff W Mod B ! Mod A: Let g W B ! A be the inverse homomorphism for f . Then g induces the covariant K-linear functor Fg W Mod A ! Mod B; and the equalities Ff Fg D 1Mod A and Fg Ff D 1Mod B hold. In particular, Ff W Mod B ! Mod A is an equivalence of categories. Properties of a module or a homomorphism are said to be Morita invariant if they are preserved under all Morita equivalences. Properties of an algebra A are also said to be Morita invariant if they are true for all algebras Morita equivalent to A. Lemma 6.4. Let A and B be Morita equivalent K-algebras and F W Mod A ! u v Y ! Z ! 0 be a short Mod B be a Morita equivalence. Moreover, let 0 ! X ! exact sequence in Mod A. Then F .u/

F .v/

0 ! F .X / ! F .Y / ! F .Z/ ! 0 is a short exact sequence in Mod B. Proof. Let G W Mod B ! Mod A be a functor quasi-inverse to F and W GF ! 1Mod A and W F G ! 1Mod B the associated equivalences of functors. Then we have the following commutative diagram in Mod A, 0

/ G.F .X //

G.F .u//

/ G.F .Y //

X

0

/X

G.F .v//

Y

/0

Z

/Y

u

/ G.F .Z//

v

/Z

/ 0,

with the vertical homomorphisms being isomorphisms. Then the upper sequence is exact, because the lower one is exact by assumption. We will show now that the sequence F .u/

F .v/

0 ! F .X / ! F .Y / ! F .Z/ ! 0 in Mod B is also exact. The exactness at F .X / is equivalent to saying that F .u/ is a monomorphism. To show it, let w W W ! F .X / be a homomorphism in Mod B with F .u/w D 0.

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Then G.F .u//G.w/ D G.F .u/w/ D 0, which implies that G.w/ D 0, because G.F .u// is injective. Since G is faithful, by Proposition 6.1, it follows that w D 0. Hence, F .u/ is indeed a monomorphism. Next, to show the exactness at F .Y /, we have to show that Im F .u/ D Ker F .v/. The inclusion Im F .u/ Ker F .v/ is clear, because F .v/F .u/ D F .vu/ D 0. We will show now that Ker F .v/ Im F .u/. Take an element y 2 Ker F .v/, and consider the right B-module W D yB and the canonical embedding w W W ! F .Y / of right B-modules. Then F .v/w D 0 because Im w Ker F .v/. Hence G.F .v//G.w/ D G.F .v/w/ D 0. Since Ker G.F .v// D Im G.F .u//, we conclude that there is a homomorphism t W G.W / ! G.F .X // in Mod A such that G.w/ D G.F .u//t. Moreover, t D G.s/ for some homomorphism s W W ! F .X / in Mod B, because the functor G W Mod B ! Mod A is full, by Proposition 6.1. Then G.w/ D G.F .u//t D G.F .u//G.s/ D G.F .u/s/ implies w D F .u/s, because G is a faithful functor. Therefore, we obtain that y 2 Im w D Im F .u/s Im F .u/, and so Ker F .v/ Im F .u/, as required. The exactness at F .Z/ is equivalent to saying that F .v/ is surjective. To show it, let w W F .Z/ ! V be a homomorphism in Mod B with wF .v/ D 0. Then G.w/G.F .v// D 0, and hence G.w/ D 0, because G.F .v// is surjective. Since the functor G is faithful, it follows that w D 0. This ensures that F .v/ is surjective. Lemma 6.4 asserts that the exactness of a sequence of homomorphisms is Morita invariant. In particular, monomorphisms, epimorphisms, and isomorphisms are Morita invariant. Thus the isomorphism classes of modules over an algebra A are in 1-1 correspondence with the isomorphism classes of modules over an algebra B Morita equivalent to A. Other important Morita invariant properties are described below. Proposition 6.5. Let A and B be finite dimensional K-algebras, and F W Mod A ! Mod B be a Morita equivalence. Then F induces an equivalence of categories F W mod A ! mod B such that `.M / D `.F .M // for every nonzero module M in mod A. Proof. It follows from Lemma I.7.7 that mod A is the full subcategory of Mod A (respectively, mod B is the full subcategory of Mod B) consisting of all modules of finite length. Therefore, it suffices to show that `.M / D `.F .M // for every nonzero module M in mod A. Let S be a simple module in mod A. We prove that F .S / is a simple module in mod B. Observe that F .S/ is a nonzero B-module. Indeed, if F .S / D 0 then the trivial homomorphism u W 0 ! S in mod A induces an isomorphism F .u/ W F .0/ ! F .S/ in Mod B, and consequently u is an isomorphism in mod A, because the functor F is faithful (Proposition 6.1). This contradicts the fact that the simple


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A-module S is nonzero. Let N be a nonzero B-submodule of F .S /, and v W N ! F .S/ be the canonical inclusion homomorphism. Since the functor F is dense (Proposition 6.1), there exists a module X in Mod A such that F .X / is isomorphic to N in Mod B. Hence, we have in Mod B a composed monomorphism g D vw W F .X/ ! F .S/, for an isomorphism w W F .X / ! N in Mod B. Further, g D F .f / for a homomorphism f W X ! S , because the functor F is also full (Proposition 6.1). Then g ¤ 0 forces f ¤ 0, and so f is an epimorphism, because S is a simple module in Mod A. Then g D F .f / is an epimorphism in Mod B, by Lemma 6.4, and hence v W N ! F .S/ is an epimorphism in Mod B. This shows that N D F .S /, and so F .S/ is a simple right B-module. Let M be a nonzero module in mod A and take a composition series 0 D M0 M1 Mm D M of M in mod A. Then `.M / D m and there exist in mod A short exact sequences ui

vi

0 ! Mi ! MiC1 ! MiC1 =Mi ! 0 such that MiC1 =Mi are simple right A-modules, for i 2 f0; 1; : : : ; m 1g. Then it follows from Lemma 6.4 and the above considerations that there exist in Mod B short exact sequences F .ui /

F .vi /

0 ! F .Mi / ! F .MiC1 / ! F .MiC1 =Mi / ! 0 such that F .MiC1 =Mi / are simple right B-modules, for i 2 f0; 1; : : : ; m 1g. Consider now the family of B-submodules of F .M / D F .Mm /, Ni D Im F .um1 / : : : F .uiC1 /F .ui / D Im F .um1 : : : uiC1 ui /; i 2 f0; 1; : : : ; m 1g. Then we obtain a chain of B-submodules 0 D N0 N1 Nm1 Nm D F .M / of F .M / such that, for i 2 f0; 1; : : : ; m 1g, Ni Š F .Mi / and NiC1 =Ni Š F .MiC1 /=F .Mi / Š F .MiC1 =Mi / in Mod B, which implies that this chain is a composition series of F .M /. In particular, we obtain `.F .M // D m D `.M / by the Jordan–Hölder Theorem I.7.5. Proposition 6.6. Let A and B be finite dimensional K-algebras and F W Mod A ! Mod B be a Morita equivalence. Then for every nonzero module M in mod A the following equivalences hold. (i) M is a simple module in mod A if and only if F .M / is a simple module in mod B.

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(ii) M is an indecomposable module in mod A if and only if F .M / is an indecomposable module in mod B. (iii) M is a projective module in mod A if and only if F .M / is a projective module in mod B. (iv) M is an injective module in mod A if and only if F .M / is an injective module in mod B. (v) M is a generator in mod A if and only if F .M / is a generator in mod B. (vi) Let M1 ; : : : ; Mn be nonzero modules in mod A. Then M Š M1 ˚ ˚ Mn in mod A if and only if F .M / Š F .M1 / ˚ ˚ F .Mn / in mod B. Proof. Let G W Mod B ! Mod A be a functor such that there exist natural isomorphisms of functors W GF ! 1Mod A and W F G ! 1Mod B . Then it follows from Proposition 6.5 that we have the induced functors F W mod A ! mod B, G W mod B ! mod A, and the equivalences of functors W GF ! 1mod A and W F G ! 1mod B . (i) This follows from Proposition 6.5 because M ¤ 0, F .M / ¤ 0 and `.M / D `.F .M //. (ii) Since the functor F is faithful and full, the K-linear homomorphism FM;M W HomA .M; M / ! HomB .F .M /; F .M //, induced by F , is an isomorphism of K-algebras. Therefore EndA .M / is a local K-algebra if and only if EndB .F .M // is a local K-algebra. Then (ii) follows from Lemma I.4.4. (iii) Assume M is projective in mod A. We show that F .M / is projective in mod B. Let u W X ! Y be an epimorphism and w W F .M / ! Y a homomorphism in mod B. Since the functor G W mod B ! mod A carries, by Lemma 6.4, epimorphisms to epimorphisms, we conclude that G.u/ W G.X / ! G.Y / is an epimorphism in mod A. Further, we have in mod A an isomorphism M W G.F .M // ! M , and so G.F .M // is a projective module in mod A. Hence there exists a homomorphism v 0 W G.F .M // ! G.X / in mod A such that G.u/v 0 D G.w/. Since the functor G W mod B ! mod A is full, we have v 0 D G.v/ for some homomorphism v W F .M / ! X in mod B. Therefore, we obtain that G.uv/ D G.u/G.v/ D G.u/v 0 D G.w/, which implies uv D w, because G is faithful. This shows that indeed F .M / is a projective module in mod B. Similarly, one proves that if F .M / is projective in mod B then G.F .M // is projective in mod A, and consequently M is projective in mod A, because G.F .M // Š M in mod A. (iv) This is proved by the dual arguments to ones used in (iii). (v) Assume M is a generator of mod A. We show that F .M / is a generator of mod B. Let Y be a module in mod B. Since F is a dense functor from mod A to mod B, there is a module X in mod A such that Y Š F .X / in mod B. It follows from the assumption on M and Proposition 5.3 that there is a positive integer n and an epimorphism f W M n ! X in mod A. Hence, by Lemma 6.4,


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we obtain the epimorphism F .f / W F .M n / ! F .X / in mod B. Then we have an epimorphism F .M /n ! Y in mod B, because F .M /n Š F .M n / and Y Š F .X / in mod B. Applying Proposition 5.3 we conclude that F .M / is a generator of mod B. Similarly, we show that, if F .M / is a generator of mod B, then M is a generator of mod A, because M Š G.F .M // in mod A. (vi) Since F is an equivalence of categories, we conclude that F .M1 /; : : : , F .Mn / are nonzero modules in mod B. Assume that M Š M1 ˚ ˚ Mn in mod A. We will show that F .M / Š F .M1 /˚ ˚F .Mn / in mod B. Observe that there exist epimorphisms pi W M ! Mi and monomorphisms ui W Mi ! M with pi ui D idMi , for i 2 f1; : : : ; ng, such that e1 D u1 p1 ; : : : ; en D un pn are pairwise orthogonal idempotents of EndA .M / and 1EndA .M / D idM D e1 C Cen . Then f1 D F .e1 /; : : : ; fn D F .en / are pairwise orthogonal idempotents of EndB .F .M // such that 1EndB .F .M // D idF .M / D f1 C C fn . Applying Lemma 2.1, we conclude that F .M / is a direct sum F .M / D f1 F .M / ˚ ˚ fn F .M / of B-submodules f1 F .M /; : : : ; fn F .M /. Hence, in order to show that F .M / Š F .M1 / ˚ ˚ F .Mn / in mod B, it is enough to prove that fi F .M / Š F .Mi / in mod B for any i 2 f1; : : : ; ng. Fix i 2 f1; : : : ; ng. From the equality idMi D pi ui , we obtain idF .Mi / D F .idMi / D F .pi ui / D F .pi /F .ui /, and hence F .ui / W F .Mi / ! F .M / is a monomorphism and F .pi / W F .M / ! F .Mi / is an epimorphism in mod B. Moreover, fi F .M / D F .ei /F .M / D F .ui pi /.F .M // D F .ui /.F .pi /.F .M /// D F .ui /.F .Mi //, and so fi F .M / is the image of F .ui /. This shows that F .ui / defines an isomorphism F .Mi / ! fi F .M / of right B-modules, as required. The sufficiency part of (vi) follows similarly by applying the quasi-inverse functor G of F . The next theorem is the first half of the Morita equivalence theorem which asserts that a progenerator defines a Morita equivalence. Theorem 6.7. Let A be a finite dimensional K-algebra and P be a progenerator of mod A. Then A is Morita equivalent to the endomorphism algebra B D EndA .P / and, regarding P as a (B; A)-bimodule, the functors HomA .P; / W Mod A ! Mod B; HomB .HomA .P; A/; / W Mod B ! Mod A define a Morita equivalence between Mod A and Mod B. Moreover, their restrictions to mod A and mod B define also an equivalence between mod A and mod B. Proof. The last assertion follows by Proposition 6.5. Let F D HomA .P; / and G D HomB .HomA .P; A/; /. We show that there are natural isomorphisms of functors GF Š 1Mod A and F G Š 1Mod B . Since P is a progenerator of mod A, it follows from Theorem 5.10 that B P is a progenerator of mod B op and PA has the double centralizer property. In particular, B P is a projective

166


module in mod B op and there is a canonical isomorphism A ! EndB op .B P /op of K-algebras. Applying Theorem 4.5 and Lemma I.8.7, we obtain isomorphisms of right A-modules HomA .P; A/ ˝B P Š HomA .HomB op .P; P /; A/ Š HomA .HomB op .P; P /op ; A/ Š HomA .A; A/ Š A: Then, for every module X in Mod A, by Theorem 4.3 we obtain isomorphisms of right A-modules G.F .X // D HomB .HomA .P; A/; HomA .P; X // Š HomA .HomA .P; A/ ˝B P; X / Š HomA .A; X / Š X; which are natural in X . This shows that there is a natural isomorphism of functors GF Š 1Mod A . It follows from Corollary 4.9 that there is a canonical isomorphism of right B-modules P ˝A HomA .P; A/ ! HomA .P; P / D B: Then, applying Theorem 4.3 and Lemma I.8.7, we obtain, for every module Y in Mod B, isomorphisms of right B-modules F .G.Y // D HomA .P; HomB .HomA .P; A/; Y // Š HomB .P ˝A HomA .P; A/; Y /; Š HomB .B; Y / Š Y; which are natural in Y . This proves that there is a natural isomorphism of functors F G Š 1Mod B . The next theorem is the second half of the Morita equivalence theorem which asserts that a Morita equivalence determines a progenerator. Theorem 6.8. Let A and B be Morita equivalent finite dimensional K-algebras and let F W Mod A ! Mod B and G W Mod B ! Mod A be a Morita equivalence pair. Then the following statements hold. (i) There is a progenerator P of mod A such that B and EndA .P / are isomorphic as K-algebras and, regarding P as a (B; A)-bimodule, there are natural isomorphisms of functors F Š HomA .P; / Š ˝A HomA .P; A/; G Š HomB .HomA .P; A/; / Š ˝B P:


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(ii) There is a progenerator Q of mod B such that A and EndB .Q/ are isomorphic as K-algebras and, regarding Q as an (A; B)-bimodule, there are natural isomorphisms of functors F Š HomA .HomB .Q; B/; / Š ˝A Q; G Š HomB .Q; / Š ˝B HomB .Q; B/: (iii) For any progenerators B PA and A QB satisfying (i) and (ii), there are (A; B)bimodule isomorphisms A QB

Š A HomA .P; A/B Š A HomB op .P; B/B ;

and there are (B; A)-bimodule isomorphisms B PA

Š B HomB .Q; B/A Š B HomAop .Q; A/A :

Proof. Observe first that, by Lemma 6.2, F is left adjoint and right adjoint for G. (i) Let P D G.B/. Applying Lemma 4.4 to the adjoint pair of functors hG; F i we conclude that the functor F is naturally isomorphic to the hom functor HomA .P; / and the functor G is naturally isomorphic to the tensor functor ˝B P . Further, by Lemma 5.4, B is a progenerator of mod B. Then, applying Proposition 6.6, we obtain that P D G.B/ is a progenerator of mod A. Observe also that the K-algebras B and EndA .P / are isomorphic. Indeed, there are canonical isomorphisms of K-algebras B ! EndB .B/, which assigns to an element b 2 B the left multiplication bL of B by b, and EndB .B/ ! EndA .G.B// D EndA .P /, induced by the faithful and full functor G (see Proposition 6.1). It follows then from Theorem 6.7 that HomA .P; / W Mod A ! Mod EndA .P / and HomB .HomA .P; A/; / W Mod EndA .P / ! Mod A form a pair of Morita equivalences between Mod A and Mod EndA .P /. Moreover, Mod B is equivalent to Mod EndA .P /, because B Š EndA .P /. In particular, Lemma 6.2 implies that the functor F Š HomA .P; / is left adjoint for the functors HomB .HomA .P; A/; / and ˝B P . Then, making use of Lemma 4.2, we conclude that the functors HomB .HomA .P; A/; / and ˝B P from Mod B to Mod A are naturally isomorphic. Finally, by Theorem 4.3, the functor ˝A HomA .P; A/ W Mod A ! Mod B is left adjoint for the functor HomB .HomA .P; A/; / W Mod B ! Mod A. Hence, by Lemma 4.1, the functors HomA .P; / and ˝A HomA .P; A/ are naturally isomorphic. (ii) This is an immediate consequence of (i), for B replacing A and Q D F .A/. (iii) Let B PA and A QB be progenerators of mod A and mod B, respectively, satisfying the conditions (i) and (ii). We will show that there are .A; B/-bimodule isomorphisms A QB

Š A HomA .P; A/B Š A HomB op .P; B/B :

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It follows from (i) and (ii) that, for every right A-module X , we have isomorphisms of right B-modules X ˝A Q Š F .X / Š X ˝A HomA .P; A/; which are natural in X , and hence are isomorphisms of .EndA .X /; B/-bimodules. Taking now X D A, we obtain isomorphisms of .A; B/-bimodules A QB

Š A .A ˝A Q/B Š A .A ˝A HomA .P; A//B Š A HomA .P; A/B :

Since B PA is a progenerator of mod A, then it follows from Theorem 5.10 that B P is a projective left B-module. Applying now Corollary 4.6 we conclude that, for every module Y in mod B, there is an isomorphism of right A-modules Y ˝B P Š HomB .HomB op .P; B/; Y /; which is natural in Y , and so the functors ˝B P and HomB .HomB op .P; B/; / from Mod B to Mod A are naturally isomorphic. Further, it follows from Theorem 4.3 that the functor ˝A HomB op .P; B/ W Mod A ! Mod B is left adjoint for the functor HomB .HomB op .P; B/; /, and hence for the functor ˝B P . On the other hand, it follows from (i) that the functor ˝A HomA .P; A/ W Mod A ! Mod B is also left adjoint for the functor ˝B P . Therefore, applying Lemma 4.1, we conclude that the functors ˝A HomB op .P; B/ and ˝A HomA .P; A/ from Mod A to Mod B are naturally isomorphic. Then we obtain isomorphisms of .A; B/-bimodules A

HomB op .P; B/B Š A .A ˝A HomB op .P; B//B Š A .A ˝A HomA .P; A//B Š A HomA .P; A/B :

The required .B; A/-bimodule isomorphisms B PA

Š B HomB .Q; B/A Š B HomAop .Q; A/A

follow from the above considerations, by exchanging A with B and B PA with A QB . Proposition 6.9. Let A and B be Morita equivalent finite dimensional K-algebras. Then there is a 1-1 correspondence between the isomorphism classes of Morita equivalences F W Mod A ! Mod B and the isomorphism classes of (B; A)-bimodules P such that P is a progenerator of mod A with the double centralizer property, via natural isomorphisms of functors F Š HomA .P; / from Mod A to Mod B. Proof. Assume that P and P 0 are (B; A)-bimodules which are progenerators of mod A and have the double centralizer property. Then it follows from Theorem 6.7 that F D HomA .P; / and F 0 D HomA .P 0 ; / are Morita equivalences from


169

Mod A to Mod B. Further, assume that P and P 0 are isomorphic as (B; A)bimodules, and let u W P 0 ! P be an isomorphism of (B; A)-bimodules. Then the family HomA .u; X / W HomA .P; X / ! HomA .P 0 ; X / of homomorphisms in Mod B, for all modules X in Mod A, defines a natural isomorphism of functors F D HomA .P; / and F 0 D HomA .P 0 ; / from Mod A to Mod B. Conversely, assume that F and F 0 are isomorphic Morita equivalences from Mod A to Mod B. Then, by Theorem 6.8, there exist (B; A)-bimodules P and P 0 such that (a) P and P 0 are progenerators of mod A; (b) B Š EndA .P / and B Š EndA .P 0 / as K-algebras; (c) the functors G D ˝B P and G 0 D ˝B P 0 from Mod B to Mod A are the quasi-inverse functors for F and F 0 , respectively. Moreover, it follows from Theorem 5.10 that P and P 0 have the double centralizer property. In particular, since the functors F and F 0 are naturally isomorphic, we conclude from (c) that the functors G and G 0 are also naturally isomorphic. Let W G ! G 0 be a natural isomorphism of functors. We will show that P and P 0 are isomorphic as (B; A)-bimodules. We know from Lemma 3.5 that there are canonical 0 isomorphisms of left B-modules ' P W B ˝B P ! P and ' P W B ˝B P 0 ! P 0 0 given by ' P .b ˝ p/ D bp and ' P .b ˝ p 0 / D bp 0 , for b 2 B, p 2 P and p 0 2 P 0 . 0 In fact, since P and P 0 are (B; A)-bimodules, ' P and ' P are isomorphisms of (B; A)-bimodules. Consider the composed isomorphism of right A-modules P

'P

B

0

f W P ! B ˝B P ! B ˝B P 0 ! P 0 ; where P D .' P /1 . We claim that f is also an isomorphism of left B-modules. Observe first that PB .1B ˝ p/ D 1B ˝ f .p/ for any p 2 P . Indeed, letting B .1B ˝ p/ D siD1 bi ˝ pi0 for some bi 2 B and pi0 2 P 0 , i 2 f1; : : : ; sg, we have s s s X X X bi ˝ pi0 D bi pi0 D 1B ˝ p 0 ; 1B ˝ bi pi0 D 1B ˝ iD1 0

for p D

iD1

Ps

0 iD1 bi pi .

iD1

Hence, we obtain B .1B ˝ p/ D 1B ˝ p 0 . But then

0

f .p/ D ' P B

P

0

0

.p/ D ' P B .1B ˝ p/ D ' P .1B ˝ p 0 / D p 0 :

Therefore, we get B .1B ˝ p/ D 1B ˝ f .p/. Further, for b 2 B, we have in Mod A the commutative diagram B ˝B P B

B ˝B P 0

bL ˝idP

/ B ˝B P B

bL ˝idP 0

/ B ˝B P 0

'P

/P f

'P

0

/ P0 ,

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where bL is the left multiplication by b. Then we obtain, for b 2 B and p 2 P , the equalities P f ' .bL ˝ idP / .1B ˝ p/ D f ' P .b ˝ p/ D f .bp/; 0 0 ' P .bL ˝ idP 0 /B .1B ˝ p/ D ' P .bL ˝ idP /.1B ˝ f .p// 0

D ' P .b ˝ f .p// D bf .p/: Hence f .bp/ D bf .p/, and so f is a homomorphism of left B-modules. Therefore, f is an isomorphism of (B; A)-bimodules. We note that it follows from the proofs of Theorems 6.7 and 6.8 that their statements hold for Mod A and Mod B replaced by mod A and mod B. We sum up below some basic properties of finite dimensional progenerators over finite dimensional K-algebras. Proposition 6.10. Let A be a finite dimensional K-algebra, P a progenerator of mod A and B D EndA .P /. Then the following statements hold. (i) HomA .P; A/ and HomB op .P; B/ are progenerators of mod B. (ii) HomA .P; A/ Š HomB op .P; B/ as (A; B)-bimodules. (iii) A Š EndB .HomA .P; A// Š EndAop .HomB op .P; B//op as K-algebras. (iv) B Š EndAop .HomA .P; A//op Š EndB .HomB op .P; B// as K-algebras. Proposition 6.11. For two finite dimensional K-algebras A and B, the following conditions are equivalent. (i) Mod A and Mod B are equivalent. (ii) Mod Aop and Mod B op are equivalent. (iii) mod A and mod B are equivalent. (iv) mod Aop and mod B op are equivalent. Proof. Assume (iii). Taking into account the note preceding Proposition 6.10, by the Morita equivalence Theorem 6.8 for mod A and mod B, the statement (iii) implies the existence of a (B; A)-bimodule P such that PA is a progenerator of mod A with B Š EndA .P /. By Proposition 5.11, the left B-module P is then a progenerator of mod B op and Aop D EndB op .P /. Hence (ii) follows by the Morita equivalence Theorem 6.7. Further, the implication (ii) ) (iv) follows by Proposition 6.5. Thus we have proved the implications (iii) ) (ii) ) (iv). This immediately implies, invoking again Proposition 6.5, (iv) ) (i) ) (iii), because .Aop /op D A and .B op /op D B.


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The following criterion makes it easy to recognize Morita equivalences. Lemma 6.12. Let A be a finite dimensional K-algebra, and let X; Y and Z be modules in mod A such that Y Š Z as right A-modules. Then the endomorphism K-algebras EndA .X ˚ Y ˚ Z/ and EndA .X ˚ Y / are Morita equivalent. Proof. Let M D X ˚ Y ˚ Z, N D X ˚ Y , B D EndA .M / and C D EndA .N /. Notice that B and C are finite dimensional K-algebras, because M and N are finite dimensional right A-modules. We denote by uX W X ! M , uY W Y ! M , uZ W Z ! M the canonical embeddings and by pX W M ! X , pY W M ! Y , pZ W M ! Z the canonical projections, for the direct sum decomposition M D X ˚ Y ˚ Z of M . Let eX D uX pX ; eY D uY pY and eZ D uZ pZ . Then eX ; eY ; eZ are pairwise orthogonal idempotents of B and 1B D eX C eY C eZ . For e D eX C eY there are canonical K-algebra isomorphisms ' W C ! eBe; W eBe ! C;

'.c/ D uX˚Y cpX˚Y .ebe/ D pX˚Y ebeuX˚Y

for c 2 C; for b 2 B;

where uX˚Y and pX˚Y are the canonical injection and canonical projection, respectively, for the direct sum decomposition M D .X ˚ Y / ˚ Z of M . It is easily verified that e D eX˚Y D uX˚Y pX˚Y ;

1C D pX˚Y uX˚Y ;

and ' and ' are identities on C and eBe, respectively. Hence C is isomorphic to EndB .eB/, because eBe Š EndB .eB/ as K-algebras (see Corollary I.8.8). On the other hand, B D eX B ˚ eY B ˚ eZ B and eY B Š eZ B as right B-modules. In fact, there are in mod B isomorphisms eY B Š HomA .M; Y / Š HomA .M; Z/ Š eZ B; where the middle isomorphism is induced by the assumption that Y and Z are isomorphic right A-modules. Putting P D eB we therefore obtain an epimorphism P 2 ! BB in mod B, and hence, by Proposition 5.3, the projective right B-module P is a generator of mod B. Thus we have proved that C is isomorphic to the endomorphism algebra EndB .P / of a progenerator P of mod B. Therefore, it follows from Theorem 6.7 that the algebras B and C are Morita equivalent. As a prototype of Morita equivalence may be considered the Wedderburn structure theorem (Theorem I.6.3). Lemma 6.13. Let F be a finite dimensional division K-algebra, n a positive integer, and Mn .F / the full n n matrix algebra over F . Then Mn .F / is Morita equivalent to F .

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Proof. The identity In of Mn .F / has the canonical decomposition In D E11 C E22 C C Enn as a sum of the diagonal elementary matrices Ei i , i 2 f1; : : : ; ng, which are pairwise orthogonal primitive idempotents of Mn .F /. It follows from the proof of Lemma I.6.2 that Mn .F / D S1 ˚ S2 ˚ ˚ Sn ˚ where Sr D Err Mn .F / D Œaij 2 Mn .F / j aij D 0 for i ¤ r , r 2 f1; : : : ; ng, are simple right Mn .F /-modules. Moreover, by Lemma I.8.23, Sr Š S1 in mod Mn .F / for any r 2 f1; : : : ; ng. Hence, Mn .F / Š S1n as right Mn .F /modules, and consequently, by Proposition 5.3, the simple module S1 is a progenerator of mod Mn .F /. Therefore, it follows from Theorem 6.7 that Mn .F / is Morita equivalent to EndMn .F / .S1 /, which is isomorphic to F . This shows that Mn .F / and F are Morita equivalent. Corollary 6.14. Let A be a finite dimensional semisimple K-algebra. Then A is Morita equivalent to a product F1 Fr of finite dimensional division Kalgebras F1 ; : : : ; Fr . Proof. It follows from the Wedderburn structure theorem (Theorem I.6.3) that there exist positive integers n1 ; : : : ; nr and finite dimensional division K-algebras F1 ; : : : ; Fr such that A Š Mn1 .F1 / Mnr .Fr / as K-algebras. Then it follows from Lemma 6.13 that A is Morita equivalent to F1 Fr (see also Exercise 8.11). Corollary 6.15. Let A be a finite dimensional simple K-algebra. Then A is Morita equivalent to a finite dimensional division K-algebra F . Proof. It is a direct consequence of Corollary I.6.5 and Lemma 6.13.

Let A be a finite dimensional K-algebra and take a minimal progenerator eA of mod A with e 2 D e 2 A (see Lemma 5.4). For example, e may be taken as follows: let si s1 sn n X X X X 1A D e1j C C enj D eij j D1

j D1

iD1

j D1

be a decomposition of the identity 1A of A into the sum of pairwise orthogonal primitive idempotents eij , where eij A Š eik A for j; k 2 f1; : : : ; si g; eij A 6Š ei 0 k A for i ¤ i 0 ; P and then let e D niD1 ei1 . By the Morita equivalence theorem (Theorem 6.7), A and eAe, isomorphic to EndA .eA/, are Morita equivalent. The algebra Ab D eAe is called the basic algebra of A, which is uniquely determined by A, up to algebra isomorphism, because it is the endomorphism algebra of a minimal progenerator


173

which is uniquely determined up to isomorphism, by Lemma 5.4. In particular, an algebra A is said to be basic if A Š Ab , which is equivalent to saying that the identity 1A of A is a sum of orthogonal primitive idempotents e1 ; : : : ; en such that ei A 6Š ej A for all i ¤ j from f1; : : : ; ng. The notion of the basic algebra was first introduced by C. Nesbitt and W. M. Scott [NeSc] in 1941. The result observed above is stated as follows. Theorem 6.16. Every finite dimensional K-algebra A is Morita equivalent to its basic algebra Ab . Lemma 6.17. Let Q be a finite quiver, I an admissible ideal of the path algebra KQ of Q over K. Then the bound quiver algebra KQ=I is a basic algebra. Proof. It follows from Lemmas I.1.3 and I.1.5 that A D KQ=I is a finite dimensional K-algebra and the cosets ea D "a C I of the trivial paths "a at the vertices a P2 Q0 of Q form a set of pairwise orthogonal primitive idempotents of A with ea D 1A . Moreover, the indecomposable projective modules ea A, a 2 Q0 , are pairwise nonisomorphic (see Corollary I.8.29). Therefore, A D KQ=I is a basic algebra. Corollary 6.18. Let Q be a finite acyclic quiver. Then the path algebra KQ of Q over K is a finite dimensional basic algebra. The following proposition gives a characterization of finite dimensional basic K-algebras. Proposition 6.19. Let A be a finite dimensional K-algebra. Then A is a basic K-algebra if and only if A= rad A is isomorphic to a product F1 Fn of finite dimensional division K-algebras F1 ; : : : ; Fn . Proof. Let e1 ; : : : ; en be a set of pairwise orthogonal idempotents of A such that 1A D e1 C C en , and B D A= rad A. Then the cosets eN1 D e1 C rad A; : : : ; eNn D en C rad A form a set of pairwise orthogonal primitive idempotents of B with 1B D eN1 C C eNn , e1 A; : : : ; en A are indecomposable projective modules in mod A with AA D e1 A ˚ ˚ en A and eN1 B; : : : ; eNn B are simple projective modules in mod B with BB D eN1 B ˚ ˚ eNn B, by Corollary I.5.10 and Proposition I.5.16. Moreover, for i; j 2 f1; : : : ; ng, we have ei A Š ej A in mod A if and only if eNi B Š eNj B in mod B (see Corollary I.8.6). Therefore, A is a basic K-algebra if and only if the simple right B-modules eN1 B; : : : ; eNn B are pairwise nonisomorphic, or equivalently, the semisimple algebra B D A= rad A is isomorphic to the product F1 Fn of finite dimensional division K-algebras F1 D EndB .eN1 B/ Š eN1 B eN1 ; : : : ; Fn D EndB .eNn B/ Š eNn B eNn . Proposition 6.20. Let A and B be finite dimensional basic K-algebras. Then A and B are Morita equivalent if and only if A and B are isomorphic.

174


Proof. It follows from Lemma 6.3 that if A and B are isomorphic K-algebras then A and B are Morita equivalent. Conversely, assume that A and B are Morita equivalent, and let F W Mod A ! Mod B be an equivalence functor. Then, by Theorem 6.7, the restriction of F to mod A induces an equivalence F W mod A ! mod B of the categories of finite dimensional modules. Since A is a basic K-algebra, it follows from Lemma 5.4 that AA is a minimal progenerator of mod A. Applying Proposition 6.6 we then conclude that F .AA / is a progenerator of mod B. Further, since F is a faithful and full Klinear functor (see Proposition 6.1) and AA is a direct sum of pairwise nonisomorphic indecomposable right A-submodules, it follows from Proposition 6.6 that F .AA / is a direct sum of pairwise nonisomorphic indecomposable right B-submodules of F .AA /, and consequently F .AA / is a minimal progenerator of mod B. Obviously then F .AA / Š BB in mod B (see Lemma 5.4). Therefore, we obtain a sequence of isomorphisms of K-algebras A Š EndA .AA / Š EndB .F .AA // Š EndB .BB / Š B;

using Lemma I.6.1 and the fact that F is faithful and full.

The following corollary shows that the Morita equivalence classes of finite dimensional K-algebras are uniquely determined by the isomorphism classes of basic finite dimensional K-algebras. Corollary 6.21. Let A and B be finite dimensional K-algebras. Then A and B are Morita equivalent if and only if the basic algebras Ab and B b are isomorphic. Proof. It follows from Theorem 6.16 that A is Morita equivalent to its basic algebra Ab and B is Morita equivalent to its basic algebra B b . Then the equivalence is a direct consequence of Proposition 6.20. We end this section with the general form of finite dimensional K-algebras which are Morita equivalent to a fixed finite dimensional basic K-algebra. Corollary 6.22. Let ƒ be a finite dimensional basic K-algebra and let ƒ D P1 ˚ ˚ Pn be a decomposition of ƒ into a direct sum of indecomposable right ƒmodules, and A be a finite dimensional K-algebra. Then A is Morita equivalent to ƒ if and only if A is isomorphic to an algebra of the form ƒ.m.1/; : : : ; m.n// WD Endƒ

n M

Pim.i/

iD1

for a sequence m.1/; : : : ; m.n/ of positive integers. Proof. It follows from Proposition 5.3 and Lemma 5.4 that P is a progenerator of L mod ƒ if and only if P is isomorphic to a module of the form niD1 Pim.i/ for a sequence m.1/; : : : ; m.n/ of positive integers. Then the required equivalence is a direct consequence of the Morita equivalence theorems (Theorems 6.7 and 6.8).

7. Morita–Azumaya duality

175

We note that ƒ Š Endƒ .ƒ/ is a basic algebra of every algebra of the form ƒ.m.1/; : : : ; m.n//.

7 Morita–Azumaya duality Let C and D be K-categories. We say that a pair of contravariant functors S W C ! D and T W D ! C defines a (functorial) duality between C and D if the composites T S and S T are naturally isomorphic to the identity functors 1C on C and 1D on D, respectively. In this case, the functors S and T are called (a pair of ) dualities. A contravariant functor S W C ! D may be regarded as a covariant functor S 0 W Cop ! D or S 00 W C ! Dop such that S.X / D S 0 .X / for each object X of C and S.f / D S 0 .f op / W S.X / ! S.X 0 / for each morphism f W X 0 ! X in C, and S.X/ D S 00 .X / for each object X 2 C and S.f / D S 00 .f /op W S.X / ! S.X 0 / for each morphism f W X 0 ! X in C. Under these notations, the contravariant functors S W C ! D and T W D ! C define a duality between C and D if and only if the covariant functors S 00 W C ! Dop and T 0 W Dop ! C define an equivalence between C and Dop . There is another useful interpretation of a duality in the case when C D mod A and D D mod B op for finite dimensional K-algebras A and B. Let D0 D HomK .; K/ be the standard duality between mod B and mod B op . Then two contravariant functors D1 W mod A ! mod B op and D2 W mod B op ! mod A define a duality if and only if the composites F D D0 D1 W mod A ! mod B and G D D2 D0 W mod B ! mod A define an equivalence between mod A and mod B, which we may visualize as mod A o

D1 D2

/

mod B op o

D0 D0

/

mod B :

It should be noted that we have to restrict the duality to the category of finite dimensional modules, while the Morita equivalence involves the category of all modules. An example suggesting this restriction is given in Exercise 8.44. S. Eilenberg and S. Mac Lane [EM] observed in 1945 that the Pontrjagin duality [Pon], which assigns to a locally compact abelian group X the locally compact abelian group Hom.X; R=Z/ of continuous homomorphisms from X to the one dimensional torus R=Z, is in fact a duality for the category of locally compact abelian groups and continuous homomorphisms. The functorial duality was established axiomatically in 1950s by S. Mac Lane [ML1] and D. A. Buchsbaum [Buc]. Let C be a K-category. An object M of C is called a cogenerator of C if, for any two different morphisms f; g 2 HomC .X; Y /, there exists a morphism h 2 HomC .Y; M / such that hf ¤ hg, or equivalently, for any nonzero morphism f 2 HomC .X; Y /, there exists a morphism h 2 HomC .Y; M / such that hf ¤ 0.

176


By the definitions of a generator and a cogenerator, an object M of C is a cogenerator of C if and only if M is a generator of Cop , and if and only if the functor HomC .; M / W C ! Mod K is faithful. The following lemma is an immediate consequence of the definition of cogenerator (see Lemma 5.1). Lemma 7.1. Let C be a K-category and M be an object of C. The following statements hold. (i) If M is a cogenerator of C and X is an object of C, then the direct sum M ˚X is a cogenerator of C. (ii) M is a cogenerator of C if M m is a cogenerator of C for some positive integer m. Let A be a finite dimensional K-algebra and let D be the standard duality HomK .; K/ between mod Aop and mod A. Recall from Proposition I.8.16 that, the duality D carries a projective or injective module X in mod Aop to an injective or projective module in mod A, respectively. Moreover, since D.f / ¤ 0 for any nonzero homomorphism f in mod Aop , a generator in mod Aop is transferred by D to a cogenerator in mod A. Applying D, the statements in the Section 5 on generators in mod Aop thus are transferred to the corresponding dual statements on cogenerators of mod A. The following lemma and proposition are the dual statements of Lemma 5.2 and Proposition 5.3. Lemma 7.2. Let A be a finite dimensional K-algebra. Then D.A/ is an injective cogenerator of mod A. Proposition 7.3. Let A be a finite dimensional K-algebra and M be a finite dimensional right A-module. The following conditions are equivalent. (i) M is a cogenerator of mod A. (ii) For any module X in mod A, there is a monomorphism from X to M n , for some positive integer n. (iii) There is a monomorphism from the right A-module D.A/ to M n , for some positive integer n, or equivalently M n Š D.A/ ˚ X in mod A, for some positive integer n and a right A-module X . (iv) M has an injective direct summand E D E1 ˚ ˚Em , where E1 ; : : : ; Em is a complete set of pairwise nonisomorphic indecomposable injective modules in mod A.


177

The following fact, a special case of Proposition 7.3, shows the structure of injective cogenerators. Corollary 7.4. Let A be a finite dimensional K-algebra, S1 ; : : : ; Sn be a complete set of pairwise nonisomorphic simple right A-modules, and M be a finite dimensional injective right A-module. Then the following conditions are equivalent. (i) M is a cogenerator of mod A. (ii) Each Si ; i 2 f1; : : : ; ng, is isomorphic to a right A-submodule of M . (iii) M is isomorphic to a direct sum E.S1 /r1 ˚ ˚ E.Sn /rn , for some positive integers r1 ; ; rn , where E.S1 /; : : : ; E.Sn / are injective envelopes of S1 ; : : : ; Sn in mod A. The following proposition is the dual of Proposition 5.11. Proposition 7.5. Let A and B be finite dimensional K-algebras and M be a finite dimensional (B; A)-bimodule. Then the following conditions are equivalent. (i) MA is an injective cogenerator of mod A and the canonical map ./L W B ! EndA .M / is an isomorphism of K-algebras. (ii)

is an injective cogenerator of mod B op and the canonical map ./R W A ! EndB op .M /op is an isomorphism of K-algebras.

BM

Considering the equivalent conditions in Corollary 7.4, we define an injective cogenerator Q of mod A to be minimal if QA is isomorphic to a direct sum of indecomposable injective right A-modules E.S1 /; : : : ; E.Sm /, where S1 ; : : : ; Sm is a complete set of pairwise nonisomorphic simple right A-modules. Then D.E.S1 //; : : : ; D.E.Sm // form a complete set of pairwise non-isomorphic indecomposable projective left A-modules, hence A P D D.E.S1 // ˚ ˚ D.E.Sm // Š D.QA / is a minimal progenerator of mod Aop . A right A-module QA is then a minimal injective cogenerator of mod A if and only if D.QA / is a minimal progenerator of mod Aop , and is uniquely determined up to isomorphism. The case when M D A in Corollary 7.4 is especially important and, as it is shown in the next proposition, the equivalent conditions in Corollary 7.4 are then always satisfied by A. Proposition 7.6. Let A be a finite dimensional K-algebra. The following conditions are equivalent. (i) The left A-module A A is injective. (ii) The right A-module AA is injective. Moreover, in these cases, the A-modules A A and AA are injective cogenerators of mod Aop and mod A, respectively.

178


Proof. Let 1A D e1 C C en be a decomposition of 1A into sum of pairwise orthogonal primitive idempotents of A, and, by changing the indices (if necessary), we may assume that e1 A; : : : ; em A (or equivalently, Ae1 ; : : : ; Aem ), m n, is a complete set of pairwise nonisomorphic indecomposable projective modules of mod A (respectively, mod Aop ). Let e D e1 C C em . Then, by Lemma 5.4, eA and Ae are minimal progenerators of mod A and mod Aop , respectively. Applying the standard duality D D HomK .; K/, we conclude that D.Ae/A and A D.eA/ are then minimal injective cogenerators of mod A and mod Aop , respectively. By the left-right symmetry, it is enough to show (i) ) (ii) only. Assume that A A is injective. Then A Ae is injective, as a direct summand of A A, and hence D.Ae/A is projective. Therefore, D.Ae/A is a direct sum of pairwise nonisomorphic indecomposable projective modules D.Aei /, i 2 f1; : : : ; mg. By Lemma 5.4, D.Ae/A is then a minimal progenerator of mod A, and hence D.Ae/ Š eA as right A-modules, which implies that eA is an injective cogenerator of mod A. This ensures that AA is an injective cogenerator of mod A as desired, because AA is isomorphic to a direct summand of an injective right A-module .eA/r , for some positive integer r, and AA contains eA (as a direct summand) being a cogenerator of mod A. A finite dimensional K-algebra A is said to be selfinjective if A A or AA is an injective A-module, or equivalently, both A A and AA are injective A-modules. Lemma 7.7. Let S W C ! D be a duality of K-categories and X be an object of C. Then the following equivalences hold. (i) X is a generator of C if and only if S.X / is a cogenerator of D. (ii) X is a cogenerator of C if and only if S.X / is a generator of D. Proof. Since the covariant functor S 00 W C ! Dop associated to S is an equivalence of K-categories, we easily conclude that X is a generator of C if and only if S 00 .X / is a generator of Dop , and if and only if S.X / is a cogenerator of D. Similarly, X is a cogenerator of C if and only if S 00 .X / is a cogenerator of Dop , and if and only if S.X/ is a generator of D. Lemma 7.8. Let S W C ! D and T W D ! C define a duality of K-categories. Then, for any objects X of C and Y of D, there is a K-linear isomorphism Y;X

W HomD .Y; S.X // ! HomC .X; T .Y //;

which is natural in X and Y . Proof. Recall that the naturality of

Y;X

in X and Y means by definition that the


179

diagram HomD .Y 0 ; S.X 0 //

Y 0 ;X 0

Hom.u;T .v//

Hom.v;S.u//

HomD .Y; S.X //

/ HomC .X 0 ; T .Y 0 //

Y;X

/ HomC .X; T .Y //

is commutative for all morphisms u W X ! X 0 of C and v W Y ! Y 0 of D. Now consider the equivalences S 00 W C ! Dop and T 0 W Dop ! C, associated to S and T . Then, by Lemma 6.2, the pair of functors hS 00 ; T 0 i is an adjunction. Hence there is a K-linear isomorphism 'X;Y W HomDop .S 00 .X /; Y / ! HomC .X; T 0 .Y //; for any objects X of C and Y of D , which is natural in X and Y . Since by definition S 00 .X/ D S.X /, T 0 .Y / D T .Y / and HomD .Y; S.X // D HomDop .S 00 .X /; Y /, we have the K-linear isomorphism Y;X

W HomD .Y; S.X // ! HomC .X; T .Y //

such that Y;X .f / D 'X;Y .f op / for all f 2 HomD .Y; S.X //, and its naturality in X and Y is a direct translation of the naturality of 'X;Y in X and Y . (Deduce the above commutative diagram from the corresponding commutative diagram for 'X;Y .) Applying Lemma 7.8 to the standard duality HomK .; K/ between mod A and mod Aop for a finite dimensional K-algebra A, we have the following lemma. Lemma 7.9. Let A be a finite dimensional K-algebra. Then, for any modules X in mod A and Y in mod Aop , there is a K-linear isomorphism HomA .X; HomK .Y; K// ! HomAop .Y; HomK .X; K//; which is natural in X and Y . The next fact shows that the standard duality D D HomK .; K/ preserves the double centralizer property of a module. Lemma 7.10. Let A and B be finite dimensional K-algebras, and M an (A; B)bimodule with the double centralizer property. Then the (B; A)-bimodule D.M / D HomK .M; K/ has the double centralizer property. Proof. By the double centralizer property of M we have isomorphisms of Kalgebras ˛ W A ! EndB .M /; ˇ W B ! EndAop .M /op

180


such that ˛.a/.m/ D am and ˇ.b/.m/ D mb for all a 2 A; b 2 B and m 2 M . Then, by the duality D D HomK .; K/ between mod A and mod Aop , we have the composed K-algebra isomorphism ˇ

D

! EndAop .M /op ! EndA .HomK .M; K// W B such that .b/.u/ D bu for all b 2 B; u 2 HomK .M; K/. Indeed, for b 2 B and u 2 HomK .M; K/, we have .b/.u/ D .D.ˇ.b///.u/ D D.bR /.u/ D Hom.bR ; K/.u/ D ubR , where bR 2 EndA .M / is the right multiplication by b 2 B, and hence . .b/.u//.m/ D .ubR /.m/ D u.mb/ D .bu/.m/ for all m 2 M , which implies that .b/.u/ D bu. Similarly, we have the composed K-algebra isomorphism ˛

D

0 W A ! EndB .M / ! EndB op .HomK .M; K//op such that 0 .a/.u/ D ua for all a 2 A; u 2 HomK .M; K/.

A typical example of dualities is provided by the correspondence of the form XR 7! S HomR .X; U /, for a bimodule S UR over rings R; S. We have already used often the duality X 7! HomK .X; K/ for finite dimensional K-vector spaces, that is, K-modules. The duality by the correspondence X 7! X D HomA .X; A/, that is, X Š X , for all finite dimensional modules X over a finite dimensional selfinjective algebra A, was developed mainly by T. Nakayama [Nak2], M. Hall [Hal], and M. Ikeda [Ike]. The module correspondence over a selfinjective algebra was slightly generalized by H. Tachikawa [Tac] to the correspondence X 7! HomA .X; U / by an injective cogenerator U over an algebra A. In 1958 K. Morita finally established the functorial duality theory for categories of modules over rings which includes the dualities given by selfinjective algebras, the Pontrjagin duality, and many other known dualities appeared in mathematics (see for example T.Y. Lam [Lam]), and the same duality theory was also obtained independently by G. Azumaya [Azu2]. Hence we call the functorial duality the Morita–Azumaya duality. The duality theorems we state in this chapter are restricted to the finite dimensional algebras, but the essence of the general duality theorems will be visible for these algebras. The following is the first half of the Morita–Azumaya duality theorem which asserts that an injective cogenerator defines a duality. Theorem 7.11. Let A be a finite dimensional K-algebra, Q be a finite dimensional right A-module and B D EndA .Q/. Assume that Q is an injective cogenerator of mod A. Then the pair of contravariant functors HomA .; Q/ W mod A ! mod B op ; HomB op .; Q/ W mod B op ! mod A define a duality between mod A and mod B op .


181

Proof. Observe first that Q is a .B; A/-bimodule. Let D D HomK .; K/ be the standard duality between mod A and mod Aop , D0 D HomK .; K/ be the standard duality between mod B and mod B op , and let D1 D HomA .; Q/ and D2 D HomB op .; Q/. Then we have the functors mod Aop o

D D

/

mod A o

D1 D2

/

mod B op o

D0 D0

/

mod B:

Let P D D.Q/. Then P is a projective generator of mod Aop , because Q is an injective cogenerator of mod A, and B Š EndAop .P /op (by Proposition 7.5 and Lemma 7.10). By the Morita equivalence theorem (Theorems 6.7 and 6.8) the (A; B)-bimodule P induces a pair of equivalence functors F D ˝A P W mod A ! mod B and G D HomB .P; / W mod B ! mod A. We show the existence of natural isomorphisms of functors D2 D1 Š 1mod A

and

D1 D2 Š 1mod B op :

First we claim that D0 D1 Š F . For all modules X in mod A, there are natural (in X) isomorphisms of right B-modules D0 D1 .X / Š HomK .HomA .X; Q/; K/ Š HomK .HomA .X; HomK .P; K//; K/ Š HomK .HomK .X ˝A P; K/; K/ Š X ˝A P D F .X /; where Q Š D.P / and the third isomorphism is by Theorem 4.3, applied to the .A; K/-bimodule P . Thus we have a natural isomorphism of functors D0 D1 Š F . Next we show that D2 D0 Š G. For all modules Y in mod B, there are natural (in Y ) isomorphisms of right A-modules D2 D0 .Y / Š HomB op .D0 .Y /; Q/ Š HomB op .D0 .Y /; D.P // Š HomB op .D0 .Y /; D0 .PB // Š HomB .P; Y / D G.Y /: Thus we have a natural isomorphism D2 D0 Š G of functors. Combining the above isomorphisms, we obtain natural isomorphisms of functors .D2 D0 /.D0 D1 / Š GF Š 1mod A and .D0 D1 /.D2 D0 / Š F G Š 1mod B . The first isomorphism then implies natural isomorphisms of functors D2 D1 Š D2 .D0 D0 /D1 Š 1mod A , where D0 D0 is naturally isomorphic to the identity functor 1mod B op on mod B op . From the second isomorphism, we have natural isomorphisms of functors D1 D2 Š D0 .D0 D1 D2 D0 /D0 Š D0 1mod B D0 Š D0 D0 Š 1mod B op : Therefore, we have natural isomorphisms of functors D2 D1 Š 1mod A and D1 D2 Š 1mod B op , and we conclude that D1 and D2 define a duality between mod A and mod B op .

182


The next theorem is the second half of the Morita–Azumaya duality theorem which characterizes the duality by an injective cogenerator. Theorem 7.12. Let A and B be finite dimensional K-algebras. Assume that a pair of contravariant functors D1 W mod A ! mod B op and D2 W mod B op ! mod A defines a duality. Then there is a (B; A)-bimodule Q such that Q satisfies the double centralizer property, is an injective cogenerator of mod A and of mod B op , and there are natural isomorphisms of functors D1 Š HomA .; Q/ and D2 Š HomB op .; Q/: Proof. Let D0 D HomK .;K/ be the standard duality between mod B and mod B op. Then the pair of functors D0 D1 W mod A ! mod B and D2 D0 W mod B ! mod A defines an equivalence between mod A and mod B. It follows from the Morita equivalence Theorem 6.8 (and its proof) that P D D0 D1 .A/ is an (A; B)-bimodule, P is a progenerator of mod B, and there are natural isomorphisms of functors D0 D1 Š ˝A P and D2 D0 Š HomB .P; /. Hence, for any module X of mod A, there is an isomorphism of right B-modules .D0 D1 /.X / Š X ˝A P , natural in X, and so we obtain isomorphisms of left B-modules D1 .X / Š HomK .X ˝A P; K/ Š HomA .X; D0 .P //; where the second isomorphism is by the adjoint theorem (Theorem 4.3) for the .A; K/-bimodule P . Thus D1 is naturally isomorphic to HomA .; D0 .P //. On the other hand, for any module Y of mod B op , we have isomorphisms of right A-modules D2 .Y / Š D2 .D0 D0 /.Y / Š.D2 D0 /.D0 .Y // Š HomB .P; D0 .Y // Š HomB op .Y; D0 .P //; where the last isomorphism follows by Lemma 7.9. Therefore, we obtain an isomorphism of right A-modules D2 .Y / Š HomB op .Y; D0 .P //; which is natural in Y . Hence D2 is naturally isomorphic to HomB op .; D0 .P //. Finally, observe that, since the (A; B)-bimodule P is a progenerator of mod B, P has the double centralizer property, by Theorem 5.10. Then the (B; A)-bimodule D0 .P / has the double centralizer property, by Lemma 7.10, and is an injective cogenerator of mod A and of mod B op , because it is the dual of A PB . Thus letting Q D D0 .P / we complete the proof.


183

By Theorems 7.11 and 7.12 we know that a Morita–Azumaya duality between mod A and mod B op is exactly determined by a finite dimensional (B; A)-bimodule Q satisfying the equivalent conditions presented in Proposition 7.5. We call such a bimodule Q a duality module. Proposition 7.13. Let A and B be finite dimensional K-algebras. Then there is a 1-1 correspondence between the isomorphism classes of dualities D from mod A to mod B op and isomorphism classes of duality (B; A)-bimodules Q, via natural isomorphisms of functors D Š HomA .; Q/ from mod A to mod B op . Proof. It follows from Lemma 7.10 that there is a 1-1 correspondence between the isomorphism classes of duality (B; A)-bimodules Q and the isomorphism classes of (A; B)-bimodules P such that P is a progenerator of mod Aop with the double centralizer property, via the isomorphism P Š HomK .Q; K/ of (A; B)-bimodules. Then, by Propositions 6.9 and 6.11, the isomorphism classes of such (A; B)bimodules P are in 1-1 correspondence with the isomorphism classes of Morita equivalences F W mod Aop ! mod B op , via natural isomorphisms of functors F Š HomAop .P; / from mod Aop to mod B op . Further, as we noticed above, the isomorphism classes of such Morita equivalences F are in 1-1 correspondence with the isomorphism classes of dualities D from mod A to mod B op , via the natural isomorphisms of functors D Š FD0 from mod A to mod B op , where D0 D HomK .; K/ is the standard duality from mod A to mod Aop . Applying now Lemma 7.9, we obtain the natural isomorphisms of functors D Š HomAop .P; / HomK .; K/ Š HomA .; HomK .P; K// Š HomA .; Q/ which completes the proof.

A duality between mod A and mod Aop is called a selfduality. For contravariant functors D1 W mod A ! mod Aop and D2 W mod Aop ! mod A defining a selfduality, both functors D1 and D2 are isomorphic to hom functors of the forms HomA .; Q/ and HomAop .; Q/, respectively, where Q is an injective cogenerator of mod A with A Š EndA .QA / and of mod Aop with A Š EndAop .A Q/op . Thus, unless no confusion occurs, we write the same notation, say D, for those contravariant functors D1 ; D2 , and also write D 2 for D2 D1 and D1 D2 . Then D 2 Š 1mod A and D 2 Š 1mod Aop . In this case, the injective cogenerator Q is isomorphic to D.A/ as an A-bimodule, and D.A/ is a selfduality module. Therefore, the A-bimodule isomorphism classes of selfduality modules are in 1-1 correspondence with the natural isomorphism classes of selfdualities. Notice that by Corollary 7.4 a selfduality module over a basic K-algebra A is nothing else than a minimal injective cogenerator of mod A or equivalently mod Aop . Corollary 7.14. For a finite dimensional K-algebra A, the standard duality D D HomK .; K/ between mod A and mod Aop is a selfduality for mod A, and its duality module Q is isomorphic to the A-bimodule HomK .A; K/.

184


Proof. This is a consequence of the following natural isomorphisms in mod Aop , for any module X in mod A, HomA .X; HomK .A; K// Š HomK .X ˝A A; K/ Š HomK .X; K/; where the first isomorphism follows by Theorem 4.3 for the .A; K/-bimodule A, and the second follows by Lemma 3.5. We shall consider the problem of determining selfduality modules over a finite dimensional basic K-algebra. Let A be a finite dimensional basic K-algebra, and Q a selfduality A-module. Since both Q and D.A/ are minimal injective cogenerators of mod Aop , we have an isomorphism A Q Š A D.A/ of left A-modules and isomorphisms of K-algebras EndAop .A Q/op Š A

and

EndAop .A D.A//op Š A:

Thus the selfduality A-modules are different only in the right operation of the elements of A. In order to describe the difference, we need automorphisms of an algebra. For a finite dimensional K-algebra A, let Aut.A/ be the group of all K-algebra automorphisms of A and Inn.A/ the subgroup of Aut.A/ of inner automorphisms. Here, an automorphism ˛ of A is said to be inner if there is an invertible element c of A such that ˛.x/ D cxc 1 for all x 2 A. Moreover, Inn.A/ is called the inner automorphism group of A. An element of Aut.A/ n Inn.A/ is called an outer automorphism of A and the factor group Aut.A/= Inn.A/ is called the outer automorphism group of A, and is denoted by Out.A/. For ˛ 2 Aut.A/, we denote by ˛Q its coset ˛ C Inn.A/ in Out.A/. We note that ˛ Inn.A/ D Inn.A/˛ for all ˛ 2 Aut.A/ (see Exercise 8.52). Let M be a right A-module and ˛ 2 Aut.A/. By M˛ we denote the right A-module such that M˛ D M as K-vector space and the right operation of an element a of A on M˛ is defined by xa D x˛.a/ for a 2 A and x 2 M: Similarly, ˛ N is defined for a left A-module N and an element ˛ 2 Aut.A/. Lemma 7.15. Let A be a finite dimensional K-algebra, D D HomK .; K/ the standard duality, and ˛, ˇ be elements of Aut.A/. The following conditions are equivalent. (i) D.A/˛ Š D.A/ˇ as A-bimodules. (ii)

˛A

Š ˇ A as A-bimodules.

Q (iii) ˛Q D ˇ.


185

Proof. Observe first that D.A/˛ D D.˛ A/ and D.A/ˇ D D.ˇ A/, and hence (i) is equivalent to (ii). We will show that (ii) and (iii) are also equivalent. Notice that, for any isomorphism W AA ! AA of right A-modules, the image .1A / of 1A is an invertible element of A. Indeed, 1A D 1 .1A / D 1 .1A .1A // D

1 .1A / .1A / and 1A D

1 .1A / D .1A 1 .1A // D .1A / 1 .1A /. Assume that there is an A-bimodule isomorphism W ˛ A ! ˇ A. Then we have the equalities

.˛.a// D .1A ˛.a// D .1A /˛.a/;

.˛.a// D .a 1A / D a .1A / D ˇ.a/ .1A /; for all a 2 A. Hence we obtain that ˇ.a/ D .1A /˛.a/ .1A /1 for all a 2 A. Take the inner automorphism of A such that .a/ D .1A /a .1A /1 for all a 2 A. Q Then we have ˇ D ˛ and consequently ˛Q D ˇ. Q Conversely, assume that ˛Q D ˇ, and let 2 Inn.A/ be such that ˇ D ˛ and .a/ D cac 1 for an invertible element c of A and all a 2 A. Note that then ˇ.a/c D c˛.a/ for all a 2 A. Now consider the mapping W ˛ A ! ˇ A defined by

.a/ D ca for all a 2 A. We claim that is an A-bimodule isomorphism. Indeed, this follows by the equalities

.˛.a/ba0 / D c˛.a/ba0 D ˇ.a/cba0 D ˇ.a/ .b/a0 ; that is, .a ba0 / D a .b/a0 for all a; b; a0 2 A.

Proposition 7.16. Let A be a finite dimensional basic K-algebra. Then there is a 1-1 correspondence between the isomorphism classes of selfduality modules Q and the outer automorphisms classes ˛Q 2 Out.A/ of ˛ 2 Aut.A/, given by the correspondence Q 7! ˛, Q satisfying Q Š D.A/˛ as A-bimodules, where D D HomK .; K/. Proof. By Lemma 7.15, there is a 1-1 correspondence between the isomorphism classes of A-bimodules D.A/˛ , ˛ 2 Aut.A/, and the elements ˛Q of the outer automorphism group Out.A/ of A. Hence, it suffices to show that a selfduality module Q is isomorphic to an A-bimodule D.A/˛ , for some ˛ 2 Aut.A/. Let ! W A Q ! A D.A/ be an isomorphism of left A-modules. If we can show that there is an automorphism ˛ of A such that !.xa/ D !.x/˛.a/ for all x 2 Q and a 2 A, then ! gives an A-bimodule isomorphism from Q to D.A/˛ . Thus we claim that such an automorphism ˛ of A exists. For any a 2 A, consider the homomorphism of left A-modules

a D !aR ! 1 W A D.A/ ! A D.A/;

186


where aR W A Q ! A Q is the right multiplication by a. Therefore the following diagram in mod Aop is commutative: aR

AQ !

A D.A/

/ AQ !

a

/ A D.A/ .

Since D induces a duality between mod A and mod Aop , there exists a unique element ˛.a/ of A such that a D Hom.˛.a/L ; K/, where ˛.a/L W AA ! AA is the left multiplication by ˛.a/. We claim that the induced map ˛ W A ! A is an algebra automorphism of A. Observe that, for all a; b 2 A, we have the commutative diagram AQ !

A D.A/

aR

/ AQ !

/ A D.A/

D.˛.a/L /

bR

/ AQ !

/ A D.A/

D.˛.b/L /

in mod Aop , and hence the equalities D.˛.ab/L / D ! .ab/R ! 1 D ! .bR aR / ! 1 D .! bR ! 1 /.! aR ! 1 / D D.˛.b/L /D.˛.a/L / D D.˛.a/L ˛.b/L / D D .˛.a/˛.b//L ; which implies that ˛.ab/L D .˛.a/˛.b//L , and so ˛.ab/ D ˛.a/˛.b/. Obviously ˛ W A ! A is a K-linear homomorphism. Therefore, indeed ˛ 2 Aut.A/. The following basic fact is obtained as an application of facts on duality modules. Lemma 7.17. Let L be a finite extension field of K and A be a finite dimensional L-algebra. Then the following statements hold. (i) HomK .; K/ and HomL .; L/ are naturally isomorphic as contravariant functors on mod L. (ii) HomK .A; K/ Š HomL .A; L/ as A-bimodules. Proof. (i) Let D D HomL .; L/ be the selfduality for mod L, where L is regarded as a K-algebra. Then, by Proposition 7.16, there exists an automorphism of the K-algebra L such that there is an isomorphism W D.L/ ! HomK .L; K/ of L-bimodules. We shall prove the assertion by showing that is the identity automorphism of L.


187

Let ' W L ! HomK .L; K/ be the composite 0

' W L ! HomL .L; L/ ! HomK .L; K/ ; where 0 W L ! HomL .L; L/ is the canonical L-bimodule isomorphism such that 0 .x/.y/ D xy for all x; y 2 L. Since ' is an L-bimodule isomorphism, '.1L / is nonzero and satisfies a'.1L / D '.1L / .a/ for all a 2 L. Indeed, for a 2 L, '.a/ D '.a1L / D a'.1L / as ' is a left L-homomorphism, and '.a/ D '.1L a/ D '.1L / a D '.1L / .a/ as ' is a right L-homomorphism. Letting f D '.1L /, we then have 0 D .af f .a//.x/ D f .xa .a/x/ D f ..a .a//x/ for all x 2 L, because xa D ax in L. Hence f ..a .a//L/ D 0. Now suppose that .a/ ¤ a for some element a 2 L. Then .a .a//L D L because L is a field. Hence f .L/ D 0, that is, f D 0 as a homomorphism from L to K. This however contradicts that '.1L / ¤ 0. Thus .a/ D a for all a 2 L, or equivalently, is the identity automorphism idL of L. Therefore, W D.L/ ! HomK .L; K/ is an isomorphism of L-bimodules, and hence we obtain the natural isomorphism of functors D. / W HomL .; D.L// ! HomL .; HomK .L; K//: Since HomL .L; D.L// Š D.L/ and HomL .L; HomK .L; K// Š HomK .L; K/ as L-bimodules, applying Theorem 7.12, we conclude that we have natural isomorphisms of functors HomL .; L/ Š HomL .; D.L// and HomK .; K/ Š HomL .; HomK .L; K//: Hence the functors HomK .; K/ and HomL .; L/ from mod L to mod Lop are naturally isomorphic. (ii) Since L is commutative, by (i) there is a natural isomorphism of functors W HomK .; K/ ! HomL .; L/ from mod L to mod L D mod Lop . Hence, for every module X in mod L, we have a natural isomorphism X W HomK .X; K/ ! HomL .X; L/ in mod L. Then A W HomK .A; K/ ! HomL .A; L/ is an L-linear isomorphism. We shall show that A is a homomorphism of A-bimodules. Let aR W A ! A be the right multiplication by a 2 A. Then, by the naturality of , we have in mod L the commutative diagram HomK .A; K/

A

HomK .aR ;K/

HomK .A; K/

/ HomL .A; L/

A

HomL .aR ;L/

/ HomL .A; L/ .

188


Take u 2 HomK .A; K/. Then we have the equalities A .uaR / D A HomK .aR ; K/ .u/ D HomK .aR ; L/A .u/ D A .u/aR ; where, for x 2 A, .uaR /.x/ D u.aR .x// D u.xa/ D .au/.x/, and hence uaR D au. Observe also that, for any x 2 A, we have the equalities .A .u/aR / .x/ D A .u/ .aR .x// D A .u/.xa/ D .aA .u// .x/; and hence A .u/aR D aA .u/. Therefore A .au/ D aA .u/. Similarly, one proves that A .ua/ D A .u/a for all a 2 A. Hence we obtain A .a0 xa/ D a0 A .xa/ D a0 A .x/a for all a; a0 ; x 2 A, which shows that A is an A-bimodule homomorphism.

8 Exercises In all exercises below K will denote a field. 1. Let F and G be finite dimensional division K-algebras, F MG an .F; G/bimodule, and assume that K acts centrally on F MG and dimK F MG < 1. Prove that ² ³ a 0 ˇˇ G 0 D a 2 G; b 2 F; m 2 F MG AD F m b F MG is a finite dimensional hereditary K-algebra. 2. Let A be a K-algebra, X; X1 ; : : : ; Xm be right A-modules and Y; Y1 ; : : : ; Ym be left A-modules. Prove that there are isomorphisms of K-vector spaces m M

m M .Xi ˝A Y / Xi ˝A Y !

iD1

and X ˝A

iD1 n M j D1

Yj

!

n M

X ˝A Yj :

j D1

3. Let V1 ; V2 ; : : : ; Vn , n 2, be finite dimensional K-vector spaces. Show that dimK .V1 ˝K V2 ˝K ˝K Vn / D .dimK V1 /.dimK V2 / : : : .dimK Vn /: 4. Let V be a K-vector space and TK .V / the tensor algebra of V over K. Prove that TK .V / has the following universal property: for any K-algebra A and a K-linear homomorphism f W V ! A there is a unique homomorphism ' W TK .V / ! A of K-algebras such that f is the restriction of ' to V D TK1 .V /.

8. Exercises

189

5. Let V be a K-vector space and SK .V / the symmetric algebra of V over K. Prove that SK .V / has the following universal property: for any commutative K-algebra A and a K-linear homomorphism f W V ! A there is a unique homomorphism W SK .V / ! A of K-algebras such that f is the restriction of to V . 6. Let V be a K-vector space with dimK V D n. Prove that (a) the K-algebra TK .V / is isomorphic to the polynomial algebra KhX1 ; : : : ; Xn i of n noncommuting variables X1 ; : : : ; Xn over K; (b) the K-algebra SK .V / is isomorphic to the polynomial algebra KŒX1 ; : : : ; Xn of n commuting variables X1 ; : : : ; Xn over K. 7. Let A and B be K-algebras and M be an .A; B/-bimodule. Prove that the induced hom functors HomB .; M / W Mod B ! Mod Aop ; HomAop .M; / W Mod Aop ! Mod B op ; HomAop .; M / W Mod Aop ! Mod B are left exact. 8. Let A and B be finite dimensional K-algebras and M be an .A; B/-bimodule of finite dimension over K. Prove the following statements. (a) If A M is a projective module in mod Aop , then the induced functor ˝A M W mod A ! mod B is exact. (b) If MB is a projective module in mod B, then the induced functor M ˝B W mod B op ! mod Aop is exact. 9. Let A and B be finite dimensional K-algebras and T W mod A ! mod B a right exact K-linear functor. Prove the following statements. (a) T .A/ is an .A; B/-bimodule. (b) There exists a natural isomorphism of functors T Š ˝A T .A/ from mod A to mod B. (c) If T .A/ is a free left A-module, then the functor T is faithful.

190


10. Let A be a finite dimensional K-algebra, e an idempotent of A and B D eAe. Consider the functors rese D ./e W mod A ! mod B; Te D ˝B eA W mod B ! mod A; Le D HomB .Ae; / W mod B ! mod A: Prove that (a) Te and Le are full and faithful K-linear functors such that rese Te Š 1mod B and rese Le Š 1mod B ; (b) Te is left adjoint to rese ; (c) Le is right adjoint to rese ; (d) rese is an exact functor; (e) Te .M / and Le .M / are indecomposable modules in mod A for any indecomposable module M in mod B; (f) Te .P / is a projective module in mod A for any projective module P in mod B; (g) Te .I / is an injective module in mod A for any injective module I in mod B. 11. Let A D A1 An be the direct product of K-algebras A1 ; : : : ; An . Let ei be the element of A whose i-th component is the identity of Ai and the other components are zero. Show the following statements. (a) Ai Š ei Aei as K-algebras. (b) There are equivalences of categories Mod A Š Mod A1 Mod An ; mod A Š mod A1 mod An : 12. Let A and B be finite dimensional K-algebras and D D HomK .; K/. Show the following statements. (a) The K-algebras A ˝K B and B ˝K A are isomorphic. (b) The K-algebras .A ˝K B/op and Aop ˝K B op are isomorphic. (c) D.A ˝K B/ Š D.A/ ˝K D.B/ as .B; A/-bimodules. (d) D.A ˝K A/ Š D.A/ ˝K D.A/ as A-bimodules.

8. Exercises

191

13. Let m, n be positive integers. Prove that the K-algebras Mm .K/ ˝K Mn .K/ and Mmn .K/ are isomorphic. 14. Let A1 ; : :Q : ; Ar and B1 ; : :Q : ; Bs be finite dimensional Qr Qs K-algebras. Prove that the r s K-algebras A B ˝ and i K j iD1 j D1 iD1 j D1 Ai ˝K Bj are isomorphic. 15. Let G and H be finite groups. Prove that the group algebra K.G H / of G H over K is isomorphic to the tensor K-algebra KG ˝K KH of the group algebras KG and KH of G and H over K. 16. Let A and B be finite dimensional K-algebras such that A˝K B is a semisimple K-algebra. Prove that A and B are semisimple K-algebras. 17. Let A and B be finite dimensional K-algebras over an algebraically closed field K. Describe (a) the Jacobson radical rad.A ˝K B/ of A ˝K B; (b) the semisimple K-algebra .A ˝K B/= rad.A ˝K B/. 18. Let A and B be finite dimensional K-algebras over an algebraically closed field K. Describe (a) the simple modules in mod A ˝K B; (b) the indecomposable projective modules in mod A ˝K B; (c) the indecomposable injective modules in mod A ˝K B. 19. Let A and B be finite dimensional K-algebras over an algebraically closed field K. Prove that A ˝K B is a semisimple K-algebra if and only if A and B are semisimple K-algebras. 20. Let A and B be finite dimensional K-algebras over an algebraically closed field K. Prove that A ˝K B is a basic K-algebra if and only if A and B are basic K-algebras. 21. Let A, B, C , D be finite dimensional K-algebras over an algebraically closed field K such that A is Morita equivalent to C and B is Morita equivalent to D. Prove that the K-algebras A ˝K B and C ˝K D are Morita equivalent. 22. Let A be a finite dimensional K-algebra and Mn .A/ the K-algebra of all square n n matrices with coefficients in A. Prove that Mn .A/ is Morita equivalent to A. 23. Let A be a finite dimensional K-algebra. For positive integers r and s, consider the K-algebra ² ˇ ³ 0 X 0 ˇ X 2 Ms .A/; Y 2 Mr .A/; Ms .A/ D ; ˇ Mrs .A/ Mr .A/ Z 2 Mrs .A/ Z Y

192


where Mrs .A/ is the .Mr .A/; Ms .A//-bimodule consisting of all r s matrices with coefficients in A, ² ³ A 0 a 0 ˇˇ T2 .A/ D D a; b; c 2 A A A c b the K-algebra ofh 2 2 lower triangular matrices with coefficients in A. Prove that i 0 s .A/ the K-algebras MMrs and T 2 .A/ are Morita equivalent. .A/ Mr .A/ 24. Let A be a finite dimensional K-algebra . Prove that the category mod T2 .A/ of finite dimensional right modules over the K-algebra T2 .A/ of 2 2 lower triangular matrices over A is equivalent to the category hom.mod A/ whose objects are homomorphisms f W M ! N in mod A, and a morphism from f W M ! N to g W X ! Y in hom.mod A/ is a pair .u; v/ consisting of homomorphisms u W M ! X and v W N ! Y in mod A such that vf D gu. 25. Let K be a field, F and G finite dimensional division K-algebras, F MG an .F; G/-bimodule, and assume that K acts centrally on F MG and dimK F MG < 1. Consider the finite dimensional K-algebra ² ³ G 0 a 0 ˇˇ AD D a 2 G; b 2 F; m 2 F MG : F m b F MG Denote by r.F MG / the category whose objects are all triples .X; Y; '/ with X a module in mod F , Y a module in mod G, and ' W X ˝F F MG ! Y a homomorphism in mod G, and morphisms .X; Y; '/ ! .X 0 ; Y 0 ; ' 0 / in r.F MG / consist of pairs .f; g/, with f W X ! X 0 a homomorphism in mod F and g W Y ! Y 0 a homomorphism in mod G, such that the following diagram in mod G X ˝F X 0 ˝F

F MG

'

f ˝F F MG F MG

/Y g

'0

/ Y0

is commutative. Prove that the categories mod A and r.F MG / are equivalent. 26. Let A and B be Morita equivalent finite dimensional K-algebras. Prove that A is semisimple if and only if B is semisimple. 27. Let A and B be Morita equivalent finite dimensional K-algebras. Prove that A is a hereditary algebra if and only if B is a hereditary algebra. 28. Let A and B be Morita equivalent finite dimensional K-algebras. Prove that A is a Nakayama algebra if and only if B is a Nakayama algebra.

8. Exercises

193

29. Let A and B be K-algebras and M be an (A; B)-bimodule. Prove that A M has the double centralizer property if and only if MB has the double centralizer property. 30. Let A D KQ=I be the bound quiver K-algebra defined by the quiver Q of the form ˛ / / 3 1 o 2 ˇ

and the ideal I of KQ generated by ˛ˇ. Let M1 ; : : : ; M5 be the right A-modules defined as follows N M1 D e1 A; M2 D e2 A=N A; M3 D e2 A=ˇA; M4 D top.e2 A/; M5 D e3 A; where each ei is the primitive idempotent of A corresponding to the vertex i of Q, N D C I 2 A and ˇN D ˇ C I 2 A. Let M D M1 ˚ ˚ M5 . (a) Show that M is a faithful right A-module but not a generator of mod A. (b) Show that there exists an exact sequence in mod A of the form 0 ! A ! M m ! M n ; for some positive integers m and n. (c) Show that the endomorphism algebra B D EndA .M / is isomorphic to the bound quiver algebra K =J , where is the quiver 1 @ 2 =^ == == == 4 ˛

== ==ˇ == = @3 ı

"

/5

and J is the ideal of K generated by ˛ and ı", where the vertices 1; : : : ; 5 correspond to the idempotents f1 ; : : : ; f5 of B such that each fi is the composition fi W M ! Mi ! M of the canonical projection and canonical injection for the direct sum decomposition M D M1 ˚ ˚ M5 . (d) Let 1 W Bf4 ! Bf2 and 2 W Bf4 ! Bf3 be the canonical embeddings of the simple left B-module Bf4 into Bf2 and Bf3 , respectively. Let N be the left B-submodule of Bf2 ˚ Bf3 such that N D f. 1 .bf4 /; 2 .bf4 // 2 Bf2 ˚ Bf3 j b 2 Bg :

194


Show that the left B-module M is isomorphic to the direct sum of three indecomposable left B-modules Bf1 ˚ .Bf2 ˚ Bf3 /=N ˚ Bf5 :

(e) Show that EndB op .M /op is isomorphic to A, that is, MA has the double centralizer property. 31. A characterization for a module to have the double centralizer property was proved by Y. Suzuki [Suz]. We shall prove it in the following way. Let A be a finite dimensional K-algebra and M be a finite dimensional right A-module. Let B D EndA .M / and C D EndB op .M /op . Consider the following two conditions. (DC1) The canonical mapping ' W B HomA .C CA ; B MA /C ! B MC ;

f 7! f .1C /;

is a .B; C /-bimodule isomorphism. (DC2) There exists an exact sequence in mod A of the form v

u

0 ! A ! M m ! M n ; for some nonnegative integers m, n. (a) Assume that MA satisfies both conditions (DC1) and (DC2). (i) Show that there exists a homomorphism j W C ! A of left A-modules j

i

A is the identity homomorphism of such that the composite A ,! C ! A, where i is the inclusion map. Hint: Consider the following diagram 0

/ HomA .C; A/

/ HomA .C; M m /

Hom.C;u/

Hom.C;v/

o

o

HomA .C; M /m

/ HomA .C; M /n

'm

0

/A

u

/ Mm

/ HomA .C; M n /

'n

v

/ M n:

(ii) Show that the inclusion map i W A ,! C is surjective, that is, M has the double centralizer property.

8. Exercises

195

(b) Assume that MA has the double centralizer property. Show that M satisfies both conditions (DC1) and (DC2). Hint: Apply the contravariant functor HomB .; B M / to an exact sequence of the form B B n ! B B m ! B M ! 0, for some nonnegative integers m, n. 32. Consider the K-subalgebra 82 9 3 1 0 0 0 ˆ > ˆ > ˆ > : ; 7 6 3 2 of the matrix algebra M4 .K/. Show the following: (a) There are orthogonal primitive idempotents e1 , e2 of A such that 1A D e1 C e2 , dimK .e1 A/ D 3 D `.e1 A/ and dimK .e2 A/ D 4 D `.e2 A/. (b) e2 A is injective and faithful in mod A. (c) There is no short exact sequence in mod A of the form 0 ! e1 A ! Am ! An for positive integers m and n. (d) e1 A does not satisfy the double centralizer property. (e) A is isomorphic to a bound quiver K-algebra KQ=I , for a finite quiver Q and an admissible ideal I of KQ. 33. Consider the K-subalgebras 82 9 3 a x z 0 0 0 ˆ > ˆ > ˆ > ˆ6y b w 0 0 0 7 > ˆ > ˆ 6 7 > ˆ 60 0 0 c u v7 > ˆ > ˆ > 40 0 0 0 a x5 ˆ > ˆ > : ; 0 0 0 0 y b 82 a x ˆ ˆ > ˇ = 0 07 ˇ 7 ˇ a; b; x; y 2 K b y5 > > ; 0 a

196


of the matrix algebras M6 .K/ and M4 .K/, respectively. Moreover, be the following elements of A: 2 3 2 3 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 60 0 0 0 0 07 60 1 0 0 0 0 7 60 0 6 7 6 7 6 60 0 0 0 0 07 60 0 0 0 0 0 7 60 0 6 6 7 7 e1 D 6 ; e2 D 6 ; e3 D 6 7 7 60 0 60 0 0 0 0 07 60 0 0 0 0 0 7 6 40 0 0 0 1 05 40 0 0 0 0 0 5 40 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0

let e1 , e2 , e3 0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 0 0

3 0 07 7 07 7: 07 7 05 0

Prove that (a) e1 , e2 , e3 are orthogonal primitive idempotents of A and 1A D e1 C e2 C e3 ; (b) dimK .ei A/ D 3 and `.ei A/ D 2 for any i 2 f1; 2; 3g; (c) e1 A Š e2 A and e1 A © e3 A in mod A; (d) A and B are Morita equivalent; (e) the basic algebra Ab of A (equivalently, the basic algebra B b of B) is isomorphic to a bound quiver algebra KQ=I , for a finite quiver Q and an admissible ideal I of KQ. 34. Let Q be the quiver ˛1 ˛2 ˛3 ˛4 ˛5 ˛6 o o o o o o ; 1 2 3 4 5 6 7 I the ideal in KQ generated by ˛3 ˛2 ˛1 , ˛4 ˛3 ˛2 , and ˛6 ˛5 ˛4 , and A D KQ=I the associated bound quiver algebra. Consider the right A-module

M D e1 A ˚ e3 A ˚ e4 A ˚ e5 A ˚ e7 A and its endomorphism algebra ƒ D EndA .M /. Prove that ƒ is isomorphic to the bound quiver algebra K =J , where is the quiver ˇ1 ˇ2 ˇ3 ˇ4 o o o o 3 4 5 1 2 and J is the ideal in K generated by ˇ2 ˇ1 and ˇ4 ˇ3 .

35. Let Q be the quiver ˛ yyy yy |yy 1 bDD DD D DD

2 5 bEE y yy EEˇ EE y E |yyy bD zz 4 DDD z D zz DD |zz ; 6 3

8. Exercises

197

I the ideal in KQ generated by ˇ˛ , and A D KQ=I the associated bound quiver algebra. Moreover, let ei D "i C I , i 2 f1; 2; 3; 4; 5; 6g, be associated primitive idempotents of A. Consider the right A-module M D e1 A ˚ e2 A ˚ e2 A ˚ e5 A ˚ e5 A ˚ e6 A ˚ e6 A ˚ e6 A and its endomorphism algebra ƒ D EndA .M /. Prove the following statements: (a) A is not a hereditary K-algebra. (b) ƒ is a hereditary K-algebra. (c) The basic algebra ƒb of ƒ is isomorphic to the path algebra K of the quiver of the form 5 yyy y y |yy ˛ bEE 1 o 2 EEE EE : 6 36. Let A be a finite dimensional K-algebra. Prove that the following statements are equivalent. (a) A is a selfinjective K-algebra. (b) HomA .S; A/ is a simple left A-module for any simple right A-module S. (c) HomAop .T; A/ is a simple right A-module for any simple left A-module T . (d) HomA .; A/ and HomAop .; A/ are a pair of dualities between the category of semisimple modules in mod A and the category of semisimple modules in mod Aop . (e) HomA .; A/ and HomAop .; A/ are a pair of dualities between the category of simple right A-modules and the category of simple left A-modules. 37. Let Q be the quiver 2 bEE yy 6 EE ˛ yyy y E y y yyˇ 3 ı EE |yyy |y o o o 7 1 bEE bE EE yy 5 EEE y E EE y EE |yyy % E 8; 4

198


I the ideal in KQ generated by ˛ C ıˇ C %, %, %, % and ı, and A D KQ=I the associated bound quiver algebra. Moreover, let ei D "i C I , i 2 f1; 2; 3; 4; 5; 6; 7; 8g, be the associated primitive idempotents of A. Consider the right A-module M D e1 A ˚ e1 A ˚ e2 A ˚ e3 A ˚ e5 A ˚ e6 A ˚ e7 A ˚ e7 A ˚ e8 A ˚ e8 A and its endomorphism algebra ƒ D EndA .M /. Prove that ƒ is Morita equivalent to the bound quiver algebra B D K =J , where is the quiver ˛ yyy yy y |y 1 bEE EE EE ˇ E

2 bEE yy 6 EE y EE y E |yyy obE 7 yy 5 EEE y E y E y E |yy ı 8 3

and J is the ideal in K generated by ˛ ıˇ, ˛ ıˇ, ı, and ˛. 38. Let A be a finite dimensional K-algebra. Prove that the following statements are equivalent. (a) A is a selfinjective K-algebra. (b) `.MA / D `.A HomA .M; A// for any module M in mod A. (c) `.A N / D `.HomAop .N; A/A / for any module N in mod Aop . 39. Let A and B be K-algebras, and let P be an (A; B)-bimodule such that P is a projective left A-module. (a) Prove Theorem 4.5 (ii). (b) Assume that B D EndAop .P /op . Prove that the following statements are equivalent. (i)

AP

is a progenerator of mod Aop .

(ii) The K-linear map W A P ˝B HomA .P; A/ ! A A defined by .p ˝ u/ D u.p/, for u 2 HomA .P; A/ and p 2 P , is an isomorphism of left Amodules. 40. Let A be a finite dimensional K-algebra and e an idempotent of A. Show that the following statements are equivalent. (a) eA is a progenerator of mod A. (b) AeA D A.

8. Exercises

199

(c) A and eAe are Morita equivalent. 41. Let A and B be finite dimensional K-algebras, and a (B; A)-bimodule P is a progenerator of mod A and of mod B op , and has the double centralizer property. (a) Show that the ordered set (by inclusion) of all A-submodules of P is isomorphic to the ordered set (by inclusion) of all left ideals of B. (b) Show that, under the equivalence functor HomA .P; / W mod A ! mod B determined by P , the (B; A)-subbimodules of P correspond to the two-sided ideals of B. (c) Show that A and B have isomorphic inclusion-ordered sets of two-sided ideals. 42. Let A and B be Morita equivalent K-algebras. Show that the centers C.A/ D fa 2 A j ax D xa for all x 2 Ag and C.B/ D fb 2 B j by D yb for all y 2 Bg of A resp. B are isomorphic as K-algebras. 43. Let A and B be commutative K-algebras. Prove that A and B are Morita equivalent if and only if A and B are isomorphic as K-algebras. 44. Let Q A be a K-algebra, and fXi gi2I a set of right A-modules. The direct product i2I Xi is the Cartesian product with the additive module structure and a right operation of the elements of A defined by the rule .xi /i C .yi /i D .xi C yi /i ; Q

.xi /i a D .xi a/i

for all .xi /i ; .yi /i 2 i2I Xi ; a 2 A, that is, Lwe define the addition and multiplication componentwise. The direct i2I Xi is by definition the right Q sum A-submodule of the right A-module i2I Xi consisting of elements whose almost all components are zero. Now letQfXi gi2I be a set of copies by L of a right A-module X. We denote Q L X and X the direct sum X and the direct product X i i, i2I i2I i2I i2I respectively . Show the following statements for an infinite index set I . L Q (a) The K-vector spaces i2I K and i2I K are not isomorphic. (b) The functor D D HomK .; K/ between Mod A and Mod Aop does not define a duality, that is, it does not hold that DD Š 1Mod A and DD Š 1Mod Aop . 45. Let A be a finite dimensional K-algebra and B a K-subalgebra of A. (a) Let f1 ; : : : ; fr be homomorphisms from AB to BB in mod B, and assume that f1 .a1 /C Cfr .ar / D 1B for Psome elements a1 ; : : : ; ar 2 A. Let ' W A ! B be the map given by '.a/ D riD1 fi .ai a/ for all a 2 A. Show the following statements.

200


(i) ' W A ! B is a homomorphism of right B-modules. (ii) ' is a retraction in mod B. (b) Show that the right B-module AB is a generator of mod B if and only if B is a direct summand of AB as a right B-module. 46. The following fact is a modification of a theorem proved by G. Azumaya [Azu3] where he characterized the rings whose all faithful right modules are generators of the category of all right modules. Let A be a finite dimensional K-algebra. Show that every finite dimensional faithful right A-module is a generator of mod A if and only if the right A-module A is injective. 47. Let A be the R-algebra

R 0 C C

defined in Exercise I.12.33, and let 1 0 0 0 e1 D ; e2 D ; 0 0 0 1

I1 D

0 0 ; R 0

I2 D

0 Ri

0 : 0

(a) Show that I1 and I2 are right ideals of A. (b) Show that the right A-modules e2 A=I1 and e2 A=I2 are isomorphic. (c) Determine a minimal injective cogenerator of mod A and count its composition length and R-dimension. (d) Determine a minimal injective cogenerator of mod Aop and count its composition length and R-dimension. 48. Let A be the R-algebra

R 0 H R

defined in Exercise I.12.34. Determine a minimal cogenerator of mod A and count its composition length and R-dimension. 49. Let A and B be finite dimensional K-algebras. Let Q and R be duality (B; A)-modules between mod A and mod B op . Assume that there is an isomorphism W B Q ! B R of left B-modules. (a) Show that there is an automorphism ˛ 2 Aut.A/ such that W Q ! R is an isomorphism of (B; A)-bimodules.

8. Exercises

201

(b) Show that the functors HomA .; Q/ and HomA .; R/ are naturally isomorphic if and only if there is an inner automorphism ˛ 2 Aut.A/ such that Q˛ and R are isomorphic as (B; A)-bimodules. 50. Give the statement for Morita equivalences and progenerators as the dual statement to Excercise 49. (See Morita [Mor], Theorem 3.5.) 51. Let A be a finite dimensional commutative K-algebra. A (self-)duality for mod A is by definition a contravariant functor D W mod A ! mod Aop D mod A such that there exists a natural isomorphism of functors W 1mod A ! D 2 : The right A-module D.A/ is a duality A-module where the left A-module structure satisfies ax D .a/.x/ for all a 2 A; x 2 D.A/, and W A ! EndA .D.A/A / is an algebra isomorphism. On the other hand, since A is commutative, D.A/ has another left A-module structure canonically defined by a x D xa for all a 2 A and x 2 D.A/, so that D.A/ becomes an A-bimodule which we denote by U . We shall examine the difference between those A-bimodules D.A/ and U , which was proved by K. Morita in [Mor], Theorem 5.1. (a) For each a 2 A, let .a/ be the unique element of A such that D.aL / D .a/R ; where aL W AA ! AA and .a/R W A D.A/ ! multiplications by a and .a/, respectively.

A D.A/

are the left and right

Show that the mapping W A ! A, a 7! .a/, is an algebra automorphism of A. (b) For each u 2 D.A/, let uR W A A ! A D.A/ be the right multiplication by u. Let v.u/ D D.uR /.A/ .1A /; w.u/ D .D.A//1 D..A//1 .v.u//; in the commutative diagram .A/

A uR

D.A/

/ D 2 .A/

D

D 2 .uR /

/ D 2 .D.A// D D.D 2 .A//

.D.A//

/ D.D.A// D.uR /

D..A//

/ D.A/ .

202


Show that the induced maps w W U ! D.A/;

u 7! w.u/;

v W D.A/ ! U;

u 7! v.u/;

are A-bimodule isomorphisms, and v is the inverse of w. Hint: Prove that w is an injection, and w.v.u// D u for all u 2 D.A/. (c) Show the equality 2 .a/w 1 .u/ D aw 1 .u/ for all a 2 A and u 2 D.A/, and that 2 is the identity automorphism of A. 52. Let A be a K-algebra. Show that ˛ Inn.A/ D Inn.A/˛ for all K-algebra automorphisms ˛ of A.

Chapter III

Auslander–Reiten theory

This chapter is devoted to presenting background on the Auslander–Reiten theory of categories of finite dimensional modules over finite dimensional algebras over a field, a concept that plays a fundamental role in the modern representation theory of algebras. We start by introducing the (Jacobson) radical of a module category and the extension spaces of finite dimensional modules, and describing their properties. Then we enter into the heart of the Auslander–Reiten theory: the theory of irreducible homomorphisms, almost split homomorphisms, and almost split sequences. Further, we introduce the Auslander–Reiten quiver of a finite dimensional algebra, which is an important combinatorial and homological invariant of its category of finite dimensional modules. In particular, the following important results are proved here: the Harada–Sai lemma, the Auslander–Reiten formulas, the existence of almost split sequences, the Auslander criterion for finite representation type, the first Brauer–Thrall conjecture, and the Bautista–Smalø theorem on sectional paths of irreducible homomorphisms.

1 The radical of a module category Let A be a finite dimensional K-algebra over a field K. For modules X and Y in mod A, the set ³ ² ˇ ˇ idX gf is invertible in EndA .X / for radA .X; Y / D f 2 HomA .X; Y / ˇ any g 2 HomA .Y; X / is said to be the Jacobson radical (briefly, radical) of HomA .X; Y /. Lemma 1.1. For modules X and Y in mod A, we have ² ³ ˇ ˇ idY fg is invertible in EndA .Y / for : radA .X; Y / D f 2 HomA .X; Y / ˇ any g 2 HomA .Y; X / Proof. Let f 2 HomA .X; Y / and g 2 HomA .Y; X /. Assume ' 2 EndA .X / is such that '.idX gf / D idX D .idX gf /'. Then idX ' C 'gf D 0X D idX ' C gf '. Take D idY Cf 'g 2 EndA .Y /. Then we obtain .idY fg/ D .idY Cf 'g/.idY fg/ D idY fg C f 'g f 'gfg D idY f .idX ' C 'gf /g D idY f 0X g D idY ; .idY fg/ D .idY fg/.idY Cf 'g/ D idY fg C f 'g fgf 'g D idY f .idX ' C gf '/g D idY f 0X g D idY :

204

Chapter III. Auslander–Reiten theory

Conversely, let 2 EndA .Y / be such that .idY fg/ D idY D .idY fg/ . Then idY C fg D 0Y D idY C fg . For ' D idX Cg f 2 EndA .X /, we obtain '.idX gf / D .idX Cg f /.idX gf / D idX gf C g f g fgf D idX g.idY C fg/f D idX g0Y f D idX ; .idX gf /' D .idX gf /.idX Cg f / D idX gf C g f gfg f D idX g.idY C fg /f D idY g0Y f D idX :

Observe that, for any module X 2 mod A, radA .X; X / is the Jacobson radical rad EndA .X/ of the endomorphism algebra EndA .X / of X (see Lemma I.3.1). The following proposition shows that radA is an ideal of the category mod A, called the Jacobson radical (briefly, radical) of mod A. Proposition 1.2. Let X; Y; Z be modules in mod A. The following statements hold. (i) radA .X; Y / is a K-vector subspace of HomA .X; Y /. (ii) For f 2 radA .X; Y / and h 2 HomA .Y; Z/, we have hf 2 radA .X; Z/. (iii) For f 2 radA .X; Y / and h 2 HomA .Z; X /, we have f h 2 radA .Z; Y /. Proof. We first show (ii) and (iii). (ii) Let f 2 radA .X; Y / and h 2 HomA .Y; Z/. Take g 2 HomA .Z; X /. Then idX g.hf / D idX .gh/f is an invertible element of EndA .X /, and consequently hf 2 radA .X; Z/. (iii) Let f 2 radA .X; Y / and h 2 HomA .Z; X /. Take g 2 HomA .Y; Z/. Then, applying Lemma 1.1, we conclude that idY .f h/g D idY f .hg/ is invertible in EndA .Y /, and hence f h 2 radA .Z; Y /. (i) Let f 2 radA .X; Y / and 2 K. Then, for g 2 HomA .Y; X /, we have idX g.f / D idX .g/f is invertible in EndA .X /. Therefore f D f 2 radA .X; Y /. Let f1 ; f2 2 radA .X; Y /. We will show that f1 C f2 2 radA .X; Y /. Take g 2 HomA .Y; X /. Since f1 2 radA .X; Y / there exists '1 2 EndA .X / such that '1 .idX gf1 / D idX D .idX gf1 /'1 . Further, since f2 2 radA .X; Y / and '1 g 2 HomA .Y; X /, there exists '2 2 EndA .X / such that '2 .idX .'1 g/f2 / D idX . Then we obtain '2 '1 .idX g.f1 C f2 // D '2 .'1 .idX gf1 / '1 gf2 / D '2 .idX .'1 g/f2 / D idX ; and consequently idX g.f1 C f2 / has a left inverse in EndA .X /. It follows from (iii) that f2 '1 2 radA .X; Y /. Then there exists '3 2 EndA .X / such that .idX g.f2 '1 //'3 D idX . Hence .idX g.f1 C f2 //'1 '3 D ..idX gf1 /'1 gf2 '1 /'3 D .idX g.f2 '1 //'3 D idX ;

1. The radical of a module category

205

and so idX g.f1 C f2 / has a right inverse in EndA .X /. Moreover, we have '2 '1 D '2 '1 idX D '2 '1 .idX g.f1 C f2 //'1 '3 D idX '1 '3 D '1 '3 : Therefore, f1 C f2 2 radA .X; Y /. Summing up, we have proved that radA .X; Y / is a K-vector subspace of HomA .X; Y /. Lemma 1.3. Let X1 ; : : : ; Xm and Y1 ; : : : ; Yn be modules in mod A. Let further fj i 2 HomA .Xi ; Yj /, for i 2 f1; : : : ; mg, j 2 f1; : : : ; ng, and f D .fj i / W

m M

Xi !

iD1

n M

Yj

j D1

Lm Ln be the induced homomorphism. Then f 2 radA iD1 Xi ; j D1 Yj if and only if fj i 2 radA .Xi ; Yj / for any i 2 f1; : : : ; mg, j 2 f1; : : : ; ng. Lm L and vs W Ys ! jnD1 Yj , s 2 Proof. Let ur W Xr ! iD1 Xi , r 2 f1; : : : ; mg, Lm f1; : : : ; ng, iD1 Xi ! Xr , r 2 f1; : : : ; mg, Lnbe the canonical injections, and pr W and qs W Y ! Y , s 2 f1; : : : ; ng, be the canonical projections. Then fj i D j s j D1 Pn P qj f ui for any i 2 f1; : : : ; mg, j 2 f1; : : : ; ng, and f D m iD1 j D1 vj fj i pi . Lm Ln Hence f 2 radA X ; Y forces, by Proposition 1.2 (ii) and (iii), i j iD1 j D1 that fj i D qj f ui 2 radA .Xi ; Yj / for any i 2 f1; : : : ; mg, j 2 f1; : : : ; ng. Conversely, if fj i 2 radA .Xi ; Yj / for any i 2 f1; : : : ; mg, j 2 f1; : : : ; ng, then, by Proposition 1.2, f D

n m X X

vj fj i pi 2 radA

iD1 j D1

m M iD1

Xi ;

n M

Yj :

j D1

Lemma 1.4. Let X and Y be indecomposable modules in mod A. Then the following statements hold. (i) radA .X; Y / is the subspace of HomA .X; Y / formed by all nonisomorphisms. (ii) radA .X; Y / D HomA .X; Y / if X © Y . Proof. Obviously (ii) follows from (i). For (i), take 0 ¤ f 2 HomA .X; Y /. Observe that if f 2 radA .X; Y / then f is not an isomorphism. Indeed, if gf D idX for some g 2 HomA .Y; X /, then idX gf D 0X is not invertible in EndA .X /. Conversely, assume that f is not an isomorphism. Let g 2 HomA .Y; X /. We claim that gf 2 EndA .X / is not invertible. Indeed, if there exists h 2 EndA .X / such that h.gf / D idX , then .hg/f D idX . Applying Lemma I.4.2, we obtain Y Š Im f ˚ Ker.hg/. Since Im f ¤ 0 and Y is indecomposable, we conclude that Y D Im f . Clearly, then f W X ! Y is an isomorphism, because f is a

206


monomorphism, as a section. Further, by Lemma I.4.4, EndA .X / is a local Kalgebra, since X is an indecomposable module in mod A. Then, it follows from Lemma I.3.8 that idX gf is an invertible element of EndA .X /. Therefore, we have proved that f 2 radA .X; Y /. Lemma 1.5. (i) Let f W L ! M be a nonzero homomorphism in mod A and L be an indecomposable A-module. The following statements are equivalent. (a) f is not a section in mod A. (b) f 2 radA .L; M /. (c) Im HomA .f; L/ radA .L; L/. (ii) Let g W M ! N be a nonzero homomorphism in mod A and N be an indecomposable A-module. The following statements are equivalent. (a) g is not a retraction in mod A. (b) g 2 radA .M; N /. (c) Im HomA .N; g/ radA .N; N /. Proof. (i) Since L is an indecomposable module in mod A, it follows from Lemmas I.3.8 and I.4.4 that EndA .L/ is a local K-algebra and radA .L; L/ D rad EndA .L/ is a unique maximal right (respectively, maximal left) ideal of EndA .L/, consisting of all noninvertible endomorphisms. Consider the homomorphism HomA .f; L/ W HomA .M; L/ ! HomA .L; L/ D EndA .L/ of K-vector spaces given by HomA .f; L/.'/ D 'f for any ' 2 HomA .M; L/. Let M D M1 ˚ ˚ Mn be a decomposition of M into a direct sum of indecomposable A-submodules M1 ; : : : ; Mn , and ui W Mi ! M and pi W M ! Mi , i 2 f1; : : : ; ng, be the associated canonical injections and projections, respectively. Then f D u1 f1 C C un fn , where fi D pi f 2 HomA .L; Mi /, for i 2 f1; : : : ; ng. Assume f … radA .L; M /. Then it follows from Lemma 1.3 that fi … radA .L; Mi /, for some i 2 f1; : : : ; ng. Hence, by Lemma 1.4, fi is an isomorphism and consequently gi fi D idL for some gi 2 HomA .Mi ; L/. But then .gi pi /f D gi fi D idL and f is a section. Therefore, (a) implies (b). It follows from Proposition 1.2 (ii) that (b) implies (c). Assume now that f is a section in mod A. Then there is g 2 HomA .M; L/ with gf D idL . Since EndA .L/ is a local K-algebra, we obtain that HomA .f; L/.g/ D gf D idL is not in radA .L; L/ D rad EndA .L/. Hence, Im HomA .f; L/ is not contained in radA .L; L/. Therefore, (c) implies (a). The proof of the equivalences in (ii) is similar.

2. The Harada–Sai lemma

207

For each natural number m 1, we define the m-th power radAm of radA such that, for modules X and Y in mod A, radAm .X; Y / is the subspace of radA .X; Y / consisting of all finite sums of homomorphisms of the form hm hm1 : : : h2 h1 with hi 2 radA .Xi1 ; Xi /, i 2 f1; : : : ; mg, for some modules X D X0 ; X1 ; : : : ; Xm1 ; Xm D Y in mod A. Moreover, the intersection radA1

D

1 \

radAm

mD1

is said to be the infinite radical of mod A. For modules X and Y in mod A, we have inclusions of K-vector spaces HomA .X; Y / radA .X; Y / radA2 .X; Y / radAm .X; Y / radA1 .X; Y /: Lemma 1.6. Let X and Y be modules in mod A. Then there exists a natural number m 1 such that radA1 .X; Y / D radAm .X; Y /. Proof. This follows from the fact that HomA .X; Y / is a finite dimensional K-vector space, as a K-vector subspace of HomK .X; Y /.

2 The Harada–Sai lemma The aim of this subsection is to prove the following important lemma established by Harada and Sai in [HaSa]. Lemma 2.1. Let A be a finite dimensional K-algebra, b a positive integer, and f1

f2

f2b 1

M1 ! M2 ! M3 ! ! M2b 1 ! M2b a chain of homomorphisms in mod A satisfying the conditions (i) For each i 2 f1; : : : ; 2b g, Mi is indecomposable with `.Mi / b. (ii) For each i 2 f1; : : : ; 2b 1g, fi 2 radA .Mi ; MiC1 /. Then f2b 1 : : : f2 f1 D 0. Proof. We may assume that the homomorphisms f1 ; f2 ; : : : ; f2b 1 are nonzero. We prove by induction on n 1 that for the chain of homomorphisms f1

f2

f2n 1

M1 ! M2 ! M3 ! ! M2n 1 ! M2n we have `.Im f2n 1 : : : f2 f1 / b n. Observe that then for n D b we obtain `.Im f2n 1 : : : f2 f1 / D 0, or equivalently f2n 1 : : : f2 f1 D 0.

208

Chapter III. Auslander–Reiten theory f1

Assume n D 1. Then we have M1 ! M2 D M2n , Im f1 M2 and hence `.Im f1 / `.M2 / b. We claim that `.Im f1 / b 1. Suppose that `.Im f1 / D b. Then Im f1 D M2 . Since b D `.Im f1 / `.M1 / b, we obtain `.M1 / D b. Hence `.M1 / D `.Im f1 / C `.Ker f1 / implies Ker f1 D 0, and so f1 is also a monomorphism. Consequently, f1 W M1 ! M2 is an isomorphism in mod A, which contradicts the assumption f1 2 radA .M1 ; M2 /, by Lemma 1.4. Thus indeed `.Im f1 / b 1. Consider now the chain of homomorphism M1

f1

f2

/ M2

/ :::

f2n 1

/ M2n 1

/ M2n

f2n

GF @A `0 M2n C1

f2n C1

/ M2n C2

/ :::

f2nC1 1

/ M2nC1 1

BC ED

/ M2nC1

and set f D f2n 1 : : : f2 f1 , g D f2n , h D f2nC1 1 : : : f2n C2 f2n C1 . By our inductive assumption for n, we have `.Im f / b n and `.Im h/ b n. Observe that if `.Im f / < bn or `.Im h/ < bn, then `.Im f2nC1 1 : : : f2 f1 / D `.Im hgf / .b n/ 1 D b .n C 1/. Hence we may assume that `.Im f / D b n > 0 and `.Im h/ D b n > 0. We prove that `.Im hgf / b n 1. Suppose that `.Im hgf / > b n 1. Since `.Im hgf / `.Im h/ D b n, we then obtain `.Im hgf / D b n. Consider the composed homomorphism j

g

v

' W Im f , ! M2n ! M2n C1 ! Im h; where j is the canonical inclusion homomorphism and v is the epimorphism given by h. Since Im ' D vgj.Im f / D Im hgf Im h and `.Im hgf / D b n D `.Im h/, we conclude that Im ' D Im h, and so ' is an epimorphism. Moreover, by our assumption, `.Im f / D b n D `.Im h/, we obtain that ' is an isomorphism. Applying Lemma I.4.1 to ' D .vg/j , we infer that j W Im f ! M2n is a section. Since M2n is an indecomposable A-module, it follows then from Lemma I.4.2 that j D idM2n . Hence Im f D M2n and ' D vg. Applying Lemma I.4.1 to the isomorphism ' D vg, we conclude that g W M2n ! M2n C1 is a section in mod A. Since M2n C1 is indecomposable, applying Lemma I.4.2, we obtain that g is an isomorphism, which contradicts the assumption imposed on g D f2n . We note that the bounds given in the Harada–Sai lemma are the best possible (see Exercise 12.5). Corollary 2.2. Let A be a finite dimensional K-algebra of finite representation type. Then there exists a positive natural number m such that radAm D 0. In particular, we have radA1 D 0.

3. The space of extensions

209

Proof. Since A is of finite representation type, mod A admits only finitely many isomorphism classes of indecomposable modules. Hence there exists a positive natural number b such that `.M / b for any indecomposable module M in mod A. Take m D 2b 1. It follow from Lemma 1.3 that every homomorphism f 2 radAm .X; Y /, for any modules X and Y in mod A, is a finite sum of compositions of m homomorphisms from radA between indecomposable modules and therefore f D 0, by Lemma 2.1. Hence, we have radAm D 0.

3 The space of extensions In this section we show that the set of equivalence classes of extensions of two finite dimensional modules has a natural structure of a finite dimensional vector space and give its homological characterizations. In fact, we will show that the extension spaces of modules are bimodules over the corresponding endomorphism algebras. Let A be a finite dimensional K-algebra over a field K. A short exact sequence f

g

E W 0 ! L ! M ! N ! 0 in mod A is called an extension of L by N . Two extensions in mod A, f

g

E W 0 ! L ! M ! N ! 0 and

f0

g0

E0 W 0 ! L ! M 0 ! N ! 0;

are said to be equivalent if there exists h 2 HomA .M; M 0 / such that the diagram 0

/L

0

/L

f

/M

idL

g

/N

g0

/N

idN

h

f0

/ M0

/0 /0

is commutative, and then we write E ' E0 . Observe that then h is an isomorphism (see Exercise I.12.17). For two modules L and N in mod A we denote by EA .N; L/ the set of all extensions of L by N in mod A. Then ' is an equivalence relation in EA .N; L/, and we may consider the set ExtA1 .N; L/ D EA .N; L/= ' of the equivalence classes ŒE D E= ' of extensions E in EA .N; L/. We will show that ExtA1 .N; L/ has a canonical K-vector space structure, natural in L and N , and give its homological interpretation. For modules L and N in mod A, we denote by ON;L the canonical splittable extension ON;L W 0 of L by N .

/L

idL 0

/L˚N

0 idN

/N

/0

210


Lemma 3.1. Let L and N be modules in mod A and f

g

E W 0 ! L ! M ! N ! 0 an extension of L by N in mod A. The following conditions are equivalent. (a) E ' ON;L . (b) f is a section in mod A. (c) g is a retraction in mod A. Proof. Observe that we have in mod A a commutative diagram of the form f

/L

0

idL

/L

0 u

idL 0

g

/M

/N idN

h

/L˚N

/0

0 idN

/N

/0

if and only if h D g for some u 2 HomA .M; L/ with uf D idL . Similarly we have in mod A a commutative diagram of the form

0

/L

0

/L

idL 0

/L˚N

0 idN

h0

idL f

/M

/N

/0

idN

g

/N

/0

if and only if h0 D Œ f v for some v 2 HomA .N; M / with gv D idN . Hence, the required equivalences hold. An extension f

g

E W 0 ! L ! M ! N ! 0 in mod A with f a section, or equivalently g a retraction, is said to be a splittable extension of L by N . Lemma 3.2. Let L and N be modules in mod A and f

g

E W 0 ! L ! M ! N ! 0 and

f0

two extensions of L by N . Then there is an extension f 00

g0

E0 W 0 ! L ! M 0 ! N ! 0

g 00

E00 W 0 ! L ! M 00 ! N ! 0


211

of L by N in mod A with M 00 D V =U , where ˇ ˚ V D .m; m0 / 2 M ˚ M 0 ˇ g.m/ D g 0 .m0 / ; ˇ ˚ U D .f .x/; f 0 .x// 2 M ˚ M 0 ˇ x 2 L ; f 00 W L ! M 00 is given by f 00 .x/ D .f .x/; 0/ C U for x 2 L, and g 00 W M 00 ! N is given by g 00 ..m; m0 / C U / D g.m/ D g 0 .m0 / for .m; m0 / 2 V . Proof. Observe that V is an A-submodule of M ˚ M 0 and U is an A-submodule of V , because gf D 0 and g 0 f 0 D 0, and consequently M 00 D V =U is a module in mod A. Moreover, g 00 is a well-defined A-homomorphism, again by gf D 0 and g 0 f 0 D 0. We first show that f 00 is a monomorphism. Take x 2 L with f 00 .x/ D 0. Then .f .x/; 0/ D .f .y/; f 0 .y// for some y 2 L. The equality 0 D f 0 .y/ then forces y D 0, since f 0 is a monomorphism, and consequently f .x/ D f .y/ D 0. Hence x D 0, because f is a monomorphism. In order to show that g 00 is an epimorphism, take an element n 2 N . Since g and g 0 are epimorphisms, we have n D g.m/ and n D g 0 .m0 / for some m 2 M and m0 2 M 0 , and then g 00 ..m; m0 / C U / D n. The equality gf D 0 forces g 00 f 00 D 0, and hence Im f 00 Ker g 00 . We claim that also Ker g 00 Im f 00 . Take an element m00 D .m; m0 / C U 2 V =U D M 00 such that g 00 .m00 / D 0. Then g.m/ D 0 D g 0 .m0 /. Since Ker g D Im f and Ker g 0 D Im f 0 , we have m D f .x/ and m0 D f 0 .x 0 / for some x; x 0 2 L. Taking x 00 D x C x 0 2 L, we obtain the equalities f 00 .x 00 / D .f .x 00 /; 0/ C U D .f .x C x 0 /; 0/ C U D .f .x/ C f .x 0 /; 0/ C U D ..f .x/; 0/ C U / C ..f .x 0 /; 0/ C U / D ..f .x/; 0/ C U / C ..0; f 0 .x 0 // C U / D .f .x/; f 0 .x 0 // C U D .m; m0 / C U D m00 ; and so m00 2 Im f 00 . Therefore, Ker g 00 Im f 00 .

The extension E00 of L by N described in the above lemma will be denoted by E C E0 . Lemma 3.3. Let L and N be modules in mod A and f

g

E W 0 ! L ! M ! N ! 0; N

f gN x W 0 ! L ! x ! E M N ! 0;

f0

g0

N0

0

E0 W 0 ! L ! M 0 ! N ! 0; f gN x 0 W 0 ! L ! x 0 ! E M N ! 0

x and E0 ' E x 0 . Then E C E0 ' E x CE x 0. extensions of L by N in mod A with E ' E

212


x and E0 ' E x 0 , we have in mod A commutative diagrams Proof. Since E ' E f

/M

idL fN

/M x

0

/L

0

/L

g

/N

/0

gN

/N

/ 0,

/ M0

idL fN 0

/M x0

/L

0

/L

idN

h

f0

0

h0

g0

/N

gN 0

/N

/0

idN

/ 0.

x CE x 0 are of the forms From Lemma 3.2 the extensions E C E0 and E f 00

g 00

E C E0 W 0 ! L ! M 00 ! N ! 0; with M 00 D V =U , where ˚ V D .m; m0 / 2 M ˚ M 0 j g.m/ D g 0 .m0 / ; ˚ U D .f .x/; f 0 .x// 2 M ˚ M 0 j x 2 L ; and

N 00

00

f gN x CE x 0 W 0 ! L ! x 00 ! M N ! 0; E

x 00 D Vx =Ux , where with M ˚ x ˚M x 0 j g. N m/ N D gN 0 .m N 0/ ; Vx D .m; N m N 0/ 2 M ˚ x ˚M x 0 j xN 2 L : Ux D .fN.x/; N fN0 .x// N 2M x 00 defined for m00 D .m; m0 / C U 2 V =U D M 00 Consider the map h00 W M 00 ! M 00 00 0 0 x . Observe that, for .m; m0 / 2 V , by h .m / D .h.m/; h .m // C Ux 2 Vx =Ux D M 0 0 0 0 0 we have gh.m/ N D g.m/ D g .m / D gN h .m /, and hence .h.m/; h0 .m0 // 2 Vx . Further, for x 2 L, we have .hf .x/; h0 f 0 .x// D .fN.x/; fN0 .x//. This shows that h00 is well defined and clearly is an A-homomorphism. Consider now the diagram 0

/L

0

/L

f 00

/ M 00

g 00

/N

gN 00

/N

h00

idL fN 00

/M x 00

/0

idN

/ 0.

xCE x 0. We claim that this diagram is commutative, and consequently E C E0 ' E 00 00 00 Indeed, for x 2 L, we have h .f .x// D h ..f .x/; 0/ C U / D .h.f .x//; 0/ C Ux D .fN.x/; 0/ C Ux D fN00 .x/, and hence h00 f 00 D fN00 . Similarly, for m00 D .m; m0 / C U 2 M 00 , we have gN 00 h00 .m00 / D gN 00 ..h.m/; h0 .m0 // C Ux / D gh.m/ N D 00 00 00 00 00 g.m/ D g .m /, and hence gN h D g .


213

The above lemma allows us to define the addition in ExtA1 .N; L/ D EA .N; L/= ' by ŒE C ŒE0 D ŒE C E0 for ŒE; ŒE0 2 ExtA1 .N; L/. We will show later that it is the addition of a K-vector space structure on ExtA1 .N; L/, and ŒO D ŒON;L is the zero element with respect to this addition. The addition C in ExtA1 .N; L/ is called the Baer sum. Let L and N be modules in mod A and f

g

E W 0 ! L ! M ! N ! 0 an extension of L by N in mod A. For a homomorphism v W V ! N in mod A there exists, by Exercise I.12.19, a commutative diagram in mod A /L

0

/V

g0

idL

/L

0

v0

/ M N V

i

v

/M

f

/0

/N

g

/0

with exact rows, where M N V is the fibered product of M and V over N , via g and v. The upper row extension will be denoted by Ev. Similarly, for a homomorphism u W L ! U in mod A there exists, by Exercise I.12.21, a commutative diagram in mod A, 0

/L

0

/U

f

g

/M

/N

f0

u u0

/ U ˚L M

/0

idN

/N

p

/0

with exact rows, where U ˚L M is the fibered sum of U and M over L, via u and f . The lower row extension will be denoted by uE. Lemma 3.4. Let L and N be modules in mod A, f

f0

g

g0

E0 W 0 ! L ! M 0 ! N ! 0

E W 0 ! L ! M ! N ! 0;

extensions of L by N in mod A such that E ' E0 and u W L ! U , v W V ! N homomorphisms in mod A. Then uE ' uE0 and Ev ' E0 v. Proof. Since E ' E0 , we have in mod A the commutative diagram 0

/L

0

/L

f

idL

/M

g

/N

g0

/N

idN

h

f0

/ M0

/0 / 0.

214


It follows from Exercise I.12.21 that p

˛

uE W 0 ! U ! U ˚L M ! N ! 0; ˇ ˚ where U ˚L M D U ˚ M=R with R D .u.x/; f .x// 2 U ˚ M ˇ x 2 L , ˛.y/ D .y; 0/ C R for y 2 U , p..z; m/ C R/ D g.m/ for .z; m/ C R 2 U ˚L M , and p0 ˛0 uE0 W 0 ! U ! U ˚L M 0 ! N ! 0; ˇ ˚ where U ˚L M 0 D U ˚ M 0 =R0 with R0 D .u.x/; f 0 .x// 2 U ˚ M 0 ˇ x 2 L , ˛ 0 .y/ D .y; 0/ C R0 for y 2 U , p 0 ..z; m0 / C R0 / D g 0 .m0 / for .z; m0 / C R0 2 U ˚L M 0 . Then we obtain the commutative diagram in mod A, 0

/U

0

/U

˛

/ U ˚L M

idU ˛0

p

/N

p0

/N

'

/ U ˚L M 0

/0

idN

/ 0,

where ' W U ˚L M ! U ˚L M 0 is defined by '..z; m/ C R/ D .z; h.m// C R0 for .z; m/ C R 2 U ˚L M . Observe that ' is a well-defined A-homomorphism because hf D f 0 . Moreover, p 0 ' D p since g 0 h D g. Therefore uE ' uE0 . It follows from Exercise I.12.19 that i

ˇ

i0

ˇ0

Ev W 0 ! L ! M N V ! V ! 0; ˇ ˚ where M N V D .m; y/ 2 M V ˇ g.m/ D v.y/ , i.x/ D .f .x/; 0/ for x 2 L, ˇ..m; y// D y for .m; y/ 2 M N V , and E0 v W 0 ! L ! M 0 N V ! V ! 0; ˇ ˚ where M 0 N V D .m0 ; y/ 2 M 0 V ˇ g 0 .m0 / D v.y/ , i 0 .x/ D .f 0 .x/; 0/ for x 2 L, ˇ 0 ..m0 ; y// D y for .m0 ; y/ 2 M 0 N V . Then we obtain the commutative diagram in mod A, 0

/L

0

/L

i

/ M N V

i0

/ M 0 N V

idL

ˇ

/V

ˇ0

/V

/0 idV

/ 0,

where W M N V ! M 0 N V is defined by ..m; y// D .h.m/; y/ for .m; y/ 2 M N V . Observe that is a well-defined homomorphism in mod A, because if .m; y/ 2 M N V then g.m/ D v.y/, and hence g 0 .h.m// D g.m/ D v.y/, implying .h.m/; y/ 2 M 0 N V . Further, ˇ 0 D ˇ is clear, and i D i 0 follows from hf D f 0 . Therefore, we have also the equivalence Ev ' E0 v.


215

We conclude from Lemma 3.4 that, if u W L ! U and v W V ! N are homomorphisms in mod A, then we have two maps ExtA1 .N; u/ W ExtA1 .N; L/ ! ExtA1 .N; U /; ExtA1 .v; L/ W ExtA1 .N; L/ ! ExtA1 .V; L/ such that ExtA1 .N; u/.ŒE/ D ŒuE and ExtA1 .v; L/.ŒE/ D ŒEv for any equivalence class ŒE in ExtA1 .N; L/. We will see later that ExtA1 .N; u/ and ExtA1 .v; L/ are in fact K-linear homomorphisms. Let L and N be modules in mod A. We will define a K-vector space ExtA1 .N; L/, involving a minimal projective resolution of N in mod A, and two K-linear homomorphisms ExtA1 .N; u/ W ExtA1 .N; L/ ! ExtA1 .N; U /; and ExtA1 .v; L/ W ExtA1 .N; L/ ! ExtA1 .V; L/ for homomorphisms u W L ! U and v W V ! N in mod A. Consider a minimal projective resolution d2

dn

d1

d0

! Pn ! Pn1 ! ! P2 ! P1 ! P0 ! N ! 0 of N in mod A (see Proposition I.8.30). Application of the contravariant functor HomA .; L/ W mod A ! mod K yields the chain of K-vector spaces HomA .d1 ;L/

HomA .d2 ;L/

HomA .P0 ; L/ ! HomA .P1 ; L/ ! HomA .P2 ; L/ with HomA .d2 ; L/ HomA .d1 ; L/ D HomA .d1 d2 ; N / D 0. This allows us to define the K-vector space ExtA1 .N; L/ D Ker HomA .d2 ; L/= Im HomA .d1 ; L/: Observe that, by Proposition I.8.30 and Lemma I.8.31, the space ExtA1 .N; L/ is well defined (does not depend on the choice of minimal projective resolution). Let u W L ! U be a homomorphism in mod A. Then we have the commutative diagram in mod K, HomA .P0 ; L/

HomA .d1 ;L/

HomA .P0 ;u/

HomA .P0 ; U /

HomA .d1 ;U /

/ HomA .P1 ; L/

HomA .d2 ;L/

HomA .P1 ;u/

/ HomA .P1 ; U /

HomA .d2 ;U /

/ HomA .P2 ; L/ HomA .P2 ;u/

/ HomA .P2 ; U / ,

from which we infer that HomA .P1 ; u/.Ker HomA .d2 ; L// Ker HomA .d2 ; U /;

216


HomA .P1 ; u/.Im HomA .d1 ; L// Im HomA .d1 ; U /: As a consequence, we may define the K-linear homomorphism ExtA1 .N; u/ W ExtA1 .N; L/ ! ExtA1 .N; U / by ExtA1 .N; u/.' C Im HomA .d1 ; L// D u' C Im HomA .d1 ; U / for ' 2 HomA .P1 ; L/ with 'd2 D HomA .d2 ; L/.'/ D 0. We note that u' D HomA .P1 ; u/.'/. Moreover, we have ExtA1 .N; t u/ D ExtA1 .N; t / ExtA1 .N; u/ for u 2 HomA .L; U / and t 2 HomA .U; T /, and clearly ExtA1 .N; idL / D idExt1 .N;L/ . A This shows that, for any module N in mod A, we have the covariant functor ExtA1 .N; / W mod A ! mod K: Let v W V ! N be a homomorphism in mod A. Consider also a minimal projective resolution d2

dn

d1

d0

! ! P2 ! P1 ! P0 ! V ! 0 ! Pn ! Pn1

of V in mod A. Then invoking the projectivity of the modules Pi , i 0, and the exactness of the projective resolution of N , we conclude that there exists a commutative diagram in mod A of the form :::

/ P n

:::

/ Pn

dn

vn dn

/ P n1

/ :::

/ P 2

/ :::

/ P2

vn1

/ Pn1

d2

v2

/ P 1

d1

v1

d2

/ P1

/ P 0

d0

v0

d1

/ P0

/V

/0 v

/N

d0

Then we obtain the commutative diagram in mod K, HomA .P0 ; L/

HomA .d1 ;L/

HomA .v0 ;L/

HomA .P0 ; L/

HomA .d1 ;L/

/ HomA .P1 ; L/

HomA .d2 ;L/

HomA .v1 ;L/

/ HomA .P ; L/ 1

HomA .d2 ;L/

/ HomA .P2 ; L/ HomA .v2 ;L/

/ HomA .P ; L/ , 2

from which we infer that HomA .v1 ; L/.Ker HomA .d2 ; L// Ker HomA .d2 ; L/; HomA .v1 ; L/.Im HomA .d1 ; L// Im HomA .d1 ; L/: We may then define the K-linear homomorphism ExtA1 .v; L/ W ExtA1 .N; L/ ! ExtA1 .V; L/

/ 0.


by

217

ExtA1 .v; L/.' C Im HomA .d1 ; L// D 'v1 C Im HomA .d1 ; L/

for ' 2 HomA .P1 ; L/ with 'd2 D HomA .d2 ; L/.'/ D 0. We note that 'v1 D HomA .v1 ; L/.'/. Unfortunately, the homomorphism v1 W P1 ! P1 , occurring in the above commutative diagram of projective resolutions, is not uniquely determined by the homomorphism v W V ! N . Assume vn0 2 HomA .Pn ; Pn /, n 0, is 0 another family of homomorphisms such that d0 v00 D vd0 and dn vn0 D vn1 dn for 0 0 n 1. Then we have d0 .v0 v0 / D d0 v0 d0 v0 D vd0 vd0 D 0, and hence Im.v0 v00 / Ker d0 D Im d1 . Consequently, by the projectivity of P0 , there exists a homomorphism s0 2 HomA .P0 ; P1 / such that v0 v00 D d1 s0 . Moreover, we have d1 .v1 v10 s0 d1 / D d1 v1 d1 v10 d1 s0 d1 D d1 v1 d1 v10 v0 d1 C v00 d1 D .d1 v1 v0 d1 / .d1 v10 v00 d1 / D 0; and so Im.v1 v10 s0 d1 / Ker d1 D Im d2 . Hence, by the projectivity of P1 , there exists a homomorphism s1 2 HomA .P1 ; P2 / such that v1 v10 s0 d1 D d2 s1 , or equivalently, v1 v10 D d2 s1 C s0 d1 . Then, for any ' 2 Ker HomA .d2 ; L/, we obtain 'v1 'v10 D '.v1 v10 / D 'd2 s1 C's0 d1 D 's0 d1 D HomA .d1 ; L/.'s0 / with 's0 2 HomA .P0 ; L/. Therefore, we have 'v1 C Im HomA .d1 ; L/ D 'v10 C Im HomA .d1 ; L/. As a consequence, the homomorphism ExtA1 .v; L/ does not depend on the choice of homomorphisms vn W Pn ! Pn , n 0. Moreover, we have ExtA1 .vw; L/ D ExtA1 .w; L/ ExtA1 .v; L/ for v 2 HomA .V; N / and w 2 HomA .W; V /, and clearly ExtA1 .idN ; L/ D idExt1 .N;L/ . This shows that, for any A module L in mod A, we have the contravariant functor ExtA1 .; L/ W mod A ! mod K: Let L and N be modules in mod A. We shall define now a map N;L from ExtA1 .N; L/ to ExtA1 .N; L/ and describe its properties. Let E 2 EA .N; L/ be an extension f g 0 ! L ! M ! N ! 0 and let d2

dn

d1

d0

! Pn ! Pn1 ! ! P2 ! P1 ! P0 ! N ! 0 be a minimal projective resolution of N in mod A. Then there exists in mod A a commutative diagram of the form P2 0

d2

/ P1

d1

t1

/L

/ P0

d0

t0

f

/M

/N

/0

idN

g

/N

/ 0;

218


where t0 is given by the projectivity of P0 and surjectivity of g, t1 by the projectivity of P1 and the facts that gt0 d1 D d0 d1 D 0 and Im f D Ker g, and the equality f t1 d2 D t0 d1 d2 D 0 forces t1 d2 D 0, because f is a monomorphism. Since t1 d2 D HomA .d2 ; L/.t1 /, we may consider the element t1 C Im HomA .d1 ; L/ 2 Ker HomA .d2 ; L/= Im HomA .d1 ; L/ D ExtA1 .N; L/: Assume tN0 2 HomA .P0 ; M / and tN1 2 HomA .P1 ; L/ are homomorphisms such that g tN0 D d0 , f tN1 D tN0 d1 . Clearly, as above, tN1 d2 D 0 holds. Further, we have g.t0 tN0 / D gt0 g tN0 D d0 d0 D 0, and so there exists s 2 HomA .P0 ; L/ such that f s D t0 tN0 , because Ker g D Im f and P0 is projective. Further, we have f .t1 tN1 / D f t1 f tN1 D t0 d1 tN0 d1 D .t0 tN0 /d1 D f sd1 , and hence t1 tN1 D sd1 D HomA .d1 ; L/.s/, because f is a monomorphism. Therefore, t1 C Im HomA .d1 ; L/ D tN1 C Im HomA .d1 ; L/. This allows us to associate to E 2 EA .N; L/ the well-defined element t1 C Im HomA .d1 ; L/ of ExtA1 .N; L/. Assume now that E0 2 EA .N; L/ is an extension f0

g0

E0 W 0 ! L ! M 0 ! N ! 0 such that E ' E0 . Then there exists in mod A a commutative diagram 0

/L

0

/L

f

idL

/M

g

/N

g0

/N

idN

h

f0

/ M0

/0 / 0.

We conclude that, in the above notation, t10 D idL t1 D t1 2 HomA .P1 ; L/ and t00 D ht0 2 HomA .P0 ; M 0 / satisfy the required commutativity conditions g 0 t00 D g 0 ht0 D gt0 D d0 and f 0 t10 D hf t1 D ht0 d1 D t00 d1 . Hence, to the extension E0 2 EA .N; L/ is assigned the same element t10 C Im HomA .d1 ; L/ D t1 C Im HomA .d1 ; L/ of ExtA1 .N; L/ as to E. Summing up, we have defined the map N;L W ExtA1 .N; L/ ! ExtA1 .N; L/ given by N;L .ŒE/ D t1 C Im HomA .d1 ; L/ for ŒE 2 ExtA1 .N; L/. The following theorem describes basic properties of the map N;L . Theorem 3.5. Let A be a finite dimensional K-algebra over a field K and L, N modules in mod A. The following statements hold. (i) The map N;L W ExtA1 .N; L/ ! ExtA1 .N; L/ is a bijection. (ii) N;L .ŒE C ŒE0 / D N;L .ŒE/ C N;L .ŒE0 / for any equivalence classes ŒE and ŒE0 in ExtA1 .N; L/.


219

(iii) N;L .ŒON;L / is the zero element of ExtA1 .N; L/. (iv) For any homomorphism u W L ! U in mod A the diagram ExtA1 .N; L/

N;L

/ Ext 1 .N; L/ A

1 ExtA .N;u/

ExtA1 .N; U /

1 .N;u/ ExtA

N;U / Ext 1 .N; U / A

is commutative. (v) For any homomorphism v W V ! N in mod A the diagram ExtA1 .N; L/

N;L

1 ExtA .v;L/

ExtA1 .V; L/

/ Ext 1 .N; L/ A 1 .v;L/ ExtA

V;L

/ Ext 1 .V; L/ A

is commutative. Proof. (i) We first show that N;L is an injection. Let ŒE, ŒE0 be elements of ExtA1 .N; L/ such that N;L .ŒE/ D N;L .ŒE0 /. Let E, E0 be the extensions f

g

E W 0 ! L ! M ! N ! 0;

f0

g0

E0 W 0 ! L ! M 0 ! N ! 0

and P2 0

d2

/ P1

d1

t1

/L

/ P0

d0

t0

f

/M

/N

/0

P2

/ 0,

0

idN

g

/N

d2

/ P1 t10

/L

d1

/ P0

d0

/N

g0

/N

t0

f0

0 / M0

/0

idN

/0

commutative diagrams in mod A, where the upper rows are the right part of a minimal projective resolution of the module N in mod A. Then we have t1 C Im HomA .d1 ; L/ D N;L .ŒE/ D N;L .ŒE0 / D t10 C Im HomA .d1 ; L/: It follows that there exists s 2 HomA .P0 ; L/ such that t1 t10 D HomA .d1 ; L/.s/ D sd1 . Let R D Im d1 , W P1 ! R the epimorphism induced by d1 and ! W R ! P0 the inclusion homomorphism. Since Im d2 D Ker d1 and induces the canonical isomorphism P1 = Ker d1 ! Im d1 D R, there exist ˛1 ; ˛10 2 HomA .R; L/ such 0 0 that t1 D ˛1 and t1 D ˛1 . Moreover, we have .˛1 ˛10 / D t1 t10 D sd1 D s!, because d1 D !. The surjectivity of then implies that ˛1 ˛10 D s!. Similarly, the equalities f ˛1 D f t1 D t0 d1 D t0 ! and f 0 ˛10 D f 0 t10 D

220


t00 d1 D t00 ! imply f ˛1 D t0 ! and f 0 ˛10 D t00 !. Therefore we have in mod A the commutative diagrams 0

/R

!

˛1

0

/L

/ P0

d0

t0

f

/M

/N

/0

/R

0

˛10

idN

g

/N

/ 0,

/L

0

!

f0

/ P0 t00

/ M0

d0

/N

g0

/N

/0

idN

/ 0.

It follows from the left diagram that M D Im f C Im t0 . Hence, every element m 2 M has a decomposition m D f .x/ C t0 .p/ for some elements x 2 L and p 2 P0 . Consider the map h W M ! M 0 defined by h.m/ D f 0 .x/ C t00 .p/ C f 0 s.p/ if m D f .x/ C t0 .p/; x 2 L; p 2 P0 : We show that h is well defined. Assume that f .x/ C t0 .p/ D m D f .y/ C t0 .q/ are two decompositions of m 2 M with x; y 2 L and p; q 2 P0 . Then t0 .p q/ D f .y x/, and so d0 .p q/ D gt0 .p q/ D gf .y x/ D 0. Since Ker d0 D Im !, we conclude that p q D !.r/ for some r 2 R. Hence, invoking the established equality ˛1 ˛10 D s!, we obtain t00 .p q/ D t00 !.r/ D f 0 ˛10 .r/ D f 0 .˛1 s!/.r/ D f 0 ˛1 .r/ f 0 s!.r/ D f 0 ˛1 .r/ f 0 s.p q/; and consequently f 0 ˛1 .r/ D .t00 C f 0 s/.p q/. Moreover, f .y x/ D t0 .p q/ D t0 !.r/ D f ˛1 .r/, and so y x D ˛1 .r/, because f is a monomorphism. Therefore, we obtain f 0 .y x/ D .t00 C f 0 s/.p q/, or equivalently the required equality f 0 .x/ C t00 .p/ C f 0 s.p/ D f 0 .y/ C t00 .q/ C f 0 s.q/. Hence, h W M ! M 0 is a well-defined map, and clearly is an A-homomorphism. Finally, we show that the diagram 0

/L

0

/L

f

idL

/M

g

/N

g0

/N

idN

h

f0

/ M0

/0 /0

is commutative, and so ŒE D ŒE0 . Indeed, for x 2 L, we have h.f .x// D h.f .x/ C t0 .0// D f 0 .x/. Further, for m D f .x/ C t0 .p/ with x 2 L and p 2 P0 , we have g 0 h.m/ D g 0 .f 0 .x/ C t00 .p/ C f 0 s.p// D g 0 t00 .p/ D d0 .p/ D gt0 .p/ D g.f .x/ C t0 .p// D g.m/. In order to prove that the map N;L W ExtA1 .N; L/ ! ExtA1 .N; L/ is a surjection, take an element ' C Im HomA .d1 ; L/ 2 ExtA1 .N; L/ for a homomorphism ' 2 HomA .P1 ; L/ with 'd2 D HomA .d2 ; L/.'/ D 0. Since d1 D ! and Ker D Im d2 , we infer that there exists ˛ 2 HomA .R; L/ such that ' D ˛. Consider now the exact sequence !

d0

0 ! R ! P0 ! N ! 0:

221


Applying Exercise I.12.21, we obtain a commutative diagram in mod A of the form /R

0

!

/ P0

f

/M

˛

ˇ

/L

EW 0

d0

/N

/0

idN

g

/N

/ 0,

where M D L ˚R P0 is the fibered sum of L and P0 over R, via ˛ and !. Since d1 D ! and ' D ˛ we obtain the commutative diagram d2

P2

/ P1

d1

'

0

/L

d0

/ P0 ˇ

/0

idN

/M

f

/N

g

/N

/ 0,

and this shows that N;L .ŒE/ D ' C Im HomA .d1 ; L/. Summing up, we have proved that N;L is a bijection. (ii) We keep the minimal projective resolution of N from (i). In particular, we have the exact sequence d0

!

0 ! R ! P0 ! N ! 0 with R D Im d1 , d1 D !, ! W R ! P0 the inclusion homomorphism, and W P1 ! R the epimorphism induced by d1 . Consider two extensions f

g

E W 0 ! L ! M ! N ! 0 and

f0

g0

E0 W 0 ! L ! M 0 ! N ! 0

of L by N in mod A. Then we have two commutative diagrams 0

/R

0

/L

!

/ P0

f

/M

˛1

d0

t0

/N

/0

0

/R

0

/L

˛10

idN

g

/N

/ 0,

!

/ P0

f0

0 / M0

d0

/N

g0

/N

t0

/0

idN

/0

with t1 D ˛1 2 HomA .P1 ; L/ and t10 D ˛10 2 HomA .P1 ; L/ satisfying f t1 D t0 d1 , f 0 t10 D t00 d1 . Observe that N;L .ŒE/ D t1 CIm HomA .d1 ; L/, N;L .ŒE0 / D t10 C Im HomA .d1 ; L/, and consequently N;L .ŒE/ C N;L .ŒE0 / D .t1 C t10 / C Im HomA .d1 ; L/. Consider the extension F of L by N given by the lower sequence of the commutative diagram 0

/R

0

/L

!

˛1 C˛10 u

/ P0

d0

/Q

/N

/0

idN

v

/N

/ 0,

222


where Q D L ˚R P0 is the fibered sum of L and P0 over R, via ˛1 C ˛10 and !. Then it follows from the proof of (i) that N;L .ŒF / D .t1 C t10 / C Im HomA .d1 ; L/, and consequently N;L .ŒF / D N;L .ŒE/ C N;L .ŒE0 /. We will show that ŒF D ŒE C ŒE0 , where ŒE C ŒE0 D ŒE C E0 . Recall that the extension E C E0 is given by the exact sequence f 00

g 00

0 ! L ! M 00 ! N ! 0 with M 00 D V =U , where ˇ ˚ V D .m; m0 / 2 M ˚ M 0 ˇ g.m/ D g 0 .m0 / ; ˇ ˚ U D .f .x/; f 0 .x// 2 M ˚ M 0 ˇ x 2 L ; f 00 W L ! M 00 is given by f 00 .x/ D .f .x/; 0/ C U for x 2 L, and g 00 W M 00 ! N is given by g 00 ..m; m0 / C U / D g.m/ D g 0 .m0 / for .m; m0 / 2 V (see Lemma 3.2). Observe that, for p 2 P0 , we have gt0 .p/ D d0 .p/ D g 0 t00 .p/, and hence .t0 .p/; t00 .p// 2 V . This allows us to define the homomorphism W P0 ! M 00 in mod A by .p/ D .t0 .p/; t00 .p// C U for any p 2 P0 . We claim that ! D f 00 .˛1 C ˛10 /. Take an element r 2 R. We have the equalities !.r/ D .t0 !.r/; t00 !.r// C U D .f ˛1 .r/; f 0 ˛10 .r// C U D ..f ˛1 .r/; 0/ C U / C ..0; f 0 ˛10 .r// C U / D ..f ˛1 .r/; 0/ C U / C ..f ˛10 .r/; 0/ C U / D .f ˛1 .r/ C f ˛10 .r/; 0/ C U D .f .˛1 C ˛10 /.r/; 0/ C U D f 00 .˛1 C ˛10 /.r/; and the claim follows. Using now the universal property of the fibered sum Q D L ˚R P0 of L and P0 over R, via ˛1 C ˛10 and ! (see Exercise I.12.21) we conclude that there is a unique A-homomorphism h W Q ! M 00 such that hu D f 00 and h D . In order to prove that F ' E C E0 , it is enough to show that the diagram in mod A, 0

/L

0

/L

u

/Q

f 00

/ M 00

idL

v

/N

g 00

/N

/0

idN

h

/ 0,

is commutative, or equivalently, that g 00 h D v, because idL f 00 D f 00 D hu. Since every element q of Q D L ˚R P0 is of the form q D u.x/ C .p/ for some x 2 L and p 2 P0 , we have g 00 h.q/ D g 00 h.u.x/ C .p// D g 00 hu.x/ C g 00 h.p/ D g 00 f 00 .x/ C g 00 .p/ D g 00 .p/ D g 00 ..t0 .p/; t00 .p// C U / D g.t0 .p// D d0 .p/ D v.p/ D vu.x/ C v.p/ D v.u.x/ C .p// D v.q/:


223

Therefore, ŒF D ŒE C E0 , and hence N;L .ŒE C ŒE0 / D N;L .ŒE/ C N;L .ŒE0 /. (iii) From Lemma 3.1 the class ŒON;L consists of all splittable extensions of L by N in mod A. Let f

g

E W 0 ! L ! M ! N ! 0 be a splittable extension in mod A. Then f is a section, and so there exists u 2 HomA .M; L/ such that uf D idL . Consider a commutative diagram in mod A, d2

P2

d1

/ P1

/ P0

t1

0

d0

/N

t0

/L

/M

f

/0

idN

/N

g

/ 0,

where the upper exact sequence is the right part of a minimal projective resolution of N in mod A. Then, for s D ut0 2 HomA .P0 ; L/, we have t1 D uf t1 D ut0 d1 D sd1 D HomA .d1 ; L/.s/, and consequently N;L .ŒON;L / D N;L .ŒE/ D t1 C Im HomA .d1 ; L/ D 0 C Im HomA .d1 ; L/: (iv) Let u W L ! U be a homomorphism in mod A and f

g

E W 0 ! L ! M ! N ! 0 an extension of L by N in mod A. By definition, ExtA1 .N; u/.ŒE/ D ŒuE where uE is the lower sequence of the commutative diagram 0

/L

0

/U

f

g

/M

/N

f0

u

idN

/ U ˚L M

u0

/0

/N

p

/0

given by the fibered sum U ˚L M of U and M over L, via u and f . Consider also a commutative diagram P2

d2

d1

/ P1 t1

0

/ P0

d0

/N

t0

/L

f

/M

/0

idN

/N

g

/ 0,

where the upper sequence is the right part of a minimal projective resolution of N in mod A. Then we obtain a commutative diagram P2 0

d2

/ P1 /U

d1

/ P0

d0

f 0 t0

ut1 u0

/ U ˚L M

/N

/0

idN

p

/N

/0

224


in mod A, and consequently the equalities N;U ExtA1 .N; u/.ŒE/ D N;U .ŒuE/ D ut1 C Im HomA .d1 ; N / D ExtA1 .N; u/.t1 C Im HomA .d1 ; N // D ExtA1 .N; u/N;L .ŒE/: Therefore, N;U ExtA1 .N; u/ D ExtA1 .N; u/N;L . (v) Let v W V ! N be a homomorphism in mod A and f

g

E W 0 ! L ! M ! N ! 0 an extension of L by N in mod A. By definition, ExtA1 .v; L/.ŒE/ D ŒEv, where Ev is the upper sequence of the commutative diagram 0

/L

0

/L

v0

/ M N V

i

g0

idL

/0 v

/M

f

/V

g

/N

/0

given by the fibered product M N V of M and V over N , via g and v. We have in mod A a commutative diagram :::

/ P n

:::

/ Pn

dn

/ P n1

vn dn

/ :::

/ P 2

/ :::

/ P2

vn1

/ Pn1

d2

v2

/ P 1

d1

/ P1

d0

0

v1

d2

/ P

/V

v0

d1

/ P0

/0 v

d0

/N

/ 0,

where the upper and lower sequences are minimal projective resolutions of V and N in mod A, respectively. Consider a commutative diagram in mod A of the form P2 0

d2

/ P1

d1

t1

/L

/ P0

d0

t0

f

/M

/N

/0

idN

g

/N

/ 0.

Since vd0 D d0 v0 D gt0 v0 , it follows from the universal property of the fibered product M N V (Exercise I.12.19) that there is a unique homomorphism t0 W P0 ! M N V in mod A such that g 0 t0 D t0 v0 and v 0 t0 D d0 . In fact, we have t0 .x/ D .t0 v0 .x/; d0 .x// for any x 2 P0 . Take now the homomorphism t1 D t1 v1 W P1 ! L. Then, for any element z 2 P1 , we have the equalities it1 .z/ D i t1 v1 .z/ D .f t1 v1 .z/; 0/ D .t0 d1 v1 .z/; 0/ D .t0 v0 d1 .z/; 0/ D .t0 v0 d1 .z/; d0 d1 .z// D t0 d1 .z/:


225

Hence, we obtain a commutative diagram in mod A, P2 0

d2

/ P 1

d1

t1

/L

/ P 0

d0

t0

i

/ M N V

v0

/V /V

/0 idV

/ 0.

Then the following equalities hold: V;L ExtA1 .v; L/.ŒE/ D V;L .ŒEv/ D t1 C Im HomA .d1 ; L/ D t1 v1 C Im HomA .d1 ; L/ D ExtA1 .v; L/.t1 C Im HomA .d1 ; L// D ExtA1 .v; L/N;L .ŒE/: Therefore, V;L ExtA1 .v; L/ D ExtA1 .v; L/N;L .

As an important consequence of Theorem 3.5 we obtain the following facts. Corollary 3.6. Let A be a finite dimensional K-algebra over a field K and L, N modules in mod A. The following statements hold. (i) The set ExtA1 .N; L/ admits a structure of K-vector space whose additive structure is given by the Baer sum, ŒON;L is the zero element, and the map N;L W ExtA1 .N; L/ ! ExtA1 .N; L/ is a K-linear isomorphism. Moreover, for ŒE 2 ExtA1 .N; L/ and 2 K, we have ŒE D Œ. idL /E D ŒE. idN / D ŒE: (ii) For any homomorphism u W L ! U in mod A, the map ExtA1 .N; u/ W ExtA1 .N; L/ ! ExtA1 .N; U / is a K-linear homomorphism. (iii) For any homomorphism v W V ! N in mod A, the map ExtA1 .v; L/ W ExtA1 .N; L/ ! ExtA1 .V; L/ is a K-linear homomorphism. In fact, we have the following stronger result. Proposition 3.7. Let A be a finite dimensional K-algebra over a field K and L, N modules in mod A. The following statements hold.

226


(i) The covariant functors ExtA1 .N; / and ExtA1 .N; / from mod A to mod K are naturally isomorphic. (ii) The contravariant functors ExtA1 .; L/ and ExtA1 .; L/ from mod A to mod K are naturally isomorphic. Proof. (i) The family of K-linear isomorphisms N;U W ExtA1 .N; U / ! ExtA1 .N; U /, with U modules in mod A, induces, by Theorem 3.5 (iv), the required isomorphism ExtA1 .N; / ! ExtA1 .N; / of covariant functors from mod A to mod K. (ii) The family of K-linear isomorphisms V;L W ExtA1 .V; L/ ! ExtA1 .V; L/, with V modules in mod A, induces, by Theorem 3.5 (v), the required isomorphism ExtA1 .; L/ ! ExtA1 .; L/ of contravariant functors from mod A to mod K. Proposition 3.8. Let A be a finite dimensional K-algebra over a field K and L, N modules in mod A. The following statements hold. (i) The K-vector space ExtA1 .N; L/ possesses a natural structure of an .EndA .L/; EndA .N //-bimodule. (ii) The K-vector space ExtA1 .N; L/ possesses a natural structure of an .EndA .L/; EndA .N //-bimodule. (iii) The K-linear isomorphism N;L W ExtA1 .N; L/ ! ExtA1 .N; L/ is an isomorphism of .EndA .L/; EndA .N //-bimodules. Proof. The left EndA .L/-module structure and the right EndA .N /-module structure on ExtA1 .N; L/ are given as follows: uŒE D ŒuE D ExtA1 .N; u/.ŒE/; ŒEv D ŒEv D ExtA1 .v; L/.ŒE/; for u 2 EndA .L/, v 2 EndA .N /, and ŒE 2 ExtA1 .N; L/. Observe that, for u; u0 2 EndA .L/, v; v 0 2 EndA .N /, ŒE 2 ExtA1 .N; L/, we have the equalities .uu0 /ŒE D ExtA1 .N; uu0 /.ŒE/ D ExtA1 .N; u/ ExtA1 .N; u0 /.ŒE/ D u.u0 ŒE/; ŒE.vv 0 / D ExtA1 .vv 0 ; L/.ŒE/ D ExtA1 .v 0 ; L/ ExtA1 .v; L/.ŒE/ D .ŒEv/v 0 ; as ExtA1 .N; / W mod A ! mod K is covariant functor and ExtA1 .; L/ W mod A ! mod K is contravariant functor. Similarly, the left EndA .L/-module structure and the right EndA .N /-module structure on ExtA1 .N; L/ are given as follows: ux D ExtA1 .N; u/.x/; xv D ExtA1 .v; L/.x/;


227

for u 2 EndA .L/, v 2 EndA .N /, and x 2 ExtA1 .N; L/. Observe that, for u; u0 2 EndA .L/, v; v 0 2 EndA .N /, x 2 ExtA1 .N; L/, we have the equalities .uu0 /x D ExtA1 .N; uu0 /.x/ D ExtA1 .N; u/.ExtA1 .N; u0 /.x// D u.u0 x/; x.vv 0 / D ExtA1 .vv 0 ; L/.x/ D ExtA1 .v 0 ; L/.ExtA1 .v; L/.x// D .xv/v 0 ; as ExtA1 .N; / W mod A ! mod K is covariant functor and ExtA1 .; L/ W mod A ! mod K is contravariant functor. By Theorem 3.5, the K-linear isomorphism N;L W ExtA1 .N; L/ ! ExtA1 .N; L/ is an isomorphism of left EndA .L/-modules and an isomorphism of right EndA .N /modules. We claim now that ExtA1 .N; L/ is an .EndA .L/; EndA .N //-bimodule. Recall that every element of ExtA1 .N; L/ is of the form 'N D ' CIm HomA .d1 ; L/ for some ' 2 HomA .P1 ; L/ with 'd2 D HomA .d2 ; L/.'/ D 0, where d2

dn

d1

d0

! Pn ! Pn1 ! ! P2 ! P1 ! P0 ! N ! 0 is a minimal projective resolution of N in mod A. Then we have, for u 2 EndA .L/ and v 2 EndA .N /, the equalities u.'v/ N D u ExtA1 .v; L/.'/ N D ExtA1 .N; u/ ExtA1 .v; L/.'/ N D ExtA1 .N; u/ .'v1 C Im HomA .d1 ; L// D u.'v1 / C Im HomA .d1 ; L/ D .u'/v1 C Im HomA .d1 ; L/ D ExtA1 .v; L/ .u' C Im HomA .d1 ; L// D ExtA1 .v; L/ ExtA1 .N; u/.'/ N D .u'/v; N where v1 2 EndA .P1 / occurs in a commutative diagram in mod A of the form P1

d1

v1

P1

/ P0

d0

v0

d1

/ P0

/N

/0

v d0

/N

/ 0.

Hence, indeed ExtA1 .N; L/ is an .EndA .L/; EndA .N //-bimodule. Now let u 2 EndA .L/, v 2 EndA .N /, and ŒE 2 ExtA1 .N; L/. Then we have the equalities N;L .u.ŒEv// D u.N;L .ŒEv// D u.N;L .ŒE/v/ D .uN;L .ŒE//v D N;L .uŒE/v D N;L ..uŒE/v/;

228


and hence u.ŒEv/ D .uŒE/v, because N;L is an isomorphism of left EndA .L/modules and right EndA .N /-modules. In particular, we obtain that ExtA1 .N; L/ is an .EndA .L/; EndA .N //-bimodule and N;L W ExtA1 .N; L/ ! ExtA1 .N; L/ is an isomorphism of .EndA .L/; EndA .N //-bimodules. Dually, for modules L and N in mod A, we may define a K-vector space e Ext .N; L/, involving a minimal injective resolution of L in mod A, and two K1 A

linear homomorphisms

e e e ext .v; L/ W Eext .N; L/ ! Eext .V; L/ E

ExtA1 .N; u/ W ExtA1 .N; L/ ! ExtA1 .N; U /; 1 A

1 A

1 A

for homomorphisms u W L ! U and v W V ! N in mod A. Consider a minimal injective resolution d0

d1

d2

dn

0 ! L ! I0 ! I1 ! I2 ! ! In1 ! In ! of L in mod A (see Proposition I.8.30). Application of the covariant functor HomA .N; / W mod A ! mod K yields the chain of K-vector spaces HomA .N;d 1 /

HomA .N;d 2 /

HomA .N; I0 / ! HomA .N; I1 / ! HomA .N; I2 / with HomA .N; d 2 / HomA .N; d 1 / D HomA .N; d 2 d 1 / D 0. This allows us to define the K-vector space

e

ExtA1 .N; L/ D Ker HomA .N; d 2 /= Im HomA .N; d 1 /:

e

Observe that, by Proposition I.8.30 and Lemma I.8.32, the space ExtA1 .N; L/ is well defined. Let u W L ! U be a homomorphism in mod A. Consider also a minimal injective resolution dQ 0

dQ 1

dQ 2

dQ n

0 ! U ! IQ0 ! IQ1 ! IQ2 ! ! IQn1 ! IQn ! of U in mod A. Then invoking the injectivity of the modules IQi , i 0, and the exactness of the injective resolution, we conclude that there exists in mod A a commutative diagram of the form 0

/L

0

/U

d0

/ I0

d1

u0

u dQ 0

/ IQ 0

/ I1

d2

u1

dQ 1

/ IQ 1

/ I2

/ :::

u2

dQ 2

/ IQ 2

/ :::

/ In1

dn

un1

/ IQ n1

dQ n

/ In

/ :::

un

/ IQ n

/ ::: .


229

Then we obtain the commutative diagram in mod K, HomA .N; I0 /

HomA .N;d 1 /

HomA .N;u0 /

HomA .N; IQ0 /

/ HomA .N; I1 /

HomA .N;d 2 /

/ HomA .N; I2 /

HomA .N;u1 /

HomA .N;dQ 1 /

/ Hom .N; IQ / A 1

HomA .N;u2 /

HomA .N;dQ 2 /

/ Hom .N; IQ / , A 2

from which we infer that HomA .N; u1 /.Ker HomA .N; d 2 // Ker HomA .N; dQ 2 /; HomA .N; u1 /.Im HomA .N; d 1 // Im HomA .N; dQ 1 /: We may then define the K-linear homomorphism

e

e

e

ExtA1 .N; u/ W ExtA1 .N; L/ ! ExtA1 .N; U / by

e

ExtA1 .N; u/.' C Im HomA .N; d 1 // D u1 ' C Im HomA .N; dQ 1 /

for ' 2 HomA .N; I1 / with d 2 ' D HomA .N; d 2 /.'/ D 0. Observe that u1 ' D HomA .N; u1 /.'/. Unfortunately, the homomorphism u1 W I1 ! IQ1 , occurring in the above commutative diagram of injective resolutions, is not uniquely determined by the homomorphism u W L ! U . Assume u0n 2 HomA .In ; IQn /, n 0, is another family of homomorphisms such that u00 d 0 D dQ 0 u and u0n d n D dQ n u0n1 for n 1. Then we have .u0 u00 /d 0 D u0 d 0 u00 d 0 D dQ 0 u dQ 0 u D 0, and hence u0 u00 factors through Coker d 0 . Consequently, by the injectivity of IQ0 , there exists a homomorphism t0 2 HomA .I1 ; IQ0 / such that u0 u00 D t0 d 1 . Moreover, we have .u1 u01 dQ 1 t0 /d 1 D u1 d 1 u01 d 1 dQ 1 t0 d 1 D dQ 1 u0 dQ 1 u00 dQ 1 u0 C dQ 1 u00 D 0; and u1 u01 dQ 1 t0 factors through Coker d 1 . Hence, by the injectivity of IQ1 , there exists a homomorphism t1 2 HomA .I2 ; IQ1 / such that u1 u01 dQ 1 t0 D t1 d 2 , or equivalently u1 u01 D t1 d 2 C dQ 1 t0 . Then, for any ' 2 Ker HomA .N; d 2 /, we obtain u1 ' u01 ' D .u1 u01 /' D .t1 d 2 C dQ 1 t0 /' D t1 d 2 ' C dQ 1 t0 ' D dQ 1 t0 ' D HomA .N; dQ 1 /.t0 '/ with t0 ' 2 HomA .N; IQ0 /. Therefore, we have u1 ' C Im HomA .N; dQ 1 / D u01 ' C Im HomA .N; dQ 1 /. As a consequence, the homomorphism ExtA1 .N; u/ does not depend on the choice of homomorphisms un W In ! IQn , n 0. Moreover, we have ExtA1 .N; wu/ D ExtA1 .N; w/ExtA1 .N; u/ for u 2 HomA .L; U / and w 2 HomA .U; W /, and clearly ExtA1 .N; idL / D idExt e A1 .N;L/ . This shows that, for any module N in mod A, we have the covariant functor

e

e

e

e e

ExtA1 .N; / W mod A ! mod K:

e

230


Let v W V ! N be a homomorphism in mod A. Then we have the commutative diagram in mod K, HomA .N; I0 /

HomA .N;d 1 /

HomA .v;I0 /

HomA .V; I0 /

/ HomA .N; I1 /

HomA .N;d 2 /

HomA .v;I1 /

HomA

.V;d 1 /

/ HomA .V; I1 /

/ HomA .N; I2 / HomA .v;I2 /

HomA

.V;d 2 /

/ HomA .V; I2 / ,

from which we conclude that HomA .v; I1 /.Ker HomA .N; d 2 // Ker HomA .V; d 2 /; HomA .v; I1 /.Im HomA .N; d 1 // Im HomA .V; d 1 /: As a consequence, we may define the K-linear homomorphism

e

e

e

ExtA1 .v; L/ W ExtA1 .N; L/ ! ExtA1 .V; L/ by

e

ExtA1 .v; L/.' C Im HomA .N; d 1 // D 'v C Im HomA .V; dQ 1 /

for ' 2 HomA .N; I1 / with d 2 ' D HomA .N; d 2 /.'/ D 0. Observe that 'v D HomA .v; L/.'/. Moreover, we have ExtA1 .vt; L/ D ExtA1 .t; L/ExtA1 .v; L/ for t 2 HomA .T; V / and v 2 HomA .V; N /, and clearly ExtA1 .idN ; L/ D idExt e A1 .N;L/ . This shows that, for any module L in mod A, we have the contravariant functor

e

e e

e

e

ExtA1 .; L/ W mod A ! mod K: We have the following fact. Proposition 3.9. Let A be a finite dimensional K-algebra over a field K and L, N be modules in mod A. The following statements hold.

e

(i) The covariant functors ExtA1 .N; / and ExtA1 .N; / from mod A to mod K are naturally isomorphic.

e

(ii) The contravariant functors ExtA1 .; L/ and ExtA1 .; L/ from mod A to mod K are naturally isomorphic. Proof. It follows from Proposition I.8.16 that d0

d1

d2

dn

0 ! L ! I0 ! I1 ! I2 ! ! In1 ! In ! is a minimal injective resolution of L in mod A if and only if ::: @A GF _/ D.I2 /

D.d 2 /

/ D.In /

D.d n /

/ D.I1 /

D.d 1 /

/ D.In1 / / D.I0 /

/ ::: D.d 0 /

/ D.L/

ED BC /0


231

is a minimal projective resolution of the dual module D.L/ in mod Aop . Hence, the duality functors D / mod A o mod Aop D

induce an isomorphism of the covariant functors

e

ExtA1 .N; / ! ExtA1 op .; D.N //D

and an isomorphism of contravariant functors

e

ExtA1 .; L/ ! ExtA1 op .D.L/; /D

from mod A to mod K. Further, the duality functors induce also an isomorphism of the covariant functors ExtA1 .N; / ! ExtA1 op .; D.N //D

and an isomorphism of the contravariant functors ExtA1 .; L/ ! ExtA1 op .D.L/; /D

from mod A to mod K (see Exercise 12.6). Applying now Proposition 3.7 to the modules D.L/ and D.N / in mod Aop , we conclude that there exist an isomorphism of the covariant functors ExtA1 op .D.L/; / ! ExtA1 op .D.L/; /

and an isomorphism of the contravariant functors ExtA1 op .; D.N // ! ExtA1 op .; D.N //

from mod Aop to mod K. Composing with the duality D W mod A ! mod Aop , we then obtain an isomorphism of covariant functors ExtA1 op .; D.N //D ! ExtA1 op .; D.N //D

and an isomorphism of contravariant functors ExtA1 op .D.L/; /D ! ExtA1 op .D.L/; /D:

e

Combining the above isomorphisms we conclude that the covariant functors ExtA1 .N; / and ExtA1 .N; / from mod A to mod K are isomorphic, and the contravariant functors ExtA1 .; L/ and ExtA1 .; L/ from mod A to mod K are isomorphic.

e

As a consequence of Propositions 3.8 and 3.9 we obtain the following facts.

232


Proposition 3.10. Let A be a finite dimensional K-algebra over a field K and L, N modules in mod A. The following statements hold.

e

(i) The K-vector space ExtA1 .N; L/ possesses a natural structure of an .EndA .L/; EndA .N //-bimodule. (ii) There is a natural isomorphism

e

ExtA1 .N; L/ ! ExtA1 .N; L/ of .EndA .L/; EndA .N //-bimodules. As an immediate consequence of Propositions 3.7 and 3.9 we obtain the following facts. Corollary 3.11. Let A be a finite dimensional K-algebra over a field K and L, N be modules in mod A. The following statements hold.

e

(i) The covariant functors ExtA1 .N; / and ExtA1 .N; / from mod A to mod K are naturally isomorphic.

e

(ii) The contravariant functors ExtA1 .; L/ and ExtA1 .; L/ from mod A to mod K are naturally isomorphic. We end this section with the following consequence of Propositions 3.8 and 3.10. Corollary 3.12. Let A be a finite dimensional K-algebra over a field K and ! L, N modules in mod A. Then there is a natural isomorphism ExtA1 .N; L/ ExtA1 .N; L/ of .EndA .L/; EndA .N //-bimodules.

e

4 The Auslander–Reiten translations In this section we introduce theAuslander–Reiten translations on the (stable) module categories of finite dimensional algebras, which plays a prominent role in further considerations. Let A be a finite dimensional K-algebra over a field K. We denote by proj A the full subcategory of mod A consisting of all projective modules and by inj A the full subcategory of mod A consisting of all injective modules. Then the standard duality D / mod A o mod Aop ; D

where D D HomK .; K/, induces the dualities proj A o

D D

/ inj Aop ;

4. The Auslander–Reiten translations

inj A o

D

233

/ proj Aop :

D

For two modules M and N in mod A we denote by PA .M; N / the subset of HomA .M; N / consisting of all homomorphisms f W M ! N which factor through a module P in proj A, that is, f D hg for some g 2 HomA .M; P / and h 2 HomA .P; N /. The following lemma shows that PA is an ideal of the category mod A. Lemma 4.1. Let M; N; U; V be modules in mod A. Then the following statements hold. (i) PA .M; N / is a K-vector subspace of HomA .M; N /. (ii) For f 2 PA .M; N / and u 2 HomA .N; U /, we have uf 2 PA .M; U /. (iii) For f 2 PA .M; N / and v 2 HomA .V; M /, we have f v 2 PA .V; N /. Proof. (i) Let f; f 0 2 PA .M; N /. Then f D hg and f 0 D h0 g 0 , for some g 2 HomA .M; P /, h 2 HomA .P; N /, g 0 2 HomA .M; P 0 /, h0 2 HomA .P 0 ; N / with P and P 0 from proj A. We may then write f C f 0 as g 0 0 0 0 ; f C f D hg C h g D h h g0 and hence f C f 0 factors through P ˚ P 0 . This shows that f C f 0 2 PA .M; N /. Moreover, for any 2 K, we have f D .hg/ D .h/g 2 PA .M; N /. Therefore, PA .M; N / is a K-vector subspace of HomA .M; N /. (ii) and (iii). Let f 2 PA .M; N / and f D hg for some g 2 HomA .M; P /, h 2 HomA .P; N /, and P a module in proj A. Then for u 2 HomA .N; U / and v 2 HomA .V; M /, we have uf D u.hg/ D .uh/g and f v D .hg/v D h.gv/, with uh 2 HomA .P; U / and gv 2 HomA .V; P /, and consequently uf 2 PA .M; U / f v 2 PA .V; N /. The above lemma allows us to define the projectively stable category mod A D mod A=PA : The objects of mod A are the modules in mod A, the K-vector space of morphisms from M to N in mod A is the quotient space HomA .M; N / D HomA .M; N /=PA .M; N /; and the composition of morphisms in mod A is induced from the composition of homomorphisms in mod A. For a homomorphism f 2 HomA .M; N /, we denote by f D f C PA .M; N / its class in HomA .M; N /.

234


For two modules M and N in mod A we may also consider the subset IA .M; N / of HomA .M; N / consisting of all homomorphisms f W M ! N which factor through a module I from inj A. The following analogue of Lemma 4.1 shows that IA is an ideal of the category mod A. Lemma 4.2. Let M; N; U; V be modules in mod A. The following statements hold. (i) IA .M; N / is a K-vector subspace of HomA .M; N /. (ii) For f 2 IA .M; N / and u 2 HomA .N; U /, we have uf 2 IA .M; U /. (iii) For f 2 IA .M; N / and v 2 HomA .V; M /, we have f v 2 IA .V; N /. This allows us to define the injectively stable category mod A D mod A=IA : The objects of mod A are the modules in mod A, the K-vector space of morphisms from M to N in mod A is the quotient space HomA .M; N / D HomA .M; N /=IA .M; N /; and the composition of morphisms in mod A is induced from the composition of homomorphisms in mod A. For a homomorphism f 2 HomA .M; N /, we denote by fN D f C IA .M; N / its class in HomA .M; N /. The following lemma is useful. Lemma 4.3. Let M; N be modules in modP A and U; V modules in modI A. The following equivalences hold. (i) M Š N in mod A if and only if M Š N in mod A. (ii) U Š V in mod A if and only if U Š V in mod A. Proof. (i) If M Š N in mod A, then there exist homomorphisms f W M ! N and g W N ! M in mod A with gf D idM and fg D idN . Then g f D gf D idM and f g D fg D idN , and hence M Š N in mod A. Conversely, assume that M Š N in mod A and f W M ! N and g W N ! M are homomorphisms in mod A such that g f D idM and f g D idN . Then we obtain that gf D idM and fg D idN and hence idM gf 2 PA .M; M / and idN fg 2 PA .N; N /. Let idM gf D vu for a module P in proj A and homomorphisms u W M ! P and v W P ! M in mod A. Since M is in modP A, it follows from Lemmas 1.3 and 1.4 that u 2 radA .M; P / and v 2 radA .M; P /, and so idM gf 2 radA .M; M /. Hence we obtain that gf D idM .idM gf / D idM idM .idM gf / is an invertible element of EndA .M /, equivalently an isomorphism in mod A. Similarly, we show that fg is an invertible element of EndA .N /, equivalently an isomorphism. In particular, there exist endomorphisms ' 2 EndA .M / and 2 EndA .N / such that


235

.'g/f D idM and fg D idN , which shows that f is simultaneously a section and a retraction in mod A. Therefore, f is an isomorphism, and consequently M Š N in mod A. The proof of (ii) is similar. Observe that the duality D D HomK .; K/ between mod A and mod Aop induces the duality D / mod A o mod Aop : D

We will introduce now a duality between mod A and mod Aop proposed by M. Auslander and M. Bridger [AB]. Consider the contravariant functor ./t D HomA .; A/ W mod A ! mod Aop ; where for a module M in mod A the left A-module structure on HomA .M; A/ is given by .af /.m/ D af .m/, for a 2 A, f 2 HomA .M; A/ and m 2 M . Lemma 4.4. The functor ./t W mod A ! mod Aop induces the duality proj A o

./t ./t

/ proj Aop :

Proof. It follows from Lemma I.8.7 that for every idempotent e of A, the Klinear map ‚Ae W HomA .eA; A/ ! Ae, given by ‚Ae .'/ D '.e/ D '.e/e for ' 2 HomA .eA; A/, is an isomorphism. Then the claim follows from Proposition I.8.2 describing the structure of modules in proj A and proj Aop . Observe that for a module P in proj A we have the canonical evaluation isomorphism "P W P ! P t t in mod A, given by "P .p/.f / D f .p/ for p 2 P and f 2 HomA .P; A/ D P t . We denote by modP A the full subcategory of mod A consisting of all modules without nonzero projective direct summands and by modI A the full subcategory of mod A consisting of all modules without nonzero injective direct summands. Observe that we have a duality modP A o

D

/

D

modI Aop :

Let M be a module in mod A. An exact sequence p1

p0

P1 ! P0 ! M ! 0 in mod A such that p0 W P0 ! M and p1 W P1 ! Im p1 D Ker p0 are projective covers is said to be a minimal projective presentation of M in mod A. For such a

236


minimal projective presentation of M in mod A, we have in mod Aop the induced exact sequence p0t

p1t

M

0 ! M t ! P0t ! P1t ! Coker p1t ! 0: The Aop -module Coker p1t is denoted by Tr.M / and called the transpose of M . We collect now basic properties of the transpose Tr. Proposition 4.5. Let M and N be modules in mod A. The following statements hold. (i) Tr.M / D 0 if and only if M is from proj A. (ii) Tr.M / is a module in modP Aop . (iii) If M is from modP A and p1

p0

P1 ! P0 ! M ! 0 is a minimal projective presentation of M in mod A, then the induced exact sequence p1t

M

P0t ! P1t ! Tr.M / ! 0 is a minimal projective presentation of Tr.M / in mod Aop . (iv) If M is from modP A, then M Š Tr.Tr.M //. (v) If M and N are from modP A, then M Š N in mod A if and only if Tr.M / Š Tr.N / in mod Aop . (vi) Tr.M ˚ N / Š Tr.M / ˚ Tr.N / in mod Aop . p1

p0

Proof. (i) Let P1 ! P0 ! M ! 0 be a minimal projective presentation of M in mod A. Assume M is a module from proj A. Then p0 is an isomorphism, P1 D 0, and consequently Tr.M / D Coker p1t D 0. Conversely, assume that Tr.M / D 0. Then p1t W P0t ! P1t is an epimorphism in mod Aop , and hence a retraction, because P1t is in proj Aop (see Lemma I.8.1). Let r W P1t ! P0t be a homomorphism in mod Aop with p1t r D idP t . Then we obtain the commutative diagram in mod A, 1

P1

p1

"P1

P1t t

p1t t

/ P0

"P0

/ P tt , 0


237

with the evaluation isomorphisms "P0 , "P1 and r t p1t t D idP t t . Then we have 1 ."P11 r t "P0 /p1 D idP1 and p1 is a section in mod A. Hence we obtain a splittable extension p1 p0 0 ! P1 ! P0 ! M ! 0 of P1 by M , and then P0 D P00 ˚ Ker p0 for some A-submodule P00 of P0 (see Lemmas 3.1 and I.4.2). Since p0 W P0 ! M is a projective cover, Ker p0 is a superfluous A-submodule of P0 , and so P0 D P00 . Thus we conclude that Ker p0 D 0, or equivalently, p0 is an isomorphism, which ensures that M is a module from proj A. (ii) Suppose that Tr.M / admits a nonzero projective direct summand Q in mod Aop . Let % W Tr.M / ! Q be the canonical projection. Then the epimorphism %M W P1t ! Q is a retraction, because Q is projective (see Lemma I.8.1), and let W Q ! P1t be a homomorphism in mod A such that %M D idQ . Apt plying the functor ./t , we obtain t .M %t / D .%M /t D idQt , and hence t tt t t M % W Q ! P1 is a section. Then it follows from Lemma I.4.2 that P1t t D t %t / ˚ X , for some right A-submodule X of P1t t . Moreover, we have Im.M t t p1t t .Im.M %t // D Im.p tt t M %t / D Im.%M p1t /t D 0, because M p1t D 0. t t Hence, Im.M % / is a direct summand of P1t t which is contained in Ker p1t t . On the other hand, since p1 W P1 ! Im p1 is a projective cover and p1t t "P1 D "P0 p1 with "P0 , "P1 isomorphisms, we conclude that p1t t W P1t t ! Im p1t t is a projective cover. Therefore, Ker p1t t is a superfluous A-submodule of P1t t . This contradicts t the fact that P1t t D Im.M %t / C X D Ker p1t t C X with X ¤ P1t t . (iii) Assume M is a module in modP A and p1

p0

P1 ! P0 ! M ! 0 is a minimal projective presentation of M in mod A. We have in mod Aop the induced exact sequence p1t

M

P0t ! P1t ! Tr.M / ! 0; where P0t and P1t are from proj Aop . We claim that it is a minimal projective presentation of Tr.M / in mod Aop . Since by (ii) Tr.M / has no nonzero projective direct summands, we then conclude that we have in mod Aop a commutative diagram p1t

P0t idP t

0

U0 ˚ V0 u

w

u0 0v

M

/ Pt 1

/ Tr.M /

idP t

idTr M

1

/ U1 ˚ V1

/0

w0

/ Tr.M /

/ 0,

where U0 ! U1 ! Tr.M / ! 0 is a minimal projective presentation of Tr.M / op in mod A , and v W V0 ! V1 is an epimorphism (possibly zero) for some direct

238


summands V0 ; V1 of P0t ; P1t , respectively, which is in fact a retraction because V1 is projective. Then it suffices to show that V0 D 0 and V1 D 0. Using again the evaluation isomorphisms "P0 W P0 ! P0t t and "P1 W P1 ! P1t t , we conclude that there exists in mod A a commutative diagram p1

P1

p1t t

P1t t Š

/M

0

"P1

U1t ˚ V1t

p0

/ Pt

"

ut 0 0 vt

"P0

#

Š

Tr.M /

/ P tt 0

/0

Š

/ Ut ˚ V t 0 0

"

0 0 0 0 0 1

/ Coker p t t 1 #

/0

Š

/ N1 ˚ N2

/ 0,

where N1 D Coker ut , N2 D Coker v t , 00 W U0t ! N1 , 10 W V0t ! N2 are canonical epimorphisms, all vertical homomorphisms are isomorphisms, and the third horizontal exact sequence is a minimal projective presentation of N D N1 ˚ N2 , because so is the first horizontal exact sequence. Since v t W V1t ! V0t is a section, it follows from Lemma 3.1 that 10 is a retraction, and hence N2 is isomorphic to a direct summand of V0t , being projective in mod A. Since N has no nonzero projective direct summand, because N is isomorphic to M , it follows that 0 t t hN2 D 0. i Hence v is an isomorphism and 1 W V0 ! N2 is zero. Moreover, since 00 0 0 10

is a projective cover of N , we get V0t D 0 and hence V1t D 0, because v t is

an isomorphism. Therefore, by using the evaluation isomorphisms "Vi W Vi ! Vit t , for i D 0; 1, we conclude that V0 D 0 and V1 D 0 as claimed. (iv) Assume M is a module in modP A. Then, by (ii), Tr.M / is a module in modP Aop , and consequently Tr.Tr.M // is a module in modP A. Then we have in mod A a commutative diagram P1

p1

"P1

P1t t

p1t t

p0

/ P0

/M

/0

/ Tr.Tr.M //

/ 0,

"P0

/ P tt 0

Tr.M /

where the upper sequence is a minimal projective presentation of M in mod A, the lower sequence is a minimal projective presentation of Tr.Tr.M // in mod A, and the vertical homomorphisms are isomorphisms. Since M D Coker p1 and Tr.Tr.M // D Coker p1t t , there exists an isomorphism M W M ! Tr.Tr.M // in mod A making the right part of the diagram commutative. (v) Assume M and N are from modP A. Assume there is an isomorphism f W M ! N in mod A. Then, applying Lemma I.8.31, we conclude that there is in


239

mod A a commutative diagram p1

P1 f1

p0

/ P0

/M

f0

Q1

q1

/ Q0

/0

f

/N

q0

/ 0,

where the upper sequence is a minimal projective presentation of M in mod A, the lower sequence is a minimal projective presentation of N in mod A, and f0 , f1 are isomorphisms. Applying the functor ./t we obtain the commutative diagram in mod Aop , q1t

Q0t

/ Qt 1

f0t

P0t

N

/ Tr.N /

/0

M

/ Tr.M /

/ 0,

f1t

/ Pt 1

p1t

where Tr.N / D Coker q1t , Tr.M / D Coker p1t , and f0t , f1t are isomorphisms. Hence there is an isomorphism g W Tr.N / ! Tr.M / in mod Aop making the right part of the diagram commutative. Therefore, M Š N in mod A implies Tr.M / Š Tr.N / in mod Aop . Assume now that Tr.M / Š Tr.N / in mod Aop . Since M and N are in modP A, the modules Tr.M / and Tr.N / are in modP Aop , and we conclude as above that Tr.Tr.M // Š Tr.Tr.N // in mod A. On the other hand, by (iv), we have M Š Tr.Tr.M // and Tr.Tr.N // Š N in mod A. Therefore, we obtain M Š N in mod A. p1 p0 q1 q0 (vi) Let P1 ! P0 ! M ! 0 and Q1 ! Q0 ! N ! 0 be minimal projective presentations of the modules M and N in mod A, respectively. Then M ˚ N admits a minimal projective presentation in mod A of the form

p1 0 0 q1

p0 0 0 q0

P1 ˚ Q1 ! P0 ˚ Q0 ! M ˚ N ! 0: Hence we obtain in mod Aop the commutative diagram

.P0 ˚ Q0 /t 0

P0t ˚ Q0t

p1 0 t 0 q1

"

t 0 p1 t 0 q1

#

/ .P1 ˚ Q1 /t

/ Tr.M ˚ N /

1

/ P t ˚ Qt 1 1

/ Tr.M / ˚ Tr.N / ,

where 0 and 1 are canonical isomorphisms in mod Aop . We conclude that there exists an isomorphism f W Tr.M ˚ N / ! Tr.M / ˚ Tr.N / in mod Aop making the right part of the diagram commutative.

240


As a direct consequence of Proposition 4.5 we obtain the following fact. Corollary 4.6. Let A be a finite dimensional K-algebra over a field K. Then the transpose Tr induces a bijection between the isomorphism classes of modules in modP A and the isomorphism classes of modules in modP Aop . Moreover, Tr carries the indecomposable modules in modP A to indecomposable modules in modP Aop . We will see below that there is no way to extend the above correspondence to a functor Tr from modP A to modP Aop , or from mod A to mod Aop . We have to pass to the stable categories. Theorem 4.7. Let A be a finite dimensional K-algebra over a field K. Then the transpose Tr induces a duality Tr

mod A o

/ mod Aop :

Tr

Proof. We will first show that there exist well-defined contravariant functors Tr W mod A ! mod Aop and Tr W mod Aop ! mod A. Let f W M ! N be a homomorphism in mod A, and p1

p0

P1 ! P0 ! M ! 0;

q1

q0

Q1 ! Q0 ! N ! 0

be minimal projective presentations of M and N in mod A, respectively. Then there exists a commutative diagram in mod A, P1

p1

f1

Q1

/ P0

p0

f0

q1

/ Q0

/M

/0

f q0

/N

/ 0.

Assume we have in mod A another commutative diagram of the form P1

p1

fQ1

Q1

/ P0

p0

fQ0

q1

/ Q0

/M

/0

f q0

/N

/ 0.

Then q0 .f0 fQ0 / D q0 f0 q0 fQ0 D fp0 fp0 D 0. Since Ker q0 D Im q1 and P0 is projective, there exists s0 2 HomA .P0 ; Q1 / such that f0 fQ0 D q1 s0 . Further, we have q1 .f1 fQ1 s0 p1 / D q1 f1 q1 fQ1 q1 s0 p1 D f0 p1 fQ0 p1 .f0 fQ0 /p1 D 0. Hence, there exists s1 2 HomA .P1 ; Ker q1 / such that f1 fQ1 s0 p1 D !s1 , for


241

the canonical embedding ! W Ker q1 ! Q1 . Therefore, we have the equality f1 fQ1 D s0 p1 C !s1 . The above two commutative diagrams in mod A induce two commutative diagrams in mod Aop , 0

/ Nt

0

/ Mt

0

/ Nt

q0t

p0t

q0t

/ Pt 0

/ Qt 0

p0t

/ Pt 0

/ Qt

N

1 f1t

p1t

q1t

/ Pt 1 / Qt

M

N

fQ1t

p1t

/ Pt 1

/ Tr.N /

/0

g

1

fQ0t

ft

0

q1t

f0t

ft

/ Mt

/ Qt 0

/ Tr.M /

/ 0,

/ Tr.N /

/0

gQ

M

/ Tr.M /

/ 0.

Moreover, we have f1t fQ1t D .f1 fQ1 /t D .s0 p1 C !s1 /t D p1t s0t C s1t ! t . Hence we obtain the equalities .g g/ Q N D gN g Q N D M f1t M fQ1t D M .f1t fQ1t / D M .p1t s0t C s1t ! t / D M p1t s0t C M s1t ! t D M s1t ! t : t On the other hand, we have ! t q1t D .q1 !/ Tr.N / D Coker q1t , then D 0. Since t t there exists r 2 HomA Tr.N /; .Ker q1 / such that ! D rN . Hence, we obtain that .g g/ Q N D .M s1t r/N , and consequently g gQ D M s1t r. Observe also t t t t op .Ker q1 / ; P /; Tr.M //, since s 2 Hom 2 that M s1 r 2 PAop .Tr.N and A M 1 1 HomAop P1t ; Tr.M / . Summing up, we have g gQ 2 PAop .Tr.N /; Tr.M //, and so g D gQ in HomAop .Tr.N /; Tr.M //. This allows us to assign to f 2 HomA .M; N / a well-defined element Tr.f / D g 2 HomAop .Tr.N /; Tr.M //. We also note that, if f 0 2 HomA .M; N / and f00 2 HomA .P0 ; Q0 /, f10 2 HomA .P1 ; Q1 / satisfy q0 f00 D f 0 p0 , q1 f10 D f00 p1 , then q0 .f0 C f00 / D .f C f 0 /p0 , q1 .f1 C f10 / D .f0 C f00 /p1 , M .f1 C f10 /t D M .f1t C .f10 /t / D M f1t C M .f10 /t D gN C g 0 N D .g C g 0 /N for g 0 2 HomAop .Tr.N /; Tr.M // with g 0 N D M .f10 /t , and consequently Tr.f C f 0 / D g C g 0 D g C g 0 D Tr.f / C Tr.f 0 /. Moreover, for 2 K, we have q0 .f0 / D .f /p0 , q1 .f1 / D .f0 /p1 , M .f1 /t D M .f1t / D .g/N , and hence Tr.f / D g D g D Tr.f /.Therefore, we have a K-linear homomorphism Tr W HomA .M; N / ! HomAop .Tr.N /; Tr.M //. We will show now that Tr vanishes on all homomorphisms from PA .M; N /. Let h 2 PA .M; N / and h D ˇ˛ for homomorphisms ˛ W M ! P and ˇ W P ! N with P from proj A. Since q0 W Q0 ! N is an epimorphism, there exists 2 HomA .P; Q0 / such that ˇ D q0 . Taking h0 D ˛p0 and h1 D 0 in

242


HomA .P1 ; Q1 /, we obtain the commutative diagram in mod A, p1

P1

h1 D0

Q1

p0

/ P0 h0

/ Q0

q1

/M

/0

h

/N

q0

/0

with h0 p1 D 0, and consequently the commutative diagram in mod Aop , q1t

Q0t

N

/ Qt 1

ht0

ht1 D0

P0t

/ Pt 0

p1t

/ Tr.N /

/0

0

M

/ Tr.M /

/ 0.

Therefore, Tr.h/ D 0. As a consequence, we obtain that the transpose induces a K-linear homomorphism Tr W HomA .M; N / ! HomAop .Tr N; Tr M / such that Tr.f / D Tr.f / for any f 2 HomA .M; N /. The above arguments show also that Tr.f 0 f / D Tr.f / Tr.f 0 / for any homomorphisms f 2 HomA .M; N / and f 0 2 HomA .N; U /, and clearly we have also Tr.idM / D idTr.M / . Summing up, we have defined a contravariant K-linear functor Tr W mod A ! mod Aop : Similarly, we have a contravariant K-linear functor Tr W mod Aop ! mod A D mod.Aop /op : We claim that 1mod A Š Tr B Tr and 1mod Aop Š Tr B Tr, as functors, and hence Tr induces a duality between mod A and mod Aop . Observe first that for a module M in mod A, there is a decomposition M D PM ˚ MP , where PM is a module from proj A and MP is a module from modP A, both uniquely determined (up to isomorphism) by M . Then the canonical section u W MP ! M and retraction v W M ! MP with vu D idMP induce the mutually inverse isomorphisms u W MP ! M and v W M ! MP in mod A. Similarly, for a module X in mod Aop , there exists a direct summand XP of X from modP Aop such that XP and X are isomorphic in mod Aop . Let f W M ! N be a homomorphism in modP A and P1

p1

f1

Q1

/ P0

p0

f0

q1

/ Q0

/M

/0

f q0

/N

/0


243

a commutative diagram in mod A, where the upper sequence is a minimal projective presentation of M in mod A and the lower sequence is a minimal projective presentation of N in mod A. It follows from Proposition 4.5 (ii) that Tr.M /, Tr.N / are modules in modP Aop , and Tr.Tr.M //, Tr.Tr.N // are modules in modP A. Moreover, by Proposition 4.5 (iii), we have a commutative diagram in mod Aop , q1t

Q0t

/ Qt 1

f0t

P0t

N

/ Tr.N /

f1t

p1t

/ Pt 1

/0

g

/ Tr.M /

M

/ 0,

where the upper sequence is a minimal projective presentation of Tr.N / in mod Aop and the lower sequence is a minimal projective presentation of Tr.M / in mod Aop , and a commutative diagram in mod A, P1t t

p1t t

/ P tt

Tr.M /

/ Tr.Tr.M //

0

f1t t

f0t t

Q1t t

q1t t

/ Qt t 0

/0

h

Tr.N /

/ Tr.Tr.N //

/ 0,

with the upper sequence a minimal projective presentation of Tr.Tr.M // in mod A and the lower sequence a minimal projective presentation of Tr.Tr.N // in mod A. It follows also from the first part of the proof that Tr.f / D g and Tr.g/ D h. Further, we have in mod A commutative diagrams p1

P1

"P1

P1t t

p0

/ P0

"P0

p1t t

/ P tt 0

q1

/ Q0

/M

/0

M

Tr.M / / Tr.Tr.M //

/0

and Q1 "Q1

Q1t t

q0

"Q0

q1t t

/ Qt t 0

/N

/0

N

Tr.N / / Tr.Tr.N //

/ 0,

where "P1 , "P0 , "Q1 , "Q0 are the evaluation isomorphisms, and M and N are the induced isomorphisms established in Proposition 4.5 (iv). Combining the above commutative diagrams, we conclude that the following diagram in mod A is com-

244


mutative: M

M

/ Tr.Tr.M //

f

N

h

/ Tr.Tr.N // .

N

Observe that then N f D Tr.Tr.f //M . Therefore, the K-linear isomorphisms M W M ! Tr.Tr.M //, for modules M in modP A, induce an isomorphism of functors 1mod A ! Tr B Tr from mod A to mod A. Similarly, the K-linear isomorphisms X W X ! Tr.Tr.X //, for modules X in modP Aop , induce an isomorphism of functors 1mod Aop ! Tr B Tr from mod Aop to mod Aop . Recall that the standard duality D D HomK .; K/ between mod A and mod Aop induces a duality between the stable categories mod A and mod Aop . Then we obtain the following consequence of Theorem 4.7. Corollary 4.8. Let A be a finite dimensional K-algebra over a field K. Then we have the mutually inverse equivalences of categories mod A o

/

D Tr Tr D

mod A .

In fact, for a module M in mod A, we have well-defined modules in mod A A M D D Tr.M / and

A1 M D Tr D.M /

called the Auslander–Reiten translations of M . The following corollary shows an important property of the Auslander–Reiten translations. Corollary 4.9. Let A be a finite dimensional K-algebra over a field K. Then the Auslander–Reiten translation A induces a bijection from the set of isomorphism classes of nonprojective indecomposable modules in mod A to the set of isomorphism classes of noninjective indecomposable modules in mod A, and A1 is the inverse bijection of A . We end this section with descriptions of the transpose and the Auslander–Reiten translations for finite dimensional hereditary algebras. Theorem 4.10. Let A be a finite dimensional hereditary K-algebra over a field K. Then the following statements hold. (i) The transpose Tr induces functors Tr W mod A ! mod Aop and

Tr W mod Aop ! mod A;


245

and a duality Tr

modP A o

/

Tr

modP Aop :

(ii) The contravariant functors Tr, ExtA1 .; A/ W mod A ! mod Aop are naturally isomorphic. Proof. (i) Recall that for a module M in mod A, there is a decomposition M D PM ˚ MP , where PM is a module from proj A and MP is a module from modP A, both uniquely determined (up to isomorphism) by M . Moreover, since A is hereditary, we have HomA .X; P / D 0 for all modules X in modP A and P 2 proj A. Indeed, suppose there exists a nonzero homomorphism f W X ! P in mod A with X in modP A and P 2 proj A. Then Im f is a nonzero right A-submodule of P , and hence Im f is in proj A, because A is right hereditary. Then the induced f

epimorphism X ! Im f is a retraction, by Lemma I.8.1, and consequently X has a decomposition X D Y ˚ Q, where Q is isomorphic to Im f (see Lemma I.4.2). This contradicts the fact that X is in modP A. Let f W M ! N be a homomorphism in mod A, and M D PM ˚ MP and N D PN ˚NP be decompositions in mod A, described above. Since HomA .MP ; PN / D 0, the restriction of f to MP gives a homomorphism fP W MP ! NP of right Amodules. Since A is right hereditary, it follows from Theorem I.9.1 that pdA MP 1 and pdA NP 1, and hence there are in mod A minimal projective resolutions of MP and NP of the forms p1

p0

q1

0 ! P1 ! P0 ! MP ! 0;

q0

0 ! Q1 ! Q0 ! NP ! 0:

Hence there exists in mod A a commutative diagram 0

/ P1

0

/ Q1

p1

.fP /1 q1

/ P0

p0

.fP /0

/ Q0

q0

/ MP

/0

fP

/ NP

/ 0,

where .fP /0 and .fP /1 are uniquely determined by fP , and hence also by f , because HomA .MP ; Q0 / D 0. This leads to a commutative diagram in mod Aop of the form Q0t

q1t

NP

1

.fP /t0

P0t

/ Qt

p1t

.fP /t1

/ Pt 1

MP

/ Tr.NP /

/0

gP

/ Tr.MP /

/ 0.

246


Observe also that Tr.M / D Tr.MP / and Tr.N / D Tr.NP /. Therefore, we obtain a well-defined homomorphism Tr.f / D gP W Tr.N / ! Tr.M / in mod Aop . Moreover, Tr.idM / D idTr.M / and Tr.f 0 f / D Tr.f / Tr.f 0 / for any homomorphisms f 2 HomA .M; N / and f 0 2 HomA .N; U /. Summing up, the transpose Tr induces a contravariant functor Tr W mod A ! mod Aop : Similarly, since A is also left hereditary, the transpose Tr induces a contravariant functor Tr W mod Aop ! mod A: Moreover, it follows from Proposition 4.5 that there are natural isomorphisms of functors 1modP A ! Tr Tr and 1modP Aop ! Tr Tr : Therefore, the transpose Tr induces a duality between modP A and modP Aop . (ii) Let M be a module in mod A. Since pdA M 1, M admits a minimal projective resolution in mod A of the form d2

d1

d0

0 D P2 ! P1 ! P0 ! M ! 0: Applying the contravariant functor HomA .; A/ W mod A ! mod Aop , we obtain the chain of left A-modules HomA .d1 ;A/

HomA .d2 ;A/

HomA .P0 ; A/ ! HomA .P1 ; A/ ! HomA .P2 ; A/; where HomA .P2 ; A/ D 0. Hence we obtain that ExtA1 .M; A/ D Ker HomA .d2 ; A/= Im HomA .d1 ; A/ D HomA .P1 ; A/= Im HomA .d1 ; A/ D Coker HomA .d1 ; A/: Therefore, we obtain an exact sequence of left A-modules d0t

d1t

0 D M t ! P0t ! P1t ! ExtA1 .M; A/ ! 0: This shows that there is a canonical isomorphism Tr.M / ! ExtA1 .M; A/ in mod Aop , which is natural in M , and consequently we obtain a natural isomorphism of contravariant functors Tr; ExtA1 .; A/ W mod A ! mod Aop :

Corollary 4.11. Let A be a finite dimensional hereditary K-algebra over a field K. Then the Auslander–Reiten translations induce the mutually inverse equivalences of categories D Tr / mod A : mod A o P

Tr D

I

5. The Nakayama functors

247

5 The Nakayama functors The aim of this section is to describe basic properties of the Nakayama functors allowing us to relate projective modules and injective modules over finite dimensional algebras as well as to provide a constructive way to compute the Auslander–Reiten translations of finite dimensional modules. Let A be a finite dimensional K-algebra over a field K. The endofunctors NA D D HomA .; A/ W mod A ! mod A; NA1 D HomAop .; A/D W mod A ! mod A are called the Nakayama functors of mod A. The following lemma exhibits important properties of the Nakayama functors. Lemma 5.1. Let A be a finite dimensional K-algebra over a field K. Then the following statements hold. (i) The functors NA and NA1 define the mutually inverse equivalences of the categories proj A o

NA 1 NA

/ inj A :

(ii) For a module P in proj A, we have top.P / Š soc.NA .P //. (iii) For a module I in inj A, we have top NA1 .I / Š soc I . Proof. We know from Proposition I.8.2 (respectively, Proposition I.8.19) that every indecomposable projective (respectively, indecomposable injective) module in mod A is isomorphic to a module eA (respectively, D.Ae/) for a primitive idempotent e of A. For a primitive idempotent e of A we have isomorphisms NA .eA/ D D HomA .eA; A/ Š D.Ae/; NA1 .D.Ae// D HomAop .; A/D.D.Ae// Š HomAop .Ae; A/ Š eA; and soc.D.Ae// Š eA=e rad A D top.eA/, by Lemma I.8.22. Hence the statements (i), (ii), (iii) follow. We note that, by Lemma 5.1, the Nakayama functors NA and NA1 induce also the functors NA W mod A ! mod A; NA1 W mod A ! mod A: We give now a useful description of the Nakayama functors.

248


Proposition 5.2. Let A be a finite dimensional K-algebra over a field. Then the following statements hold. (i) The functors NA ; ˝A D.A/ W mod A ! mod A are naturally isomorphic. (ii) The functors NA1 ; HomA .D.A/; / W mod A ! mod A are naturally isomorphic. Proof. (i) For a module M in mod A, consider the K-linear homomorphism

M W M ˝A D.A/ ! D HomA .M; A/ D NA .M / defined by M .m ˝ f /.g/ D f .g.m// for m 2 M , f 2 D.A/, and g 2 HomA .M; A/. We claim that M is an isomorphism. Observe that A is the canonical isomorphism A ˝A D.A/ ! D.A/. Hence F is an isomorphism for every module F D Am , with m a positive integer. It follows from Lemma I.8.1 that every module P from proj A is a direct summand of a free module F D Am , and hence

P is a K-linear isomorphism (see Theorem II.4.5). For an arbitrary module M in mod A, consider a minimal projective presentation p1

p0

P1 ! P0 ! M ! 0 of M in mod A. Since both functors ˝A D.A/ and NA are right exact, we obtain the commutative diagram of K-vector spaces P1 ˝A D.A/

p1 ˝A D.A/

/ P0 ˝A D.A/ p0 ˝A D.A// M ˝A D.A/

P1

NA .P1 /

P0

NA .p1 /

/ NA .P0 /

/0

M

NA .p0 /

/ NA .M /

/0

with exact rows, where P0 and P1 are isomorphisms. Then M is also an isomorphism. Observe also that for a homomorphism h W M ! N in mod A the diagram M ˝A D.A/

M

NA .h/

h˝A D.A/

N ˝A D.A/

/ NA .M /

N

/ NA .N /

is commutative. Therefore the family of K-linear isomorphisms M , M modules in mod A, defines a required isomorphism of functors ˝A D.A/ ! NA . (ii) For a module M in mod A, we have the composed K-linear isomorphism M

W HomA .D.A/; M / ! HomA .D.A/; DD.M // ! HomAop .D.M /; A/ D NA1 .M /:


249

Moreover, for a homomorphism h W M ! N in mod A, the diagram M

HomA .D.A/; M /

/ N1 .M / A

HomA .D.A/;h/

HomA .D.A/; N /

N

1 .h/ NA

/ N1 .N / A

is commutative. Hence, the family of K-linear isomorphisms M , M modules in mod A, defines a required isomorphism of functors HomA .D.A/; / ! NA1 . We exhibit now an important connection between the Nakayama functors and the Auslander–Reiten translations. i1 i0 For a module M in mod A, an exact sequence 0 ! M ! I0 ! I1 in mod A such that i0 W M ! I0 and iN1 W Coker i0 ! I1 , induced by i1 , are injective envelopes is said to be a minimal injective copresentation of M in mod A. Proposition 5.3. Let A be a finite dimensional K-algebra over a field K and M a module in mod A. The following statements hold. p1

p0

(i) Let P1 ! P0 ! M ! 0 be a minimal projective presentation of M in mod A. Then there exists an exact sequence in mod A of the form NA .p1 /

NA .p0 /

0 ! A M ! NA .P1 / ! NA .P0 / ! NA .M / ! 0: i0

i1

(ii) Let 0 ! M ! I0 ! I1 be a minimal injective copresentation of M in mod A. Then there exists an exact sequence in mod A of the form 1 .i / NA 0

1 .i / NA 1

0 ! NA1 .M / ! NA1 .I0 / ! NA1 .I1 / ! A1 M ! 0: Proof. (i) Observe that NA D D HomA .; A/ D D./t . From the definition of Tr.M / we have the following exact sequence in mod Aop : p0t

p1t

M

0 ! M t ! P0t ! P1t ! Tr.M / ! 0: Applying the standard duality functor D W mod Aop ! mod A, we obtain an exact sequence in mod A, D.M /

D.p1t /

D.p0t /

0 ! D.Tr.M // ! D.P1t / ! D.P0t / ! D.M t / ! 0; which is the required exact sequence.

250


(ii) Applying the standard duality D W mod A ! mod Aop to the given minimal injective copresentation of M in mod A, we obtain, by Proposition I.8.16, a minimal projective presentation D.i1 /

D.i0 /

D.I1 / ! D.I0 / ! D.M / ! 0 of the module M in mod Aop , and consequently an exact sequence in mod A of the form D.i0 /t

D.i1 /t

0 ! D.M /t ! D.I0 /t ! D.I1 /t ! Tr.D.M // ! 0: Since NA1 D HomAop .; A/D D ./t D, it is the required exact sequence.

The above proposition allows us to prove an easy criterion for a module to have a projective, or injective, dimension at most 1. Proposition 5.4. Let A be a finite dimensional K-algebra over a field K and M a module in mod A. The following equivalences hold. (i) pdA M 1 if and only if HomA .D.A/; A M / D 0. (ii) idA M 1 if and only if HomA .A1 M; A/ D 0. Proof. (i) From Proposition 5.3 (i), we have in mod A an exact sequence NA .p1 /

NA .p0 /

0 ! A M ! NA .P1 / ! NA .P0 / ! NA .M / ! 0; induced by a minimal projective presentation p1

p0

P1 ! P0 ! M ! 0 of M in mod A. Since NA1 D HomAop .; A/D is a left exact functor, we obtain a commutative diagram in mod A of the form 0

/ N1 .A M / A

/ N1 .NA .P1 // A

/ Ker p1

/ P1

1 .N .p // NA A 1

/ N1 .NA .P0 // A

Š

0

Š

p1

/ P0

p0

/M

/ 0.

Hence Ker p1 is isomorphic to NA1 .A M / in mod A. On the other hand, it follows from Proposition 5.2 (ii) that NA1 .A M / Š HomA .D.A/; A M /. Clearly, pdA M 1 if and only if Ker p1 D 0. This shows that the equivalence (i) holds. (ii) It follows from Proposition I.8.16 that idA M 1 if and only if pdAop D.M / 1. The second inequality is, from the first part of the proof,


251

equivalent to HomAop .D.AA /; Aop D.M // D 0. Observe now that we have isomorphisms of K-vector spaces HomAop .D.AA /; Aop D.M // D HomAop .D.A/; D Tr.D.M /// Š HomA .Tr.D.M //; A/ D HomA .A1 M; A/. Therefore, the equivalence (ii) also holds. We present also an example showing that Proposition 5.3 gives a constructive method to compute the Auslander–Reiten translations of a module. Example 5.5. Let A D KQ be the path algebra of the quiver

QW

1 _@ @@ @@˛ @@

2

o ~3 ~ ~ ~~ ~~ ˇ

o 4

5

over a field K. We identify mod A D repK .Q/ and consider the indecomposable A-module (representation of Q over K)

MW

0 _?? ?? ?? ??

0

Ko

1

Ko

0.

We use the description of the indecomposable projective and indecomposable injective A-modules given in Proposition I.8.27. Then we have in mod A a minimal projective presentation of M of the form p1

p0

P .1/ ˚ P .2/ ! P .4/ ! M ! 0; where p1 is the canonical inclusion homomorphism of soc.P .4// D S.1/ ˚ S.2/ D P .1/ ˚ P .2/ ! P .4/ and p0 is the canonical epimorphism P .4/ ! P .4/= soc.P .4// D M . Applying Lemma 5.1 and Proposition 5.3, we obtain an exact sequence in mod A of the form NA .p1 /

0 ! A M ! NA .P .1/ ˚ P .2// ! NA .P .4//; where NA .P .1/ ˚ P .2// D I.1/ ˚ I.2/ and NA .P .4// D I.4/. Since M is indecomposable nonprojective in mod A, the module A M is indecomposable noninjective in mod A, by Corollary 4.9. Further, HomA .I.1/; I.4// and HomA .I.2/; I.4//

252


are the one-dimensional K-vector spaces generated by the canonical epimorphisms u1 W I.1/ ! I.1/= rad2 I.1/ D I.4/ and u2 W I.2/ ! I.2/= rad2 I.2/ D I.4/, respectively. Hence A M D Ker NA .p1 / is isomorphic to the kernel of the epimorphism Œ u1 u2 W I.1/ ˚ I.2/ ! I.4/ which is the projective A-module P .3/. Therefore, we have A M Š P .3/. Dually, we have in mod A a minimal injective copresentation of M of the form i0

i1

0 ! M ! I.3/ ! I.5/; where i0 is the inclusion homomorphism M D rad I.3/ ! I.3/ and i1 is the canonical epimorphism I.3/ ! top.I.3// D S.5/ D I.5/. Applying Lemma 5.1 and Proposition 5.3, we obtain an exact sequence in mod A of the form 1 .i / NA 1

NA1 .I.3// ! NA1 .I.5// ! A1 M ! 0 , where NA1 .I.3// D P .3/ and NA1 .I.5// D P .5/. Since M is indecomposable noninjective in mod A, we know from Corollary 4.9 that A1 M is indecomposable nonprojective in mod A. Moreover, HomA .P .3/; P .5// is the one-dimensional Kvector space generated by the inclusion homomorphism v W P .3/ D rad2 P .5/ ! P .5/. Hence A1 M D Coker NA1 .i1 / is isomorphic to Coker v, which is the injective module I.4/. Therefore, we have A1 M Š I.4/.

6 The Auslander–Reiten formulas The aim of this section is to prove the Auslander–Reiten formulas which allow us to describe the extension spaces between finite dimensional modules by the corresponding stable homomorphism spaces. Let A be a finite dimensional K-algebra over a field K. For modules X and Y in mod A, we consider the K-linear homomorphism 'YX W Y ˝A X t ! HomA .X; Y / given by 'YX .y ˝ f /.x/ D yf .x/, for y 2 Y , f 2 X t D HomA .X; A/, and x 2 X. Observe that for homomorphisms u W U ! X and v W Y ! V in mod A the diagrams Y ˝A X t

X 'Y

Y ˝ut

Y ˝A U t

/ HomA .X; Y / HomA .u;Y /

U 'Y

/ HomA .U; Y /

Y ˝A X t and

X 'Y

v˝X t

V ˝A X t

/ HomA .X; Y / HomA .X;v/

X 'V

/ HomA .X; V /

are commutative, that is, the homomorphisms 'YX are natural in X and Y . We have also the following fact.

6. The Auslander–Reiten formulas

253

Lemma 6.1. Let X and Y be modules in mod A. Then we have the exact sequence of K-vector spaces X Y

X 'Y

Y ˝A X t ! HomA .X; Y / ! HomA .X; Y / ! 0 where the right homomorphism YX assigns to f 2 HomA .X; Y / its stable class f in HomA .X; Y /. In particular, if X or Y is projective, then 'YX is an isomorphism. Proof. Let f W P ! Y be a projective cover of Y in mod A. We show first that the induced sequence X Y

HomA .X;f /

HomA .X; P / ! HomA .X; Y / ! HomA .X; Y / ! 0 is exact, or equivalently Im HomA .X; f / D PA .X; Y / D Ker YX . Clearly, we have Im HomA .X; f / PA .X; Y /. Take now a homomorphism g 2 PA .X; Y /. Then there exist a module P 0 in proj A and g1 2 HomA .P 0 ; Y /, g2 2 HomA .X; P 0 / such that g D g1 g2 . Then we have in mod A a commutative diagram P0 } h }}} g1 }} } ~} f /Y P since f is an epimorphism. Hence, we obtain that g D g1 g2 D f hg2 D HomA .X; f /.hg2 / 2 Im HomA .X; f /. Therefore, indeed we have the required equality Im HomA .X; f / D PA .X; Y /. We claim now that 'PX W P ˝A X t ! HomA .X; P / is an isomorphism. Observe that 'AX is the canonical isomorphism A ˝A X t ! X t D HomA .X; A/. Then, for a positive integer m, the homomorphism 'AXm W Am ˝A X t ! HomA .X; Am / is also an isomorphism. It follows also from Lemma I.8.1 that there exists a module Q in mod A such that P ˚ Q Š Am for some positive integer m. Hence, 'PX is also an isomorphism. We have the commutative diagram in mod K of the form P ˝A X t

f ˝X t

X 'P

HomA .X; P /

/ Y ˝A X t

/0

X 'Y

HomA .X;f /

/ HomA .X; Y /

X Y

/ Hom .X; Y / , A

with 'PX an isomorphism and f ˝ X t an epimorphism, because the functor ˝A X t W mod A ! mod K is right exact. Then we obtain Im 'YX D 'YX .Y ˝A X t / D 'YX .f ˝ X t /.P ˝A X t / D HomA .X; f /.'PX .P ˝A X t // D HomA .X; f /.HomA .X; P // D PA .X; Y /:

254


Therefore, Im 'YX D Ker YX , as required. For X D Q projective, 'YQ is an isomorphism by Corollary II.4.9. We have also a useful consequence of the above lemma. Corollary 6.2. Let X and Y be modules in mod A. Then there exists an exact sequence of K-vector spaces X/ D.Y

X !Y

0 ! D HomA .X; Y / ! D HomA .X; Y / ! HomA .Y; NA .X //; which is natural in X and Y . In particular, if X or Y is projective, then !YX is an isomorphism. Proof. Applying the functor D D HomK .; K/ to the exact sequence in Lemma 6.1, we obtain the exact sequence of K-vector spaces X/ D.Y

X/ D.'Y

0 ! D HomA .X; Y / ! D HomA .X; Y / ! D.Y ˝A X t /: Moreover, we have the adjoint K-linear isomorphism ! HomA .Y; D.X t // D HomA .Y; NA .X //: YX W D.Y ˝A X t /

Then, for !YX D YX D.'YX / W D HomA .X; Y / ! HomA .Y; NA .X //, we have Ker !YX D Ker D.'YX / D Im D.YX /. Observe that !YX is natural in X and Y , because 'YX and YX have this property. If X or Y is projective then, by Lemma 6.1, 'YX is an isomorphism, and consequently !YX D YX D.'YX / is an isomorphism. We are now in position to establish the Auslander–Reiten formulas. Theorem 6.3. Let A be a finite dimensional K-algebra over a field K and M , N be modules in mod A. Then there exist isomorphisms of K-vector spaces D HomA .A1 N; M / Š ExtA1 .M; N / Š D HomA .N; A M /; which are natural in M and N . Proof. We first prove that ExtA1 .M; N / Š D HomA .A1 N; M /. Observe that, if I is a module in inj A, then A1 I D Tr D.I / D 0, because D.I / is in proj Aop , and clearly ExtA .M; I / Š ExtA1 .M; I / D 0, by Lemma I.8.13. Similarly, if P is a module in proj A, then ExtA1 .P; N / D 0 and HomA .A1 N; P / D 0. Therefore, we may assume that M belongs to modP A and N belongs to modI A. In particular, then we have L D A1 N in modP A and N Š A L in mod A (see Corollary 4.9). Let p1 p0 P1 ! P0 ! L ! 0 be a minimal projective presentation of the module L in mod A.

6. The Auslander–Reiten formulas

255

It follows from Proposition 5.3 (i) that we have in mod A the exact sequence NA .p1 /

D.L /

NA .p0 /

0 ! A L ! NA .P1 / ! NA .P0 / ! NA .L/ ! 0; obtained by applying the duality D W mod Aop ! mod A to the exact sequence in mod Aop p0t

p1t

L

0 ! Lt ! P0t ! P1t ! Tr L ! 0: Since L belongs to modP A, it follows from Proposition 4.5 (iii) that p1t

L

P0t ! P1t ! Tr L ! 0 is a minimal projective presentation of Tr L in mod A. Hence D.L /

NA .p1 /

0 ! A L ! NA .P1 / ! NA .P0 / is a minimal injective copresentation of A L in mod A. This implies that A L admits in mod A a minimal injective resolution d0

d1

d2

d nC1

0 ! A L ! I0 ! I1 ! I2 ! ! In ! InC1 ! such that I0 D NA .P1 /, I1 D NA .P0 /, d 0 D D.L /, d 1 D NA .p1 /, and d 2 is the composition of NA .p0 / with an injective envelope u W NA .L/ ! E.NA .L// D I2 of NA .L/ in mod A. It follows from Corollary 3.11 (i) that ExtA1 .M; A L/ is isomorphic to ExtA1 .M; A L/ D Ker HomA .M; d 2 /= Im HomA .M; d 1 /. Moreover, d 2 D uNA .p0 / with u a monomorphism, and hence Ker HomA .M; d 2 / D Ker HomA .M; NA .p0 //. Summing up, we have proved that there are isomorphisms of K-vector spaces

e

ExtA1 .M; N / Š ExtA1 .M; A L/ Š Ker HomA .M; NA .p0 //= Im HomA .M; NA .p1 //: In the following we abbreviate pN0 D HomA .M; NA .p0 // D HomA .M; D.p0t // and pN1 D HomA .M; NA .p1 // D HomA .M; D.p1t //. Applying the right exact functor D HomA .; M / to the exact sequence in mod A, p0 p1 P1 ! P0 ! L ! 0; we obtain the exact sequence in mod K, pQ1

pQ0

D HomA .P1 ; M / ! D HomA .P0 ; M / ! D HomA .L; M / ! 0; where pQ1 D D HomA .p1 ; M / and pQ0 D D HomA .p0 ; M /.

256


It follows from Corollary 6.2 that we have the following commutative diagram in mod K, D HomA .P1 ; M /

pQ1

/ D HomA .P0 ; M /

P

/ D HomA .L; M /

P

!M1

HomA .M; NA .P1 //

pQ0

L !M

!M0

pN 1

/ HomA .M; NA .P0 //

pN 0

/ HomA .M; NA .L// ,

P1 P0 where !M and !M are isomorphisms, and L Ker !M Š D HomA .L; M / D D HomA .A1 N; M /:

On the other hand, we know that ExtA1 .M; N / Š Ker pN0 = Im pN1 : P0 1 / W HomA .M; NA .P0 // ! Consider the K-linear homomorphism pQ0 .!M P P0 1 L pQ0 .!M0 /1 D pN0 , pQ0 .!M / induces a K-linear D HomA .L; M /. Since !M P0 1 L homomorphism ˛ W Ker pN0 ! Ker !M , and Ker pQ0 .!M / Ker pN0 . Observe L that ˛ is an epimorphism. Indeed, take f 2 Ker !M . Since pQ0 is an epimorP0 1 phism, we have f D pQ0 .!M / .g/ for some g 2 HomA .N; NA .P0 //. Moreover, P0 1 L L pQ0 .!M / .g/ D !M .f / D 0, and so g 2 Ker pN0 . Hence f D ˛.g/ pN0 .g/ D !M and the claim follows. As a consequence, ˛ induces a K-linear isomorphism L : Ker pN0 = Ker ˛ ! Ker !M

Hence, in order to prove that ExtA1 .M; N / Š D HomA .A1 N; M /, it is enough to P0 1 / pN1 D show that Ker ˛ D Im pN1 . Observe that Im pN1 Ker ˛, because pQ0 .!M P1 1 P0 1 pQ0 pQ1 .!M / , and clearly pQ0 pQ1 D 0. Take h 2 Ker pQ0 .!M / . Then we have P0 1 P0 1 / .h/ 2 Ker pQ0 D Im pQ1 , and .!M / .h/ D pQ1 .w/ for some homothat .!M P0 P0 1 .!M / .h/ D morphism w 2 D HomA .P1 ; M /. Then we obtain that h D !M P0 P1 !M pQ1 .w/ D pN1 !M .w/ 2 Im pN1 . This shows that Ker ˛ D Im pN1 . Therefore, we established a K-linear isomorphism ExtA1 .M; N / Š D HomA .A1 N; M /, which is natural in M and N . Applying the duality D D HomK .; K/ between mod A and mod Aop , we obtain also K-linear isomorphisms D ExtA1 .M; N / Š D ExtA1 .M; N / Š ExtA1 op .D.N /; D.M // Š ExtA1 op .D.N /; D.M // Š D HomAop .A1 op D.M /; D.N // Š D HomAop .Tr D.D.M //; D.N // Š D HomAop .Tr M; D.N // Š HomA .DD.N /; D Tr M / Š HomA .N; A M /:

7. Irreducible and almost split homomorphisms

257

Therefore, we have a K-linear isomorphism ExtA1 .M; N / Š D HomA .N; A M /, which is natural in M and N . Corollary 6.4. Let A be a finite dimensional K-algebra over a field K and M , N be modules in mod A. Then the following statements hold. (i) If pdA M 1, then there exists a K-linear isomorphism ExtA1 .M; N / Š D HomA .N; A M /: (ii) If idA N 1, then there exists a K-linear isomorphism ExtA1 .M; N / Š D HomA .A1 N; M /: Proof. (i) It follows from Theorem 6.3 that there exists a K-linear isomorphism ExtA1 .M; N / Š D HomA .N; A M /. By statement (i) of Proposition 5.4, the assumption pdA M 1 forces HomA .D.A/; A M / D 0. Moreover, it follows from Proposition I.8.19 that for every module I in inj A there exists a module I 0 in mod A such that I ˚ I 0 is isomorphic to D.A/m for some positive integer m. Hence HomA .D.A/; A M / D 0 implies that IA .N; A M / D 0, or equivalently HomA .N; A M / D HomA .N; A M /. Therefore, we have an isomorphism ExtA1 .M; N / Š D HomA .N; A M /. (ii) It follows from Theorem 6.3 that there exists a K-linear isomorphism ExtA1 .M; N / Š D HomA .A1 N; M /. Further, the assumption idA N 1 forces, by Proposition 5.4 (ii), that HomA .A1 N; A/ D 0. Finally, it follows from Proposition I.8.2, that for every module P in proj A there exists a module P 0 in mod A such that P ˚ P 0 is isomorphic to Am for some positive integer m. Then we conclude that PA .A1 N; M / D 0, or equivalently HomA .A1 N; M / D HomA .A1 N; M /. Therefore, we have a required isomorphism ExtA1 .M; N / Š D HomA .A1 N; M /.

7 Irreducible and almost split homomorphisms Let A be a finite dimensional K-algebra over a field K, and L; M; N modules in mod A. A homomorphism f W L ! M in mod A is said to be left minimal if every homomorphism h 2 EndA .M / with hf D f is an isomorphism. A homomorphism g W M ! N in mod A is said to be right minimal if every homomorphism h 2 EndA .M / with gh D g is an isomorphism. A homomorphism f W L ! M in mod A is said to be left almost split if the following conditions are satisfied: (i) f is not a section in mod A;

258


(ii) for every homomorphism u W L ! U in mod A which is not a section, there exists a homomorphism u0 W M ! U such that u D u0 f , that is, making the diagram f

L@ @@ @@ u @@

U

/M } } }} }} u0 } ~}

commutative. A homomorphism g W M ! N in mod A is said to be right almost split if the following conditions are satisfied: (i) g is not a retraction in mod A; (ii) for every homomorphism v W V ! N in mod A which is not a retraction, there exists a homomorphism v 0 W V ! M in mod A such that v D gv 0 , that is, making the diagram M À AA AA A v 0 AA

g

V

/N }> } } }}v }}

commutative. A homomorphism f W L ! M in mod A is said to be left minimal almost split if f is left minimal and left almost split in mod A. A homomorphism g W M ! N in mod A is said to be right minimal almost split if g is right minimal and right almost split in mod A. Lemma 7.1. (i) Let f W L ! M and f 0 W L ! M 0 be left minimal almost split homomorphisms in mod A. Then there exists an isomorphism h W M ! M 0 such that f 0 D hf . (ii) Let g W M ! N and g 0 W M 0 ! N be right minimal almost split homomorphisms in mod A. Then there exists an isomorphism h W M ! M 0 such that g D g 0 h. Proof. (i) Since f and f 0 are left almost split homomorphisms in mod A, there exist homomorphisms h W M ! M 0 and h0 W M 0 ! M such that f 0 D hf and f D h0 f 0 . Then we obtain f D h0 hf and f 0 D hh0 f 0 . Moreover, f and f 0 are left minimal homomorphisms in mod A. Hence h0 h 2 EndA .M / and hh0 2 EndA .M 0 / are isomorphisms. In particular, h W M ! M 0 is a monomorphism and an epimorphism, and consequently an isomorphism in mod A. The proof of (ii) is similar.


259

Lemma 7.2. (i) Let f W L ! M be a left almost split homomorphism in mod A. Then L is an indecomposable A-module. (ii) Let g W M ! N be a right almost split homomorphism in mod A. Then N is an indecomposable A-module. Proof. (i) Observe first that L ¤ 0 because f is not a section. Assume L D L1 ˚ L2 for some nonzero A-submodules L1 and L2 of L, and let p1 W L ! L1 and p2 W L ! L2 be the canonical projections. Then Ker p1 D L2 ¤ 0 and Ker p2 D L1 ¤ 0 imply that p1 and p2 are not sections. Since f is a left almost split homomorphism in mod A, there exist homomorphisms u1 W M ! L1 and u2 W M ! L2 in mod A such that p1 D u1 f and p2 D u2 f . Consider the A-homomorphism u u D 1 W M ! L1 ˚ L2 D L: u2 Then, for each x 2 L, we have .uf /.x/ D u .f .x// D .u1 .f .x// ; u2 .f .x/// D .p1 .x/; p2 .x// D x D idL .x/, and hence uf D idL . Hence f is a section in mod A, a contradiction because f is a left almost split homomorphism in mod A. Therefore, L is an indecomposable A-module. The proof of (ii) is dual. Lemma 7.3. Let f W X ! Y be the zero homomorphism between two modules X and Y in mod A. Then the following equivalences hold. (i) f is a left minimal almost split homomorphism in mod A if and only if X is a simple injective A-module and Y D 0. (ii) f is a right minimal almost split homomorphism in mod A if and only if X D 0 and Y is a simple projective A-module. Proof. (i) Assume f is a left minimal almost split homomorphism in mod A. Then X is nonzero and, for every homomorphism v W X ! V in mod A which is not a section, we have v D v 0 f for some homomorphism v 0 W Y ! V in mod A, and so v D 0, because f D 0. Observe that then X is a simple module, because otherwise we have a proper nonzero epimorphism X ! X=L for an A-submodule L of X different from X and 0. Further, the injective envelope u W X ! E.X / of X in mod A is a nonzero homomorphism, and hence a section. Moreover, by Corollary I.8.21, E.X / is an indecomposable injective A-module. Hence it follows from Lemma I.4.2 that u is an isomorphism, and consequently X is a simple injective module in mod A. Finally, for the zero endomorphism 0Y W Y ! Y we have f D 0Y f , so, by the left minimality of f , 0Y is an isomorphism, or equivalently, Y D 0. Conversely, if X is a simple injective A-module and Y D 0, then f W X ! 0 is a left minimal homomorphism in mod A, because every nonzero homomorphism u W X ! M in mod A is a section, by Lemma I.8.13.

260


The proof of (ii) is similar.

A homomorphism f W X ! Y in mod A is said to be irreducible if the following conditions are satisfied: (i) f is neither a section nor a retraction in mod A; (ii) if f D f1 f2 for some homomorphisms f2 W X ! Z and f1 W Z ! Y in mod A, then either f2 is a section or f1 is a retraction in mod A. Observe that, by the property (i), for an irreducible homomorphism f W X ! Y in mod A, we have X ¤ 0 and Y ¤ 0. The following lemma will be useful. Lemma 7.4. Let f W X ! Y be a homomorphism and u W U ! X, v W Y ! V isomorphisms in mod A. Then the following equivalences hold. (i) f is a section in mod A if and only if vf u is a section in mod A. (ii) f is a retraction in mod A if and only if vf u is a retraction in mod A. (iii) f is an irreducible homomorphism in mod A if and only if vf u is an irreducible homomorphism in mod A. (iv) f is a left minimal almost split homomorphism in mod A if and only if vf u is a left minimal almost split homomorphism in mod A. (v) f is a right minimal almost split homomorphism in mod A if and only if vf u is a right minimal almost split homomorphism in mod A. Proof. Let t W X ! U and w W V ! Y be homomorphisms in mod A with ut D idX , tu D idU , wv D idY , and vw D idV . (i) Assume f is a section in mod A and rf D idX for some r 2 HomA .Y; X /. Then t rw 2 HomA .V; U / and .t rw/.vf u/ D t .r.wv/f /u D t .rf /u D idU , and so vf u is a section in mod A. Conversely, assume that vf u is a section in mod A and s.vf u/ D idU for some s 2 HomA .V; U /. Then .usv/f D .usv/f .ut / D u.svf u/t D ut D idX , and so f is a section in mod A. Therefore, the equivalence (i) holds. The proof of (ii) is similar to the proof of (i). (iii) It follows from (i) and (ii) that f is neither a section nor a retraction in mod A if and only if vf u is neither a section nor a retraction in mod A. Assume vf u D hg for some homomorphisms g W U ! Z and h W Z ! V in mod A. Then we have f D .wv/f .ut / D w.vf u/t D w.hg/t D .wh/.gt / with gt W X ! Z and wh W Z ! Y . Conversely, if f D qp for some homomorphisms p W X ! W and q W W ! Y in mod A, then vf u D vqpu D .vq/.pu/ with pu W U ! W and vq W W ! V . Therefore, applying (i) and (ii), we conclude that the equivalence (iii) holds.


261

(iv) It follows from (i) that f is not a section in mod A if and only if vf u is not a section in mod A. We claim that f is left minimal if and only if vf u is left minimal. Assume f is left minimal and h 2 EndA .V / satisfies h.vf u/ D vf u. Then .whv/f D f , and so whv is an isomorphism, which implies that h D v.whv/w is an isomorphism. Conversely, assume that vf u is left minimal and gf D f for some g 2 EndA .Y /. Then .vgw/.vf u/ D v.gf /u D vf u, and hence vgw is an isomorphism, which implies that g D w.vgw/v is an isomorphism. Finally, we show that f is left almost split in mod A if and only if vf u is almost split in mod A. Assume f is left almost split in mod A, and h W U ! M be a homomorphism in mod A which is not a section. Then, by (i), ht W X ! M is not a section in mod A and, by assumption on f , there exists a homomorphism ' W Y ! M in mod A such that ht D 'f . Then g D 'w W V ! M is a homomorphism in mod A and gvf u D .'w/.vf u/ D .'f /u D ht u D h. This shows that vf u is left almost split in mod A. Conversely, assume that vf u is left almost split in mod A, and let ˛ W X ! N be a homomorphism in mod A which is not a section. Then, by (i), ˛u W U ! N is not a section in mod A, and, by assumption on vf u, we conclude that there is a homomorphism W V ! N such that ˛u D .vf u/. But then ˇ D v W Y ! N is a homomorphism in mod A and ˇf D . v/f .ut / D . vf u/t D ˛ut D ˛. This shows that f is left almost split. The proof of (v) is similar to the proof of (iv). Lemma 7.5. Let f W X ! Y be an irreducible homomorphism in mod A. Then either f is a proper monomorphism or a proper epimorphism. Proof. Consider the factorization of f , f

/Y XD z= DD z DD zz D zz p DD zz j ! Im f ,

where p is the epimorphism induced by f and j is the inclusion map. Since f is an irreducible homomorphism, we conclude that either p is a section or j is a retraction. Moreover, f is not an isomorphism in mod A. Assume f is not a proper epimorphism. Then j W Im f ! Y is not a retraction in mod A, because otherwise Im f D Y and f is a proper epimorphism. Therefore, p W X ! Im f is a section in mod A, and hence an isomorphism. This implies that f W X ! Y is a proper monomorphism. Lemma 7.6. Let P be an indecomposable projective module in mod A and let u W rad P ! P be the inclusion homomorphism. Then (i) u is a right minimal almost split homomorphism in mod A;

262


(ii) if P is a nonsimple A-module, then u is an irreducible homomorphism in mod A. Proof. It follows from Propositions I.5.16 and I.8.2 that rad P is a unique maximal right A-submodule of P . In particular, u W rad P ! P is not a retraction, since u is not an epimorphism. Let v W V ! P be a homomorphism in mod A which is not a retraction. Then v is not an epimorphism, by Lemma I.8.1. Hence Im v is a proper A-submodule of P , and so Im v rad P . Therefore, we obtain v D uv 0 , for the A-homomorphism v 0 W V ! rad P induced by v. This shows that u is a right almost split homomorphism in mod A. Observe also that, if uh D u for some h 2 EndA .rad P /, then h D idrad P . Hence u W rad P ! P is also right minimal. Therefore, u is a right minimal almost split homomorphism in mod A. (ii) Since P is indecomposable and rad P is a proper submodule of P , we infer that u is neither a section nor a retraction. Assume u D gf for some homomorphisms f W rad P ! M and g W M ! P in mod A. Moreover, assume that g is not a retraction. Then g is not an epimorphism, by the projectivity of P , and so Im g rad P . Then there exists a homomorphism h W M ! rad P such that g D uh. Hence we obtain u idrad P D u D gf D uhf , and then hf D idrad P since u is a monomorphism. This shows that f is a section. Therefore, u is an irreducible homomorphism. Lemma 7.7. Let I be an indecomposable injective module in mod A and v W I ! I = soc I the canonical epimorphism. Then (i) v is left minimal almost split homomorphism in mod A; (ii) if I is nonsimple A-module, then v is an irreducible homomorphism in mod A. Proof. Consider the duality D W mod A ! mod Aop . The canonical short exact sequence in mod A, v

! I = soc I ! 0; 0 ! soc I , !I gives a short exact sequence in mod Aop , D.v/

0 ! D.I = soc I / ! D.I / ! D.soc I / ! 0; where D.I / is an indecomposable projective module in mod Aop and D.soc I / Š top D.I /, by Proposition I.8.16. Therefore D.v/ induces an isomorphism D.I = soc I / ! rad D.I /. Moreover, v is a left minimal almost split homomorphism (respectively, irreducible homomorphism) in mod A if and only if D.v/ is a right minimal almost split homomorphism (respectively, irreducible homomorphism) in mod Aop . Then the lemma follows from Lemma 7.6. The following important fact has been proved by R. Bautista in [Bau].


263

Lemma 7.8. Let X and Y be indecomposable modules in mod A and let f 2 HomA .X; Y /. Then f is an irreducible homomorphism in mod A if and only if f 2 radA .X; Y / n radA2 .X; Y /. Proof. Assume f is an irreducible homomorphism in mod A. Then f is not an isomorphism, because f is neither a section nor a retraction. Applying Lemma 1.4 we then infer that f 2 radA .X; Y /. We show that f … radA2 .X; Y /. Suppose f D gh for some homomorphisms h 2 radA .X; Z/ and g 2 radA .Z; Y /. Let Z D Z1 ˚ ˚ Z t be a decomposition of Z into a direct sum of indecomposable A-submodules. Let 2 3 h1 t M 6 :: 7 Zi ; h D 4 : 5 W X ! iD1 ht for hi 2 HomA .X; Zi /, i 2 f1; : : : ; tg, and g D Œg1 ; : : : ; g t W

t M

Zi ! Y;

iD1

for gi 2 HomA .Zi ; Y /, i 2 f1; : : : ; tg. Since f D gh is an irreducible homomorphism in mod A, we conclude that either h is a section or g is a retraction in mod A. Lt Suppose h is a section in mod A, and h0 D Œh01 ; : : : ; h0t W iD1 Zi ! X , with h0i 2 HomA .Zi ; X /, i 2 f1; : : : ; tg, is a homomorphism in mod A such that P idX D h0 h D tiD1 h0i hi . Since h 2 radA .X; Z/ it follows from Lemma 1.3 that hi 2 radA .X; Zi / for any i 2 f1; : : : ; tg. Then, by Proposition 1.2, we P conclude that h0i hi 2 radA .X; X / for any i 2 f1; : : : ; tg, and consequently idX D tiD1 h0i hi 2 radA .X; X/ D rad EndA .X /. This is a contradiction, because X is an indecomposable module in mod A and then rad EndA .X / is a unique maximal right ideal of the local K-algebra EndA .X /. Suppose g is a retraction in mod A, and 2 03 g1 t M 6 :: 7 0 g D 4 : 5 W Y ! Zi ; 0 iD1 gt for gi0 2 HomA .Y; Zi /, i 2 f1; : : : ; tg, a homomorphism in mod A such that P idY D gg 0 D tiD1 gi gi0 . Since g 2 radA .Z; Y /, applying Lemma 1.3 again, we conclude that gi 2 radA .Zi ; Y / for any i 2 f1; : : : ; tg. Hence, by Proposition 1.2, we have gi gi0 2 radA .Y; Y / for any i 2 f1; : : : ; tg, and consequently P idY D tiD1 gi gi0 2 radA .Y; Y / D rad EndA .Y /. This is again a contradiction, because Y is an indecomposable module in mod A and so rad EndA .Y / is a unique

264


maximal right ideal of the local K-algebra EndA .Y /. Summing up, we have proved that f 2 radA .X; Y / n radA2 .X; Y /. Conversely, assume that f 2 radA .X; Y / n radA2 .X; Y /. Since f 2 radA .X; Y / and X; Y are indecomposable, we infer that f is not an isomorphism, and hence is neither a section nor a retraction in mod A, by Lemma I.4.2. Suppose f has a factorization XA AA AA A u AA

f

M

/Y > }} } } }} v }}

Lr

in mod A. Let M D iD1 Mi be a decomposition of M into a direct sum of indecomposable A-submodules and 2 3 u1 r r M M 6 7 u D 4 ::: 5 W X ! Mi and v D Œv1 ; : : : ; vr W Mi ! Y iD1 iD1 ur with ui 2 Hom P A .X; Mi / and vi 2 HomA .Mi ; Y /, for i 2 f1; : : : ; rg. Then we have f D vu D riD1 vi ui . Because f … radA2 .X; Y /, there exists i 2 f1; : : : ; rg such that ui … radA .X; Mi / or there exists j 2 f1; : : : ; rg such that vj … radA .Mj ; Y /. Applying Lemma 1.4 we then obtain that either ui W X ! Mi is an isomorphism, for some i 2 f1; : : : ; rg, or vj W Mj ! Y is an isomorphism, for some j 2 f1; : : : ; rg. Hence either u is a section or v is a retraction in mod A. Therefore, f W X ! Y is an irreducible homomorphism in mod A. We mention that there exist finite dimensional K-algebras A and irreducible homomorphisms f W X ! Y and g W Y ! Z between indecomposable modules X, Y , Z in mod A such that 0 ¤ gf 2 radA1 .X; Z/ (see Exercise 12.9). We will exhibit now some properties of irreducible homomorphisms in module categories. f

g

Lemma 7.9. Let 0 ! L !M !N ! 0 be a nonsplittable exact sequence in mod A. Then the following equivalences hold. (i) f is an irreducible homomorphism in mod A if and only if for every v 2 HomA .V; N / there exists v1 2 HomA .V; M / such that v D gv1 or there exists v2 2 HomA .M; V / such that g D vv2 . (ii) g is an irreducible homomorphism in mod A if and only if for every u 2 HomA .L; U / there exists u1 2 HomA .M; U / such that u D u1 f or there exists u2 2 HomA .U; M / such that f D u2 u.


265

Proof. (i) Assume f is an irreducible homomorphism in mod A. For every v 2 HomA .V; N / there exists a commutative diagram in mod A, 0

/L

0

/L

f0

/Q

g0

/V

w

idL f

/M

/0 v

g

/N

/ 0,

with exact rows, where Q is the fibered product M N V of M and V over N given by g and v (see Exercises I.12.18 and I.12.19). In particular, we have f D wf 0 . Since f is irreducible in mod A, either f 0 is a section or w is a retraction in mod A. If f 0 is a section, then the upper exact sequence is splittable, g 0 is a retraction in mod A and hence g 0 v 0 D idV for some v 0 2 HomA .V; Q/. Then v1 D wv 0 2 HomA .V; M / satisfies gv1 D gwv 0 D vg 0 v 0 D v. If w is a retraction in mod A, then ww 0 D idM for some w 0 2 HomA .M; Q/, and v2 D g 0 w 0 2 HomA .M; V / satisfies vv2 D vg 0 w 0 D gww 0 D g. Conversely, assume now that f satisfies the stated condition. Suppose that f D f1 f2 for some module W in mod A and f1 2 HomA .W; M / and f2 2 HomA .L; W /. Since f is a monomorphism, f2 is also a monomorphism, and we have in mod A a commutative diagram 0

/L

0

/L

f2

/W

h

/V

g

/N

v

f1

idL f

/M

/0 / 0,

where V D Coker f2 D W = Im f2 , h W W ! V is the canonical epimorphism, and v.w C Im f2 / D gf1 .w/ for w 2 W . Then W is isomorphic to the fibered product M N V of M and V over N , given by g and v (see Exercise I.12.19). It follows from our assumption on f that there exists v1 2 HomA .V; M / such that v D gv1 or there exists v2 2 HomA .M; V / such that g D vv2 . In the first case, applying the universal property of the fibered product to the homomorphisms v1 W V ! M and idV W V ! V , we infer that there exists h0 2 HomA .V; W / such that f1 h0 D v1 and hh0 D idV . Hence, h is a retraction in mod A, and consequently f2 is a section in mod A. In the second case, applying the universal property of the fibered product to the homomorphisms idM W M ! M and v2 W M ! V , we conclude that there exists h00 2 HomA .M; W / such that f1 h00 D idM and hh00 D v2 , and so f1 is a retraction in mod A. Therefore, f is an irreducible homomorphism in mod A. The proof of (ii), invoking the fibered sums of modules (see Exercises I.12.20 and I.12.21), is similar. Proposition 7.10. (i) Let f W L ! M be an irreducible monomorphism in mod A. Then N D Coker f is an indecomposable A-module.

266


(ii) Let g W M ! N be an irreducible epimorphism in mod A. Then Ker g is an indecomposable A-module. Proof. (i) It follows from Lemma 7.5 that f is a proper monomorphism, and hence N D Coker f ¤ 0. Then we have the induced exact sequence f

g

0 ! L ! M ! N !0 in mod A. Suppose N D N1 ˚ N2 for some nonzero A-submodules N1 and N2 of N . Let w1 W N1 ! N and w2 W N2 ! N be canonical embeddings. Since g is an epimorphism and w1 ; w2 are proper monomorphisms, we have g ¤ wi ui for all ui 2 HomA .M; Ni /, i 2 f1; 2g. Then, applying Lemma 7.9 (i), we conclude that there exist v1 2 HomA .N1 ; M / and v2 2 HomA .N2 ; M / such that gv1 D w1 and gv2 D w2 . Then the homomorphism v D Œ v1 v2 W N1 ˚ N2 ! M satisfies gv D idN , and so g is a retraction in mod A. This forces f to be a section in mod A, a contradiction because f is irreducible in mod A. Therefore N is an indecomposable A-module. The proof of (ii) is similar. The following theorems establish a strong relation between the irreducible homomorphisms and almost split homomorphisms in the module categories. Theorem 7.11. Let f W L ! M be a nonzero left minimal almost split homomorphism in mod A. Then the following statements hold. (i) f is an irreducible homomorphism in mod A. (ii) A homomorphism f 0 W L ! M 0 in mod A is irreducible if and only if M 0 ¤ 0, M Š M 0 ˚M 00 for a module M 00 in mod A, and there exists a homomorphism f 00 W L ! M 00 in mod A such that the induced homomorphism 0 f W L ! M 0 ˚ M 00 f 00 is a left minimal almost split homomorphism in mod A. Proof. (i) Since f is a left almost split homomorphism in mod A, f is not a section in mod A and L is an indecomposable A-module, by Lemma 7.2 (i). Then it follows from Lemma I.4.2 that f is not a retraction in mod A. Let f D f1 f2 for some module U in mod A and f2 2 HomA .L; U /, f1 2 HomA .U; M /. Assume f2 is not a section in mod A. Then there exists f20 2 HomA .M; U / such that f2 D f20 f , since f is left almost split in mod A. Then f D f1 f2 D .f1 f20 /f , and so f1 f20 2 EndA .M / is an automorphism, because f is left minimal in mod A. Hence there exists h 2 EndA .M / with f1 f20 h D idM , and f1 is a retraction in mod A. Therefore, f is an irreducible homomorphism in mod A.


267

(ii) Let f 0 W L ! M 0 be an irreducible homomorphism in mod A. Observe that then M 0 ¤ 0 because otherwise f 0 is trivially a retraction. Since f 0 is not a section in mod A and f is a left almost split homomorphism in mod A, we have f 0 D hf for some h 2 HomA .M; M 0 /. Then the irreducibility of f 0 forces that h is a retraction in mod A, because f is not a section. Consider the exact sequence u

h

0 ! M 00 , !0 !M ! M0 where M 00 D Ker h and u is the canonical embedding. Since h is a retraction, there exists g 2 HomA .M 0 ; M / with hg D idM 0 . Moreover, then u is a section in mod A 00 (see Lemma I.4.2), and so there A .M; M / such that vu D idM 00 . exists v 02 Hom 00 h Then it follows that ' D v W M ! M ˚ M is an isomorphism in mod A. Let f 00 D vf . Then 0 f D 'f W L ! M 0 ˚ M 00 f 00 is a left minimal almost split homomorphism in mod A. Conversely, assume now that f 0 W L ! M 0 satisfies the stated conditions in (ii). We will prove that f 0 is an irreducible homomorphism in mod A. Suppose 0 f 0 is a section in mod A. Then hf 0 D id i some h 2 HomA .M ; L/, and, for h L ,0 for Œ h 0 W M 0 ˚ M 00 ! L, we obtain Œ h 0 ff 00 D hf 0 D idL , a contradiction, since h 0i by our assumption ff 00 W L ! M 0 ˚ M 00 is a left almost split homomorphism in mod A. In particular, we infer that f 0 is not an isomorphism. Since L is an indecomposable A-module, applying Lemma I.4.2, we then conclude that f 0 is not a retraction in mod A. Let f 0 D uv for some module Z in mod A and v 2 HomA .L; Z/, u 2 HomA .Z; M 0 /. Assume that v is not a section in mod A. We show that u is a retraction in mod A. Consider the homomorphisms v u 0 00 W L ! Z ˚ M W Z ˚ M 00 ! M 0 ˚ M 00 : ; f 00 0 idM 00 Then we obtain

u 0 0 idM 00

0 v uv f : 00 D 00 D f 00 f f

By our assumption, v is not a section in mod A, and then Im HomA .v; L/ radA .L; L/, by Lemma 1.5. Observe that f 00 W L ! M 00 is also not a sec00 00 tion in i A, because otherwise gf D idL for some g 2 Hom h mod h 0Ai.M ; L/, 0 Œ 0 g ff 00 D gf 00 D idL , and we have a contradiction because ff 00 is a left almost split homomorphism in mod A. Hence, applying Lemma 1.5 again, we obtain that Im HomA .f 00 ; L/ radA .L; L/. Then v Im HomA f 00 ; L D Im HomA .v; L/ C Im HomA .f 00 ; L/ radA .L; L/;

268


and consequently

v f 00

h is not a section in mod A, again by Lemma 1.5. Since

f0 f 00

i

h 0i is a left minimal almost split homomorphism in mod A, it follows from (i) that ff 00 0 W Z ˚ M 00 ! is an irreducible homomorphism in mod A. Therefore, u0 idM 00 0 00 M ˚ M is a retraction in mod A, and hence there exist g11 2 HomA .M 0 ; Z/, g21 2 HomA .M 0 ; M 00 /, g12 2 HomA .M 00 ; Z/, g22 2 HomA .M 00 ; M 00 / such that 0 u 0 g11 g12 ug11 0 idM 0 D D : 0 idM 00 0 g22 0 idM 00 g21 g22

Then ug11 D idM 0 , and hence u is a section in mod A. Summing up, we have proved that f 0 W L ! M 0 is an irreducible homomorphism in mod A. Theorem 7.12. Let g W M ! N be a nonzero right minimal almost split homomorphism in mod A. Then the following statements hold. (i) g is an irreducible homomorphism in mod A. (ii) A homomorphism g 0 W M 0 ! N is an irreducible homomorphism in mod A if and only if M 0 ¤ 0, M Š M 0 ˚ M 00 for some module M 00 in mod A, and there exists a homomorphism g 00 W M 00 ! N in mod A such that the induced homomorphism 0 00 g g W M 0 ˚ M 00 ! N is a right minimal almost split homomorphism in mod A. Proof. Similar to the proof of Theorem 7.11.

We end this section with an application of Lemma 7.8, showing the importance of the irreducible homomorphisms for the description of the module category of a finite dimensional algebra of finite representation type. Proposition 7.13. Let A be a finite dimensional K-algebra of finite representation type. Then any nonzero nonisomorphism between indecomposable modules in mod A is a finite sum of compositions of irreducible homomorphisms. Proof. Since A is of finite representation type, by Corollary 2.2, there exists a positive integer m such that radAmC1 D 0. Let M and N be indecomposable modules in mod A and f W M ! N be a nonzero nonisomorphism of right A-modules. Then f 2 radA .M; N /, by Lemma 1.4. Hence there exists n 2 f1; : : : ; mg such that f 2 radAn .M; N / n radAnC1 .M; N /. Applying Lemmas 1.3 and 1.4, we conclude that f is a finite sum of compositions hn : : : h2 h1 of nonzero homomorphisms h1 ; : : : ; hn from radA between indecomposable modules. If n D m, then the homomorphisms h1 ; : : : ; hn do not belong to radA2 , and hence are irreducible homomorphisms in mod A, by Lemma 7.8, and so f has the required presentation. Assume n m 1. Denote by f 0 the sum of all compositions hn : : : h2 h1 , occurring in the

8. Almost split sequences

269

decomposition of f , in which all the homomorphisms h1 ; : : : ; hn are in radA n radA2 , or equivalently, are irreducible. Then f 00 D f f 0 2 radAnC1 . If f 00 D 0, then f D f 0 has the required presentation. If f 00 ¤ 0, then, by induction, f 00 is a finite sum of compositions of irreducible homomorphisms between indecomposable modules in mod A, and consequently f D f 0 C f 00 has the same property.

8 Almost split sequences The main aim of this section is to prove the Auslander–Reiten theorem on the existence of almost split sequences in the module categories of finite dimensional algebras, and provide their characterizations. Let A be a finite dimensional K-algebra over a field. Two short exact sequences in mod A, f

g

0 ! L ! M ! N ! 0

and

f0

g0

0 ! L0 ! M 0 ! N 0 ! 0;

are said to be isomorphic if there is a commutative diagram in mod A 0

/L

0

/ L0

f

u

/M

g

v

f0

/ M0

/N

/0

w

g0

/ N0

/ 0,

where u, v, w are isomorphisms. We note that v is an isomorphism if u and w are isomorphisms (see Exercise I.12.17). We have also the following useful facts. Lemma 8.1. Let 0

/L

0

/L

f

u

/M

g

v

f

/M

/N

/0

w

g

/N

/0

be a commutative diagram in mod A, where the rows are exact and not splittable. The following statements hold. (i) If L is indecomposable and w is an isomorphism, then u and v are isomorphisms. (ii) If N is indecomposable and u is an isomorphism, then v and w are isomorphisms.

270


Proof. (i) Assume L is indecomposable and w is an isomorphism in mod A. Suppose that u is not an isomorphism. Since L is indecomposable, EndA .L/ is a local algebra, by Lemma I.4.4, and hence u 2 rad EndA .L/ D radA .L; L/. In particular, u is a nilpotent endomorphism of L, say um D 0 for some m 1. Then f u D vf forces v m f D f um D 0, and so there exists a homomorphism h W N ! M in mod A such that v m D hg. Hence gv D wg implies w m g D gv m D ghg, and then w m D gh, because g is an epimorphism. Further, by the assumption, w is an isomorphism, and so ww1 D idN for some w1 2 EndA .N /. Then we obtain that g.hw1m / D w m w1m D idN , which shows that g is a retraction in mod A. This contradicts the fact that the exact sequence given by f and g is not splittable (see Lemma I.4.2). Hence u is an isomorphism, and consequently v is an isomorphism (see Exercise I.12.17). The proof of (ii) is similar. A short exact sequence in mod A f

g

0 ! L ! M ! N ! 0 is called an almost split sequence if f is a left minimal almost split homomorphism and g is a right minimal almost split homomorphism in mod A. Observe that then the sequence is not splittable because f is not a section and g is not a retraction in mod A The following lemma also shows that an almost split sequence is uniquely determined (up to isomorphism) by each of its end terms. Lemma 8.2. Let f

g

f0

g0

0 ! L ! M ! N ! 0 and 0 ! L0 ! M 0 ! N 0 ! 0 be almost split sequences in mod A. The following statements are equivalent. (i) The exact sequences are isomorphic. (ii) L and L0 are isomorphic in mod A. (iii) N and N 0 are isomorphic in mod A. Proof. Obviously (i) implies (ii) and (iii). Assume that there exists an isomorphism u W L ! L0 in mod A. Since f 0 is not a section, then f 0 u W L ! M 0 is not a section, by Lemma 7.4 (i). Moreover, since f W L ! M is a left almost split homomorphism in mod A, we conclude that there exists v 2 HomA .M; M 0 / such that f 0 u D vf . Let u0 W L0 ! L be the inverse of u in mod A. Then f u0 W L0 ! M is not a section in mod A, and hence there exists v 0 2 HomA .M 0 ; M / such that f u0 D v 0 f 0 , because f 0 is a left almost split homomorphism in mod A. We have also the equalities v 0 vf D v 0 f 0 u D f u0 u D f and vv 0 f 0 D vf u0 D f 0 uu0 D f 0 . Since f and


271

f 0 are left minimal homomorphisms in mod A, we conclude that v 0 v and vv 0 are isomorphisms in mod A. Then v is an isomorphism in mod A. Moreover, we have in mod A a commutative diagram 0

/L

0

/ L0

f

u

/M

g

v

f0

/ M0

/N

/0

w

g0

/ N0

/ 0,

because N Š Coker f and N 0 Š Coker f 0 , where u and v are isomorphisms. Then w is also an isomorphism, and consequently the given almost split sequences are isomorphic. This proves that (ii) implies (i). The proof that (iii) implies (i) is similar. We provide now several characterizations of almost split sequences. f

g

Theorem 8.3. Let 0 ! L !M ! N ! 0 be a short exact sequence in mod A. The following statements are equivalent. f

g

(i) 0 ! L !M ! N ! 0 is an almost split sequence in mod A. (ii) L is an indecomposable A-module and g is a right almost split homomorphism in mod A. (iii) N is an indecomposable A-module and f is a left almost split homomorphism in mod A. (iv) f is a left minimal almost split homomorphism in mod A. (v) g is a right minimal almost split homomorphism in mod A. (vi) L and N are indecomposable A-modules, and f and g are irreducible homomorphisms in mod A. Proof. Observe that (i) implies (iv) and (v) by definition of an almost split sequence in mod A. It follows also from Lemma 7.2 that (i) implies (ii) and (iii). Similarly, (i) implies (vi), by Lemma 7.2 and Theorems 7.11 and 7.12. We show now that (v) implies (ii). Assume that g is a right minimal almost split homomorphism in mod A. Then, applying Theorem 7.12, we conclude that g is an irreducible epimorphism in mod A. Hence, by Proposition 7.10 (ii), L D Ker g is an indecomposable A-module, and so (ii) follows. Similarly, we prove that (iv) implies (iii). We claim now that (ii) and (iii) are equivalent. Assume that L is an indecomposable A-module and g is a right almost split homomorphism in mod A. Then, applying Lemma 7.2, we conclude that N is an indecomposable A-module. We

272


will show that f is a left almost split homomorphism in mod A. Since g is not a retraction in mod A the exact sequence f

g

0 ! L ! M ! N ! 0 is not splittable, and consequently f is not a section in mod A (see Lemma I.4.2 and Exercise I.12.16). Suppose u W L ! U is a homomorphism in mod A such that u ¤ hf for any homomorphism h W M ! U in mod A. We will show that then u is a section. Consider the commutative diagram in mod A, 0

/L

0

/U

f

u u0

/M /V

g

f0

/N

/0

idN

p

/N

/0

with exact rows, where the lower sequence is given by the fibered sum V D U ˚L M of U and M over L, via u and f (see Exercise I.12.21). Observe that then u0 is not a section. Indeed, if u00 u0 D idU for some u00 2 HomA .V; U /, then we obtain .u00 f 0 /f D u00 .f 0 f / D u00 .u0 u/ D .u00 u0 /u D u, which contradicts the choice of u. Since u0 is not a section, we infer that the lower exact sequence is not splittable, and hence p is not a retraction in mod A. By the assumption, g is a right almost split homomorphism in mod A, so there exists a homomorphism g 0 W V ! M in mod A such that p D gg 0 . Therefore, there exists in mod A a commutative diagram u0

0

/U

0

/L

p

/V g0

uN f

/M

/N

/0

idN

g

/N

/ 0,

/N

/0

and hence a commutative diagram of the form 0

/L

0

/L

f

/M

g

g0f 0

uu N f

/M

g

idN

/N

/ 0.

Since L is indecomposable and idN is an isomorphism, applying Lemma 8.1, we conclude that uu N is an isomorphism in mod A. Then u is a section in mod A (see Lemma 4.1). This proves that (ii) implies (iii). The proof that (iii) implies (ii) is similar. Assume that (ii) and (iii) hold. We claim that then (i) also holds. We have to show that f is left minimal and g is right minimal in mod A. Assume h 2 EndA .M /


273

satisfies hf D f . Then we have in mod A a commutative diagram 0

/L

0

/L

f

idL

/M

g

/M

/0

w

h

f

/N

g

/N

/ 0.

Since N is indecomposable and idL is an isomorphism, it follows from Lemma 8.1 (ii) that h is an isomorphism. Hence f is a left minimal homomorphism in mod A. Similarly, if h 2 EndA .M / satisfies gh D g, we have in mod A a commutative diagram 0

/L

0

/L

f

u

/M

g

f

/0

idN

h

/M

/N

g

/N

/ 0.

Since L is indecomposable and idN is an isomorphism, applying Lemma 8.1 (i) again, we obtain that h is an isomorphism. This shows that g is a right minimal homomorphism in mod A. In the final step of the proof we show that (vi) implies (ii). Assume (vi) holds. Then L is an indecomposable A-module and g is an irreducible homomorphism in mod A. In particular, g is not a retraction in mod A. Consider a homomorphism v W V ! N in mod A which is not a retraction. We claim that there exists a homomorphism v 0 W V ! M in mod A such that gv 0 D v. Clearly, we may assume that V is an indecomposable A-module. Since, by (vi), f is an irreducible homomorphism in mod A, applying Lemma 7.9 (i), we conclude that there exists v1 2 HomA .V; M / such that v D gv1 or there exists v2 2 HomA .M; V / such that g D vv2 . In the first case, for v 0 D v1 , we have v D gv 0 . Assume the second case. Since g is an irreducible homomorphism and v is not a retraction, the equality g D vv2 forces v2 to be a section in mod A. But V is, by our assumption, indecomposable and then v2 is an isomorphism. Then for the inverse homomorphism v 0 2 HomA .V; M / of v2 , we have gv 0 D .vv2 /v 0 D v.v2 v 0 / D v, as required. We will prove now the existence theorem for almost split sequences established by M. Auslander and I. Reiten in [AR1], called the Auslander–Reiten theorem. Theorem 8.4. Let A be a finite dimensional K-algebra over a field K. The following statements hold. (i) For any nonprojective indecomposable module M in mod A, there exists an almost split sequence in mod A of the form f

g

0 ! A M ! E ! M ! 0:

274


(ii) For any noninjective indecomposable module N in mod A, there exists an almost split sequence in mod A of the form u

v

0 ! N ! F ! A1 N ! 0: Proof. (i) Assume M is a nonprojective indecomposable module in mod A. It follows from Lemma I.4.4 that EndA .M / is a local K-algebra, and consequently FM D EndA .M /= rad EndA .M / is a division K-algebra (see Lemma I.3.8). Further, it follows from Proposition 3.8 that, for every module L in mod A, the K-vector spaces ExtA1 .M; L/ and ExtA1 .M; L/ are right EndA .M /-modules, and the K-linear isomorphism M;L W ExtA1 .M; L/ ! ExtA1 .M; L/ is an isomorphism of right EndA .M /-modules. For a module L in mod A, consider also the K-vector space SA .L; M / D HomA .L; M /= radA .L; M /: We note that, by Lemma 1.4, radA .L; M / is the subspace of HomA .L; M / consisting of all nonisomorphisms whose restriction to any direct summand of L is nonisomorphism. Observe also that PA .L; M / radA .L; M /. Indeed, if P is a module in proj A, u 2 HomA .L; P /, v 2 HomA .P; M / and vu … radA .L; M /, then, by Lemma 1.5 (ii), vu is a retraction, and so v is a retraction. Consequently, M is projective, a contradiction. Therefore, we may define the K-linear epimorphism pL;M W HomA .L; M / ! SA .L; M /; defined by pL;M .f C PA .L; M // D f C radA .L; M / for f 2 HomA .L; M /, which is in fact an epimorphism of left EndA .M /-modules. Applying D D HomK .; K/ we obtain a monomorphism D.pL;M / W DSA .L; M / ! D HomA .L; M / of right EndA .M /-modules. Since M is an indecomposable nonprojective module in mod A, we know from Corollary 4.9 that A M is an indecomposable noninjective module in mod A and M Š A1 .A M /. Applying now Theorem 6.3 we conclude that there is an isomorphism

M;M W D HomA .M; M / ! ExtA1 .M; A M / of right EndA .M /-modules. We claim that ExtA1 .M; A M /, considered as a right EndA .M /-module, has simple socle. It is enough to show that the right EndA .M /module D HomA .M; M / D D EndA .M / has simple socle. Observe that we have an epimorphism of K-algebras pM;M W EndA .M / ! SA .M; M / D EndA .M /= rad EndA .M / D FM , and EndA .M /= rad EndA .M / Š


275

EndA .M /= rad EndA .M /. Since the local algebra EndA .M /, considered as a left EndA .M /-module, has simple top FM , the left EndA .M /-module EndA .M / has also FM as the top. But then the right EndA .M /-module D HomA .M; M / has simple socle. Moreover, D.pM;M / W DSA .M; M / ! D HomA .M; M / induces an isomorphism from the simple right EndA .M /-module D.SA .M; M // D D.FM / to the socle of the right EndA .M /-module D HomA .M; M /. In particular, the right EndA .M /-module ExtA1 .M; A M / has simple socle, which coincides with

M;M D.pM;M /.D.FM //. Take now the extension f

g

E W 0 ! A M ! E ! M ! 0 of A M by M in mod A such that M;A M .ŒE/ is a nonzero element of the socle of the right EndA .M /-module ExtA1 .M; A M /. Notice that M;A M .ŒE/ 2

M;M D.pM;M / .D.FM //. We claim that E is an almost split sequence in mod A. Since M;A M .ŒE/ ¤ 0, it follows from Lemma 3.1 and Corollary 3.6 that E is a nonsplittable exact sequence in mod A. Moreover, A M is an indecomposable A-module. Hence, in order to prove that E is an almost split sequence in mod A, it is enough to show, by Theorem 8.3, that g is a right almost split homomorphism in mod A. Let v W V ! M be a homomorphism in mod A which is not a retraction. Since M is an indecomposable A-module, it follows from Lemma 1.5 (ii) that v 2 radA .V; M /. We have the commutative diagrams of K-vector spaces D.pM;M /

/ D Hom .M; M / A

DSA .M; M /

D.pM;V /

/ D Hom .M; V / A

and ExtA1 .M; A M /

M;A M

1 ExtA .v;A M /

ExtA1 .V; A M /

V;A M

/ Ext 1 .M; A M / A 1 .v; M / ExtA A

D HomA .M;v/

DSA .M;v/

DSA .M; V /

M;M

M;V

/ Ext 1 .V; A M / A

/ Ext 1 .M; A M / A 1 .v; M / ExtA A

/ Ext 1 .V; A M / , A

where M;V is the natural K-linear isomorphism from Theorem 6.3. Since v 2 radA .V; M /, we have DSA .M; v/ D 0, and consequently ExtA1 .v; A M / M;M D.pM;M / D M;V D.pM;V /DSA .M; v/ D 0: Then we obtain that V;A M ExtA1 .v; A M /.ŒE/ D ExtA1 .v; A M /M;A M .ŒE/ 2 ExtA1 .v; A M / M;M D.pM;M /.D.FM // D 0;

276


and hence V;A M ExtA1 .v; A M /.ŒE/ D 0 and ExtA1 .v; A M /.ŒE/ D ŒOV;A M , because V;A M is an isomorphism. But ExtA1 .v; A M /.ŒE/ D ŒEv, where Ev is the upper sequence of the commutative diagram 0

/ A M

/ E0

i

g0

idA M

0

/ A M

v0

/E

f

/V

/0 v

g

/M

/0

given by the fibered product E 0 D E M V of E and V over M , via g and v. Since ŒEv D ŒOV;A M , the sequence Ev is splittable (see Lemma 3.1), and so v 0 is a retraction in mod A. Let v 00 W V ! E 0 be a homomorphism in mod A with v 0 v 00 D idV . Then h D g 0 v 00 2 HomA .V; E/ and gh D gg 0 v 00 D vv 0 v 00 D v. This shows that g is a right almost split homomorphism in mod A, and consequently E is an almost split sequence in mod A. (ii) Assume N is a noninjective indecomposable module in mod A. Then D.N / is a nonprojective indecomposable module in mod Aop . Applying (i) to the Aop module D.N /, we conclude that there exists an almost split sequence in mod Aop of the form ˇ

˛

0 ! Aop D.N / ! X ! D.N / ! 0: Applying D W mod Aop ! mod A, we obtain an almost split sequence in mod A of the form D.ˇ /

D.˛/

0 ! D.D.N // ! D.X / ! D.Aop D.N // ! 0: Note that D.D.N // Š N and D.Aop D.N // D D.D Tr.D.N /// Š Tr.D.N // D A1 N in mod A. Therefore, we conclude that there exists in mod A an almost split sequence of the form u

v

0 ! N ! F ! A1 N ! 0; where F Š D.X /.

Corollary 8.5. Let A be a finite dimensional K-algebra over a field K. The following statements hold. (i) For every indecomposable module M in mod A, there exists a right minimal almost split homomorphism g W E ! M in mod A. (ii) For every indecomposable module N in mod A, there exists a left minimal almost split homomorphism u W N ! F in mod A.


277

Proof. (i) Assume M is an indecomposable module in mod A. If M is projective, then it follows from Lemma 7.6 that the canonical inclusion homomorphism g W rad M ! M is a right minimal almost split homomorphism in mod A. Assume M is not projective. Then, by Theorem 8.4 (i), there exists in mod A an almost split sequence f

g

0 ! A M ! E ! M ! 0; where g is a right minimal almost split homomorphism in mod A. (ii) Assume N is an indecomposable module in mod A. If N is injective, then it follows from Lemma 7.7 that the canonical epimorphism u W N ! N= soc N is a left minimal almost split homomorphism in mod A. Assume N is not injective. Then, by Theorem 8.4 (ii), there exists in mod A an almost split sequence u

v

0 ! N ! F ! A1 N ! 0; where u is a left minimal almost split homomorphism in mod A.

We describe now a class of almost split sequences playing a prominent role in our further considerations Proposition 8.6. Let A be a finite dimensional K-algebra over a field K and P be an indecomposable projective-injective module in mod A of length at least two. Then the sequence Œ uq Œ j v 0 ! rad P ! .rad P = soc P / ˚ P ! P = soc P ! 0; where u; j are the inclusion homomorphisms and q, v are the canonical epimorphisms, is an almost split sequence in mod A. In particular, we have rad P Š A .P = rad P / and P = soc P Š A1 .rad P /: Proof. Observe that Œ uq is a monomorphism, since u is a monomorphism. Similarly, Œ j v is an epimorphism, since v is an epimorphism. Clearly, Œ j v Œ uq D j q C vu D 0. Assume x 2 rad P and y 2 P are such that .x C soc P; y/ 2 Ker Œ j v . Then x C soc P D y C soc P , and hence y 2 rad P and .x C soc P; y/ D .y C soc P; y/ D Œ uq .y/. Hence the sequence is exact. Observe that rad P has simple socle equal to soc P , and so rad P is an indecomposable A-module. Then, in order to prove that the given sequence is an almost split sequence in mod A, it is enough to show, by Theorem 8.3, that Œ j v is a right almost split homomorphism in mod A. Observe that Œ j v is not a retraction in mod A. Indeed, otherwise P = soc P is an indecomposable direct summand of .rad P = soc P / ˚ P , by Lemma I.4.2, and this contradicts the Krull–Schmidt theorem, because P is indecomposable

278


with `.P / > `.P = soc P / and rad P = soc P is a direct sum of indecomposable A-modules of length smaller than `.P = soc P /. Let f W X ! P = soc P be a homomorphism in mod A which is not a retraction. We will show that f D Œ j v f 0 for some homomorphism f 0 W X ! .rad P = soc P / ˚ P in mod A. We may assume that X is an indecomposable Amodule. Since P is indecomposable projective, it follows from Propositions I.5.16 and I.8.2 that rad P is a unique maximal right A-submodule of P . Moreover, since P is also indecomposable injective, soc P is a unique simple A-submodule of P , and hence is contained in every nonzero right A-submodule of P . Hence, rad P = soc P is a unique maximal right A-submodule of P = soc P . Then, if f is not an epimorphism, then Im f is contained in rad P = soc P , and we obtain that f D Œ j v f . Assume f is an epimorphism. Then we have a commutative 0 diagram P vv w vvv v v vv zvv f / P = soc P , X because P is projective. We claim that then w is a monomorphism. Indeed, if Ker w ¤ 0, then soc P Ker w, and there is a homomorphism w 0 W P = soc P ! X such that w D w 0 v. But then v D f w D f w 0 v implies f w 0 D idP = soc P , which contradicts the fact that f is not a retraction. Hence, indeed w W P ! X is a monomorphism, and then w is a section, because P is injective (see Lemma I.8.13). Since X is indecomposable in mod A, applying Lemma I.4.2, we conclude that w W P ! X is an isomorphism. Then, for the homomorphism g W X ! P converse to w, we obtain that Œ j v g0 D vg D f wg D f . Therefore, we have proved that Œ j v is a right almost split homomorphism in mod A. We present now a complete description of the almost split sequences in the module categories of Nakayama algebras which were the motivating class of examples for the whole theory developed by M. Auslander and I. Reiten in [AR1], [AR2], [AR3], [AR4], and exhibited in the book [ARS]. Recall that, by Theorem I.10.5, if M is an indecomposable right module of Loewy length m over a Nakayama algebra A, then there exists a unique, up to isomorphism, indecomposable projective A-module P (the projective cover of M ) such that M Š P = radm P . Moreover, M is nonprojective if and only if m < ``.P /. Theorem 8.7. Let A be a finite dimensional Nakayama K-algebra over a field K, P be an indecomposable projective module in mod A, and m a positive integer such that m < ``.P /. Then the sequence 0 ! rad P = radmC1 P Œ uq Œ j v ! .rad P = radm P / ˚ .P = radmC1 P / ! P = radm P ! 0;


279

where u, j are the inclusion homomorphisms and q, v are the canonical epimorphisms, is an almost split sequence in mod A. Proof. Applying arguments in the proof of Proposition 8.6, we conclude that the above sequence is exact. Moreover, rad P = radmC1 P D rad.P = radmC1 P / is indecomposable as an A-submodule of the uniserial A-module P = radmC1 P . Hence, in order to prove that the above exact sequence is an almost split sequence in mod A, it is sufficient to show, by Theorem 8.3, that Œ j v is a right almost split homomorphism in mod A. Observe that Œ j v is not a retraction in mod A. Indeed, rad P = radm P D rad.P = radm P /, P = radmC1 P and P = radm P are indecomposable A-modules with `.rad P = radm P / < `.P = radm P / < `.P = radmC1 P /, and so P = radm P is not isomorphic to a direct summand of the middle of the exact sequence. Let f W V ! P = radm P be a homomorphism in mod A which is not a retraction. We will show that f D Œ j v f 0 for some homomorphism f 0 W V ! .rad P = radm P / ˚ .P = radmC1 P / in mod A. We may assume that V is an indecomposable A-module. Since P = radm P is a uniserial A-module, it follows that rad P = radm P D rad.P = radm P / is a unique maximal A-submodule of P = radm P . Hence, if f is not an epimorphism, then Im f rad P = radm P , and we have the f factorization f D Œ j v 0 . Assume f is an epimorphism. It follows from Theorem I.10.5 that there is an isomorphism g W V ! Q= rads Q in mod A, for some indecomposable projective A-module Q and a positive integer s ``.Q/. Since f is epimorphism, the homomorphism top.f / W top.V / ! top.P = radm P / is an epimorphism, by Lemma I.5.18, and consequently an isomorphism, because top.V / Š top.Q/ and top.P = radm P / Š top.P / are simple A-modules. Hence, Q Š P and s m C 1, since f is not an isomorphism. Then we conclude that mC1 there is an epimorphism h W V ! P = rad P such that f D vh. This gives a 0 required factorization f D Œ j v h in mod A. Therefore, indeed Œ j v is a right almost split homomorphism in mod A. Examples 8.8. (a) Let n 2 be a natural number and Q.n/ be the quiver 1 v cHHH ˛1 v vv 2 HH vvˇ2 hh kVVV˛2 HHH v VVVVVH {vvhhhhh n C 1. 0 sheKKK :: KK ˇn : ˛n ssss KK s KK sss ys n ˇ1

Consider the bound quiver algebra ƒ.n/ D KQ.n/=I.n/ over a field K, where I.n/ is the ideal in KQ.n/ generated by the elements ˛i ˇi ˛1 ˇ1 for all i 2 f2; : : : ; ng. We identify mod ƒ.n/ with repK .Q.n/; I.n//. Applying Proposition I.8.27, we

280


conclude that the indecomposable projective ƒ.n/-module P .n C 1/ coincides with the indecomposable injective ƒ.n/-module I.0/, and is of the form K x x xx xixiii1iii K {x P .n C 1/ W K tieK : KK KK 1 :: KK KK K 1 xxx

cFF FF1 FF jUUU1U FFFF UUUU K. ss 1 sss ss yss

Moreover, soc.P .n C 1// D S.0/, top.P .n C 1// D S.n C 1/, K ~ ~ ~ ~~ K ~ul~lllll P .n C 1/=S.0/ W 0 bF : FF FF :: FF

K } _@@@ } } @ }}1 iR @@ ~}uk}kkkkk K RRRRR@ rad P .n C 1/ W K 0, : cGG xx GG 1 :: x x GG x |xx K 1

K

ÀA AA1 iSSS1SAAA SS K, w 1 ww w {ww

and hence rad P .n C 1/= soc P .n C 1/ D S.1/ ˚ S.2/ ˚ ˚ S.n/. Therefore, applying Proposition 8.6, we conclude that we have in mod ƒ.n/ an almost split sequence of the form 0 ! rad P .n C 1/ ! S.1/ ˚ ˚ S.n/ ˚ P .n C 1/ ! P .n C 1/=S.0/ ! 0: This shows that the middle terms of almost split sequences may have arbitrary large number of indecomposable direct summands. (b) Let K be a field, Q the quiver 1 ?~ @@ @@˛ ~~ @@ ~ ~ @ ~~ 2 , 3 o ˇ

I the ideal in KQ generated by ˛ˇ, and A D KQ=I the associated bound quiver algebra. It follows from Theorem I.10.3 that A is a Nakayama algebra. We identify mod A D repK .Q; I /. Then, by Proposition I.8.27, the indecomposable projective A-modules are of the form AK = ===1 = P .1/ W 0 o K,

K A >>>> 01 >> P .2/ W K o K2 , 1

10


281

@K = ===1 = P .3/ W K o K, 1

0

and hence ``.P .1// D 2, ``.P .2// D 4, ``.P .3// D 3. Applying now Theorem I.10.5 we conclude that the remaining indecomposable modules in mod A are (up to isomorphism) of the form

S.1/ D P .1/= rad P .1/ W

? K >> >>

0o

0,

A0> >>> S.2/ D P .2/= rad P .2/ W K, 0o

I.3/ D P .2/= rad2 P .2/ W K o

? 0 >> >> 1

K,

> K AA 0 AA }} } } I.1/ D P .2/= rad3 P .2/ W K o K, 1 1

S.3/ D P .3/= rad P .3/ W

Ko

? 0 `.Y /. In particular Y is nonprojective, and we have in mod A an Auslander–Reiten sequence f

g

0 ! A Y ! E ! Y ! 0: Since g is a minimal right almost split homomorphism in mod A, we conclude 0 that E Š X dX Y ˚ E 0 in mod A. Then we obtain that `.A Y / D `.E/ `.Y / 0 0 dXY `.X/ `.Y / > `.X /, because dXY 2. Applying Lemma 9.1 (i) and Proposition 9.6 (i), we infer that there is in A the valued arrow .dA YX ;d0

A YX

/

A Y ! X 0 2 and d0A YX D dXY 2. Moreover, there is an irreducible with dA YX D dXY homomorphism w W A Y ! X which is a proper epimorphism, since `.A Y / > `.X/. Hence X is nonprojective and there exists in mod A an almost split sequence

0 ! A X ! U ! X ! 0; where U Š .A Y /dXY ˚ U 0 in mod A. Then we get `.A X / D `.U / `.X / dXY `.A Y /`.X / > `.A Y /. Iterating the above arguments, we obtain inductively two families of indecomposable nonprojective A-modules An X , n 2 N, and An Y , n 2 N, such that `.AnC1 X / > `.AnC1 Y / > `.An X / > `.An Y / for any n 2 N.

9. The Auslander–Reiten quiver

291

For h W X ! Y an irreducible monomorphism, so `.X / < `.Y /, the proof is similar. We have also the following facts on the Auslander–Reiten quivers of finite dimensional algebras over algebraically closed fields. Proposition 9.9. Let A be a finite dimensional K-algebra over an algebraically 0 .dXY ;dXY /

0 closed field K and X ! Y be an arrow in A . Then dXY D dXY .

Proof. Since K is algebraically closed, the canonical K-algebra homomorphisms K ! FX and K ! FY of division K-algebras are isomorphisms (see Exercise I.12.38). Moreover, K acts centrally on irrA .X; Y /. Hence, applying Corollary 9.4, we obtain the equalities dXY D dimFY irrA .X; Y / D dimK irrA .X; Y / D dimFX irrA .X; Y / 0 : D dXY

As a direct consequence of Propositions 9.8 and 9.9 we obtain the following result. Corollary 9.10. Let A be a finite dimensional K-algebra over an algebraically 0 .dX Y ;dX Y/

closed field K of finite representation type and X ! Y be an arrow of 0 A . Then dXY D 1 D dXY . In the representation theory of finite dimensional algebras over an algebraically closed field K, instead of a valued arrow .m;m/

X ! Y of an Auslander–Reiten quiver A , usually one writes a multiple arrow / / :: Y X : / consisting of m arrows from X to Y (see [ASS], [SS1], [SS2]). By a component of A we mean a connected component of the quiver A . We note that usually A consists of many components whose shapes give important information on A and mod A. This leads to the notion of a translation quiver taking into account combinatorial aspects of the Auslander–Reiten quiver of an algebra. Let be a valued quiver with the set of vertices 0 and the set of arrows 1 . Moreover, we have two valuation maps d; d 0 W 1 ! N1 D N n f0g such that each arrow ˛ 2 1 has the valuation .d˛ ; d˛0 /. For each vertex x of , we consider the set

292


x D fy 2 0 j there is an arrow in from y to xg of all immediate predecessors of x in and the set x C D fy 2 0 j there is an arrow in from x to yg of all immediate successors of x in . Then is said to be locally finite if, for each vertex x of , the sets x and x C are finite. A valued translation quiver is a pair .; / consisting of a locally finite valued quiver , with the vertex set 0 and the arrow set 1 , and an injective map from a subset of 0 into 0 , satisfying the following conditions: (1) has no loops and no multiple arrows. (2) For each vertex x of such that .x/ is defined, we have x D .x/C . .a;b/

(3) For each valued arrow x ! y in such that .y/ is defined, we have in .b;a/

the valued arrow .y/ ! x. The partially defined map W 0 ! 0 is called the translation of the translation quiver .; /. A vertex x of is called a projective vertex if x is not in the domain of and called an injective vertex if x is not in the image of . Let 00 (respectively, 000 ) be the subset of 0 consisting of all projective vertices (respectively, injective vertices). Then the translation defines a bijection W .0 n 00 / ! .0 n 000 / and the inverse bijection is denoted by 1 . Since has no loops and no multiple arrows, an arrow ˛ 2 1 can be written as 0 / .dxy ;dxy

x ! y; where x D s.˛/ is the source of ˛, y D t .˛/ is the target of ˛, and dxy D d˛ , .a;b/

0 dxy D d˛0 . A valued arrow x ! y in with a D 1 D b is said to be an arrow with trivial valuation, and will be simply denoted by x ! y. If .; / and . ; / are valued translation quivers, then a valued translation quiver morphism f W .; / ! . ; / is a pair of maps f0 W 0 ! 0 and f1 W 1 ! 1 such that the following conditions are satisfied:

(1) f is a morphism of valued quivers, that is, for any arrow ˛ 2 1 with valuation .a; b/, f1 .˛/ 2 1 is an arrow with s.f1 .˛// D f0 .s.˛//, t .f1 .˛// D f0 .t.˛// and valuation .a; b/. (2) f0 . .x// D .f0 .x// for all nonprojective vertices x 2 0 n 00 . Moreover, a valued translation quiver morphism f W .; / ! . ; / is said to be an isomorphism if the maps f0 W 0 ! 0 and f1 W 1 ! 1 are bijections. Further, a valued translation quiver morphism f W .; / ! . ; / is said to be a covering if the following conditions are satisfied: (1) f0 .0 / D 0 and f1 .1 / D 1 .


293

(2) f0 .00 / D 00 and f1 .000 / D 000 . (3) For each vertex x 2 0 , f0 induces bijections x ! f0 .x/ and x C ! C f0 .x/ preserving the valuations of the arrows. A group G of valued translation quiver automorphisms of a valued translation quiver .; / is said to be admissible if each G-orbit Gx D fg.x/ j g 2 Gg, x 2 0 , in 0 intersects y [fyg in at most one vertex and fyg[y C in at most one vertex, for any vertex y 2 0 . For an admissible group G of automorphisms of avalued translation quiver .; /, we may form the orbit valued translation quiver =G; =G such that the set .=G/0 of vertices of =G is the set 0 =G of G-orbits in 0 , the set .=G/1 of arrows of =G is the set 1 =G of G-orbits in 1 , and the canonical surjection of the valued translation quivers .; / ! =G; =G is a covering of valued translationquivers. We will frequently write briefly instead of .; / and =G instead of =G; =G . We may also consider an additional combinatorial invariant of a valued translation quiver, motivated by the (composition) length of a finite dimensional module. Namely, a length function of a valued translation quiver .; / is a function ` W 0 ! N1 satisfying the following conditions: (i) For each nonprojective vertex y 2 0 , we have X `.v/dvy : `.y/ C `. .y// D v2y

(ii) For each projective vertex p 2 0 , we have X 0 `.x/dxp : `.p/ D 1 C x2p

(iii) For each injective vertex i 2 0 , we have X `.x/dix : `.i / D 1 C x2i C

The Auslander–Reiten quiver A of a finite dimensional K-algebra A over a field K, or more generally a component C of A , is obviously a valued translation quiver with a length function. A valued translation quiver .; / is said to be stable if the sets 00 and 000 are empty, or equivalently, its translation is a bijection from 0 to 0 . An important class of stable valued translation quivers can be constructed as follows. Let be a locally finite valued quiver without loops and multiple arrows, 0 the set of vertices of , 1 the set of arrows of , and d; d 0 W 1 ! N1 the valuation maps. We associate to the valued translation quiver Z as follows. The set

294


.Z /0 D Z 0 D f.i; x/ j i 2 Z; x 2 0 g is the set of vertices of Z . The set .Z /1 of arrows of Z consists of the valued arrows 0 / .dxy ;dxy

.i; ˛/ W .i; x/ ! .i; y/ and 0 ;d .dxy xy /

.i; ˛ 0 / W .i C 1; y/ ! .i; x/ 0 / .dxy ;dxy

i 2 Z, for all arrows ˛ W x ! y in 1 , where x D s.˛/, y D t .˛/, dxy D 0 d˛ , dxy D d˛0 . The translation W Z 0 ! Z 0 is defined by .i; x/ D .i C 1; x/ for all i 2 Z, x 2 0 . Observe that .Z ; / is a stable valued translation quiver. For a subset I of Z, we denote by I the full translation subquiver given by the vertex set .I /0 D I 0 . In particular, we have the valued translation quivers N and .N/ associated to the valued quiver . We also note that for any positive integer m, the infinite cyclic group . m /, generated by the iterated translation m of Z , is an admissible group of automorphisms of the valued translation quiver Z , and we may consider the orbit stable valued translation quiver Z =G D Z =G; Z=G . Consider the infinite quiver A1 W

0

/1

/2

/3

/ ::::

Then ZA1 is the translation quiver of the form .i C 1; 0/ .i 1; 0/ .i 2; 0/ 6 .i; 0/ OOO QQ( NNN& 8 QQ( mm6 mmm oo7 ' ppp :: . .i C 1; 1/ .i; 1/ .i 1; 1/ . : . LLL 7 8 7 OOO OOO o o q o o q O' & q ' oo oo :: . .i C 1; 2/ .i; 2/ O : .. 8 MM OOO o7 r o r M o r & o ' :: :: . : .. : with the translation defined by .i; j / D .i C 1; j / for i 2 Z, j 2 N. For an integer r 1, denote by ZA1 =. r / the orbit translation quiver obtained from ZA1 by identifying each vertex x of ZA1 with r x and each arrow x ! y in ZA1 with r x ! r y. Then ZA1 =. r / is a stable translation quiver consisting of -periodic vertices of period r, called a stable tube of rank r. The set of vertices of a stable tube T D ZA1 =. r / having exactly one immediate predecessor (equivalently, exactly one immediate successor) is said to be the mouth of T.


295

For example, a stable tube of rank 3 is of the form x1 c

2 x1 \ x1 L a 2 x2

x 2 :

c 2 x3

#

\ x3

x3 " L

2 x4 ? ? ??? :: :: : :

_? ?? x4 ? :: :: : :

x2 M

z

x4 # ? ? ??? :: :: : :

and x1 , x1 and 2 x1 form its mouth. Examples 9.11. (a) Let A D KQ=I be the Nakayama algebra over a field K given by the quiver Q of the form 1 @ === ==˛ ==

3o

ˇ

2

and the ideal I of KQ generated by ˛ˇ. Then it follows from Example 8.8 (b) that

296


the Auslander–Reiten quiver A of A is of the form S.2/ S.1/ S.3/ S.2/ @@ ?? > : ? II II uu ?? @@ }} u I } u I u ? @@ } II u ?? } @ II uu }} uu $ P .1/ I.3/ : M II AA ? II uu u AA II uu I AA u II uu A I$ uu P .3/ I.1/ II u: II u u II uu II uu I$ u u P .2/ D I.2/ . (b) Let Q be the quiver 3 4 2 @@ @@ ~~ ~ @ˇ ~~ ˛ @@ ~~ 1 and A D KQ the path algebra of Q over a field K. We identify mod A D repK .Q/. It follows from Lemma I.8.25 and Proposition I.8.27 that 0> 0 >> >> K,

0

S.1/ D P .1/ W

0> K >> >>1 K,

0

P .3/ W

P .2/ W

0 K@ @@ @@ 1 K,

0

P .4/ W

0 K 0? ?? }} ?? } ~}} 1 K

are the indecomposable projective modules in mod A, and I.1/ W

S.3/ D I.3/ W

K@ K K @@ ~ @@1 ~~~ ~~ 1 1 K, 0 >> K >> >> 0,

S.2/ D I.2/ W

0 K> 0 >> >> 0,

S.4/ D I.4/ W

K 0= 0 == == 0

0

are the indecomposable injective modules in mod A. Since S.1/ is simple projective noninjective A-module and S.1/ D rad P .2/, S.1/ D rad P .3/ and S.1/ D


297

rad P .4/, it follows from Lemma 7.6 and Corollary 9.7 (i) that we have in mod A an almost split sequence of the form f

g

0 ! S.1/ ! P .2/ ˚ P .3/ ˚ P .4/ ! A1 S.1/ ! 0; where f is induced by the inclusion homomorphisms S.1/ ! P .2/, S.1/ ! P .3/, S.1/ ! P .4/, and is easily seen that A1 S.1/ is isomorphic to the A-module (representation of Q) MW

KB K K BB 0 || BB 1 | || BB 1 B ~||| 1 0 1 K2 .

Further, since S.2/, S.3/ and S.4/ are simple injective nonprojective A-modules and I.1/= soc I.1/ D I.1/=S.1/ Š S.2/˚S.3/˚S.4/, it follows from Lemma 7.7 and Corollary 9.7 (ii) that we have in mod A almost split sequences of the forms 0 ! A S.2/ ! I.1/ ! S.2/ ! 0; 0 ! A S.3/ ! I.1/ ! S.3/ ! 0; 0 ! A S.4/ ! I.1/ ! S.4/ ! 0; and hence A S.2/ Š X2 , A S.3/ Š X3 , A S.4/ Š X4 , for representations X2 , X3 , X4 of the form 0> K K >> ~ ~ > ~ X2 W >1 ~~~ 1 K,

K@ 0 K @@ ~ ~ @ ~ X3 W 1 @ ~~~ 1 K,

KA K 0 AA A 1 X4 W 1 A K.

Applying Lemma 9.1 we conclude that there are in mod A almost split sequences of the forms 0 ! P .2/ ! M ! A1 P .2/ ! 0; 0 ! P .3/ ! M ! A1 P .3/ ! 0; 0 ! P .4/ ! M ! A1 P .4/ ! 0; and it is easily seen that A1 P .2/ Š A S.2/, A1 P .3/ Š A S.3/, A1 P .4/ Š A S.4/. Therefore, applying Theorems 7.11 and 7.12, we conclude that there is in mod A an almost split sequence of the form 0 ! M ! A S.2/ ˚ A S.3/ ˚ A S.4/ ! I.1/ ! 0:

298


Summing up, we conclude that A admits a component of the form S.2/ P .2/ X2 C DD CC }> x< y< x } y D CC } DD xx yy CC }} DD xx yy C! D! }}} yy xx / X3 / S.3/ /M / I.1/ / P .3/ S.1/ = AA FF = EE z { AA EE FF zz {{ AA EE FF zz {{ z EE A FF { A {{ " " zz P .4/ S.4/ . X4 In fact, it follows from Exercise I.12.7 (and also from Theorem 10.2) that this is the whole Auslander–Reiten quiver A of A. Observe that A is isomorphic to the full translation subquiver f0; 1; 2g Qop of ZQop , where Qop is the opposite quiver to Q. (c) Let H be the R-algebra of quaternions and ³ ² ˇ R 0 a 0 ˇ AD D 2 M2 .H/ a 2 R; b; c 2 H : H H c b Then A is an the R-algebra with canonical orthogonal primitive idempotents e1 D 10R 00 and e2 D 00 10H such that 1A D e1 C e2 . Hence, we have in mod A two indecomposable projective modules P1 D e1 A, P2 D e2 A and two indecomposable injective modules I1 D D.Ae1 /, I2 D D.Ae2 /, where D D HomR .; R/ is the standard duality between mod A and mod Aop , and dimR P1 D 1, dimR P2 D 8, dimR I1 D 5, dimR I2 D 4. In particular, there are no nonzero projective-injective modules in mod A. We have isomorphisms of R-vector spaces HomA .P1 ; P2 / Š e2 Ae1 Š H, HomA .P2 ; P1 / Š e1 Ae2 D 0, HomA .I1 ; I2 / Š HomAop .Ae2 ; Ae1 / Š e2 Ae1 Š H, and HomA .I2 ; I1 / Š HomAop .Ae1 ; Ae2 / Š e1 Ae2 D 0. Moreover, we have isomorphisms of Ralgebras EndA .P1 / Š e1 Ae1 D R and EndA .P2 / Š e2 Ae2 D H, and hence FP1 D EndA .P1 /= rad EndA .P1 / Š R and FP2 D EndA .P2 /= rad End 0A .P2 / Š H, 0 is a twosince R and H are division R-algebras. The R-subspace J D H 0 sided ideal of A such that J 2 D 0 and A=J Š R H, and hence J D rad A, by Lemma I.3.5. Hence, rad P1 D P1 .rad A/ D e1 rad A D 0 and rad P2 D P2 rad A D e2 rad A D e2 Ae1 Š H. Therefore, P1 is a simple projective right A-module and all indecomposable direct summands of rad P2 are isomorphic to P1 . More precisely, we have irrA .P1 ; P2 / D HomA .P1 ; P2 /, and hence, applying Corollary 9.4, we obtain dP1 P2 D dimFP2 irrA .P1 ; P2 / D dimH H D 1 and dP0 1 P2 D dimFP1 irrA .P1 ; P2 / D dimR H D 4. Obviously, irrA .P2 ; P1 / D 0 because HomA .P2 ; P1 / D 0. In particular, we have in A the valued arrow .1;4/

P1 ! P2 :


299

Since P1 is a simple projective noninjective module in mod A, applying Theorem 8.4 (ii) and Corollary 9.7 (i), we conclude that there exists in mod A an almost split sequence 0 ! P1 ! P2 ! A1 P1 ! 0: Moreover, we have in mod A a right minimal almost split homomorphism of the form (see Lemma 7.6) P14 ! P2 : Further, since P2 is noninjective it follows from Lemma 9.1 (ii) and Proposition 9.6 (ii), that we have in A the valued arrow .4;1/

P2 ! A1 P1 : In fact, it is the unique valued arrow in A with the target A1 P1 , because P2 is the target of a unique valued arrow A with the source P1 (see Lemma 9.1). Hence we have in mod A an almost split sequence of the form 0 ! P2 ! .A1 P1 /4 ! A1 P2 ! 0: Repeating the arguments we conclude that A contains an infinite component P.A/ of the form A1 P2 A2 P2 A3 P2 P B 2 :: A A A ;; ;; ;; : .1;4/ .1;4/ .1;4/ .4;1/ .4;1/ .4;1/ ; ;; ;; :: .1;4/ ;; ;; :: : ; ; 1 2 3 A P1 A P1 A P1 P1

:::

isomorphic to the full translation subquiver .N/ of Z , where is the valued quiver .1;4/ / : 1 2 In fact, since dimR P1 D 1, dimR P2 D 8, dimR I1 D 5 and dimR I2 D 4, we conclude inductively on n that we have in mod A almost split sequences of the forms 0 ! An P1 ! An P2 ! An1 P1 ! 0;

0 ! An P2 ! .An1 P1 /4 ! An1 P2 ! 0; and dimR An P1 D 6n C 1 and dimR An P2 D 12n C 8, for any n 2 N. Clearly, the indecomposable A-modules corresponding to different vertices of P.A/ are nonisomorphic, since they have different dimensions over R. Dually, we infer that Ae2 is a simple projective module in mod Aop and is isomorphic to the radical of the indecomposable projective Aop -module Ae1 . Then

300


I2 D D.Ae2 / is a simple injective module in mod A and I1 D D.Ae1 / is an injective module in mod A such that soc.I1 / Š P1 and I1 = soc.I1 / Š I2 . In particular, we have in mod A a left minimal almost split homomorphism of the form (see Lemma 7.7) I1 ! I2 , and irrA .I1 ; I2 / D HomA .I1 ; I2 / Š e2 Ae1 , and clearly irrA .I2 ; I1 / D 0, because HomA .I2 ; I1 / D 0. Applying Corollary 9.4, we conclude that dI1 I2 D dimFI2 irrA .I1 ; I2 / D dimH H D 1 and dI01 I2 D dimFI1 irrA .I1 ; I2 / D dimR H D 4, because EndA .I1 / Š D.e1 Ae1 / Š R and EndA .I2 / Š D.e2 Ae2 / Š H. Thus we have in A the valued arrow .1;4/

I1 ! I2 : Since I2 is a simple injective nonprojective A-module, applying Theorem 8.4 (i) and Corollary 9.7 (ii), we conclude that there exists in mod A an almost split sequence 0 ! A I2 ! I14 ! I2 ! 0: Further, since I1 is nonprojective, it follows from Lemma 9.1 (i) and Proposition 9.6 (i), that we have in A the valued arrow .4;1/

A I2 ! I1 : In fact, it is the unique valued arrow in A with target I1 , because I1 is the source of a unique valued arrow A with the target in I2 . Hence we have in mod A an almost split sequence of the form 0 ! A I1 ! A I2 ! I1 ! 0: Repeating the arguments, we conclude that A contains an infinite component Q.A/ of the form

:::

3 2 A A A I2 I2 B I2:: B I2:: B ;;; @ : : ; ::.4;1/ ::.4;1/ ;;.4;1/ :: :: .1;4/ :: .1;4/ ;; .1;4/ :: .1;4/ ; : : A3 I1 A2 I1 A I1 I1

isomorphic to the full translation subquiver N of Z . Moreover, since we have dimR P1 D 1, dimR P2 D 8, dimR I1 D 5 and dimR I2 D 4, by induction on n, we conclude that we have in mod A almost split sequences of the forms 0 ! AnC1 I2 ! .An I1 /4 ! An I2 ! 0; 0 ! AnC1 I1 ! AnC1 I2 ! An I1 ! 0; and dimR An I1 D 6n C 5 and dimR An I2 D 12n C 4, for n 2 N. Observe also that the indecomposable A-modules corresponding to different vertices of Q.A/ are nonisomorphic, since they have different dimensions over R.

10. The Auslander theorem

301

We note also the following fact. Proposition 9.12. Let A be a finite dimensional K-algebra over a field K and A D A1 A2 Ar be a decomposition of A into a product of indecomposable K-algebras (blocks). Then the Auslander–Reiten quiver A of A is a disjoint union A D A1 [ A2 [ [ Ar of the Auslander–Reiten quivers of A1 ; A2 ; : : : ; Ar . Proof. There is an equivalence mod A Š mod A1 mod A2 mod Ar of categories. Then for any modules Mi in mod Ai , i 2 f1; : : : ; rg, we have HomA .Mi ; Mj / D 0 for i ¤ j . Clearly, then the required decomposition of A is a consequence of the definition of the Auslander–Reiten quiver of an algebra.

10 The Auslander theorem In this section we establish a criterion for an indecomposable finite dimensional algebra to be of finite representation type, due to M. Auslander. As an application, we obtain the validity of the first Brauer–Thrall conjecture on the finite representation type of finite dimensional algebras having indecomposable finite dimensional modules of bounded dimension. Let A be a finite dimensional K-algebra over a field K. A chain f1

f2

ft

M D M0 ! M1 ! M2 ! ! M t1 ! M t D N of homomorphisms in mod A is said to be a path of irreducible homomorphisms of length t from M to N if the modules M0 ; M1 ; : : : ; M t are indecomposable and the homomorphisms f1 ; : : : ; f t are irreducible. Proposition 10.1. Let M and N be indecomposable modules in mod A with radA .M; N / ¤ 0, and t be a positive integer. Assume that there is no path of irreducible homomorphisms in mod A from M to N of length < t . Then the following statements hold. (i) There exists a path of irreducible homomorphisms f1

f2

ft

M D M0 ! M1 ! M2 ! ! M t1 ! M t and a homomorphism h t W M t ! N in mod A such that h t f t : : : f1 ¤ 0.

302


(ii) There exists a path of irreducible homomorphisms gt

g2

g1

N t ! N t1 ! ! N2 ! N1 ! N0 D N and a homomorphism v t W M ! N t in mod A such that g1 : : : g t v t ¤ 0. Proof. (i) We proceed by induction on t. Assume t D 1. Take a nonzero homomorphism h 2 radA .M; N /. Since M is indecomposable, we then conclude by Lemma 1.5 (i) that h is not a section in mod A. In particular, we obtain that M is not a simple injective A-module. Then it follows from Lemma 7.3 and Corollary 8.5 (ii) that there is in mod A a left minimal almost split homomorphism 2 3 w1 n M 6 :: 7 w D 4 : 5 W M ! Fj ; j D1 wn where w ¤ 0, F1 ; : : : ; Fn are indecomposable A-modules and wj 2 HomA .M; Fj /, for j 2 f1; : : : ; ng. Further, it follows from Theorem 7.11 that w1 ; : : : ; wn are irreducible homomorphisms. Since h W M ! N isLnot a section, we infer that n there exists a homomorphism g D Œ g1 ::: gn W N , with g1 2 j D1 Fj ! Pn HomA .F1 ; N /; : : : ; gn 2 HomA .Fn ; N /, such that h D gw D j D1 gj wj . Then gj wj ¤ 0 for some j 2 f1; : : : ; ng, because h ¤ 0. Taking f1 D wj and h1 D gj , we obtain the claim for t D 1. Assume t 2 and, for a positive integer r with r < t , we have by induction hypothesis a path of irreducible homomorphisms f1

f2

fr

M D M0 ! M1 ! M2 ! ! Mr1 ! Mr ; and a homomorphism hr W Mr ! N in mod A such that hr fr : : : f1 ¤ 0. Observe that hr is not an isomorphism since by the assumption there is no path of irreducible homomorphisms in mod A of length r from M to N . Since Mr and N are indecomposable A-modules, we then conclude that radA .Mr ; N / ¤ 0, and hence hr is not a section in mod A, by Lemma 1.5 (i). Obviously then Mr is not a simple injective A-module. Then, applying Lemma 7.3 and Corollary 8.5 (ii) again, we conclude that there is a nonzero left minimal almost split homomorphism in mod A with left end Mr , 2 3 u1 m M 6 :: 7 Ei ; u D 4 : 5 W Mr ! iD1 um where E1 ; : : : ; Em are indecomposable A-modules and ui 2 Hom .Mr ; Ei /, for LAm i 2 f1; : : : ; mg. Then there exists a homomorphism Œ v1 ::: vm W iD1 Ei ! N , with v1 2 HomA .E1 ; N /; : : : ; vm 2 HomA .Em ; N /, such that hr D vu D


303

Pm vi ui . Then 0 ¤ hr fr : : : f1 D iD1 vi ui fr : : : f1 , and consequently vi ui fr : : : f1 ¤ 0 for some i 2 f1; : : : ; mg. Take frC1 D ui and hrC1 D vi . Observe that frC1 is an irreducible homomorphism in mod A, by Theorem 7.11. Therefore, we have a path of irreducible homomorphisms in mod A, Pm

iD1

f1

f2

fr

frC1

M D M0 ! M1 ! M2 ! ! Mr1 ! Mr ! MrC1 ; and a homomorphism hrC1 W MrC1 ! N in mod A such that hrC1 frC1 fr : : : f1 ¤ 0. This shows (i). The proof of (ii) is similar. The following theorem is the announced criterion of M. Auslander (see [Au1], [Au2]) for the finiteness of the representation type of a finite dimensional algebra. Theorem 10.2. Let A be an indecomposable finite dimensional K-algebra over a field K and C be a component of A . Assume that there exists a positive integer b such that `.X/ b for every indecomposable A-module X with fX g in C . Then C is a finite component and C D A . In particular, A is of finite representation type. Proof. Let M and N be indecomposable modules in mod A with HomA .M; N / ¤ 0. We claim that fM g 2 C0 if and only if fN g 2 C0 (C0 is the set of vertices of the translation quiver C ). Clearly, if M Š N , then fM g D fN g and the claim follows. Hence, assume that radA .M; N / ¤ 0. Suppose fM g 2 C0 but fN g … C0 . Then there is no finite path of irreducible homomorphisms in mod A from M to N . Applying Proposition 10.1, we conclude that, for each positive integer t , there exists a path of irreducible homomorphisms f1

f2

ft

M D M0 ! M1 ! M2 ! ! M t1 ! M t and a homomorphism h t W M t ! N in mod A such that h t f t : : : f1 ¤ 0. Clearly, fi 2 radA .Mi1 ; Mi / for any i 2 f1; : : : ; tg, because f1 ; : : : ; f t are irreducible homomorphisms. Since `.Mi / b for i 2 f1; : : : ; tg, taking t D 2b 1, we obtain a contradiction to the Harada–Sai lemma (Lemma 2.1). Hence fM g 2 C0 implies fN g 2 C0 , and similarly fN g 2 C0 implies fM g 2 C0 . Let X be an indecomposable module in mod A with fX g in C0 . Then there exists an indecomposable projective A-module P such that HomA .P; X / ¤ 0, because X admits a projective cover P .X / ! X in mod A. It follows from the above discussion that fP g belongs to C0 . Take now an arbitrary indecomposable projective A-module Q. Since A is indecomposable as K-algebra, we know (see Excercise I.12.29) that there exists a sequence of indecomposable projective A-modules P D P1 ; P2 ; : : : ; Ps D Q such that HomA .Pi ; PiC1 / ¤ 0 or HomA .PiC1 ; Pi / ¤ 0 for any i 2 f1; : : : ; s 1g. Then fP g 2 C0 forces that fP1 g; : : : ; fPs g 2 C0 , and hence fQg 2 C0 . Hence C contains all projective vertices of A .

304


Let M be an arbitrary indecomposable module in mod A. Since we have HomA .Q; M / ¤ 0 for an indecomposable projective A-module Q, and fQg belongs to C0 , we conclude that fM g belongs to C0 . Moreover, it follows from the first part of the proof that there is in C a path of length smaller than 2b 1 from fQg to fM g. Then the local finiteness of A , and hence of C , implies that C D A and is finite. In particular, A is of finite representation type. Corollary 10.3. Let A be an indecomposable finite dimensional K-algebra over a field K. Then A is of finite representation type if and only if A admits a finite component. Proof. It is a direct consequence of Theorem 10.2, since there is a common bound on the length of indecomposable modules representing the vertices of a finite component C of A . In particular, we obtain the following consequence of Proposition 9.12 and Corollary 10.3. Corollary 10.4. Let A be an indecomposable finite dimensional K-algebra over a field K such that A is connected and finite. Then A is an indecomposable algebra of finite representation type. The next consequence of Theorem 10.2 is the validity of the first Brauer–Thrall conjecture, proved originally by A. V. Roiter [Ro]. Corollary 10.5. Let A be a finite dimensional K-algebra over a field K such that there is a common bound on the length of indecomposable modules in mod A. Then A is of finite representation type. Proof. Let A D A1 A2 Ar be decomposition of A into a product of indecomposable K-algebras (blocks of A). Then every indecomposable module in mod A is an indecomposable module in mod Ai , for some i 2 f1; : : : ; rg. Then it follows from the assumption on A and Theorem 10.2 that the algebras A1 ; A2 ; : : : ; Ar are of finite representation type, and hence A is also of finite representation type. Examples 10.6. (a) Let A be the following R-subalgebra of the matrix algebra M2 .C/ ² ³ ˇ R 0 a 0 ˇ D 2 M2 .C/ a 2 R; b; c 2 C : C C c b We will show, applying Theorem 10.2, that A is of finite representation type and there are in mod A exactly 4 isomorphism classes of indecomposable modules. The R-algebra A has the standard idempotents 0 1 0 0 e1 D R and e2 D 0 0 0 1C


305

such that 1A D e1 C e2 . Hence we have in mod A two indecomposable projective modules P1 D e1 A, P2 D e2 A and two indecomposable injective modules I1 D D.Ae1 /, I2 D D.Ae2 /, where D D HomR .; R/ is the standard duality between mod A and mod Aop , dimR P1 D 1, dimR P2 D 4, dimR I1 D 3, dimR I2 D 2. Also, we have isomorphisms of R-vector spaces HomA .P1 ; P2 / Š e2 Ae1 D C, HomA .P2 ; P1 / Š e1 Ae2 D 0, HomA .I1 ; I2 / Š HomAop .Ae2 ; Ae1 / Š e2 Ae1 D C, and HomA .I2 ; I1 / Š HomAop .Ae1 ; Ae2 / Š e1 Ae2 D 0. Further, we have isomorphisms of R-algebras EndA .P1 / Š e1 Ae1 D R and EndA .P2 / Š e2 Ae2 D C, and FP1 D EndA .P1 /= rad EndA .P1 / Š R and FP2 D EndA .P2 /= rad EndA .P2 / Š C, since R and C are fields. Observe also that 0 0 rad A D ; C 0 0 0 2 because J D C 0 is a two-sided ideal of A with J D 0 and A=J Š R C (see Lemma I.3.5). Hence we obtain rad P1 D P1 rad A D e1 rad A D 0 and rad P2 D P2 rad A D e2 rad A D e2 rad Ae1 Š C. Therefore, P1 is a simple projective right A-module and all indecomposable direct summands of rad P2 are isomorphic to P1 . More precisely, we have irrA .P1 ; P2 / D HomA .P1 ; P2 /, and applying Corollary 9.4, we conclude that dP1 P2 D dimFP2 irrA .P1 ; P2 / D dimC C D 1 and dP0 1 P2 D dimFP1 irrA .P1 ; P2 / D dimR C D 2. Clearly, irrA .P2 ; P1 / D 0 because HomA .P2 ; P1 / D 0. In particular, we have in A the valued arrow .1;2/

P1 ! P2 : Since P1 is a simple projective noninjective module in mod A, applying Theorem 8.4 (ii) and Corollary 9.7 (i), we conclude that we have in mod A an almost split sequence 0 ! P1 ! P2 ! A1 P1 ! 0; where dimR A1 P1 D dimR P2 dimR P1 D 4 1 D 3. Dually, Ae2 is a simple projective module in mod Aop and is the radical of the indecomposable projective Aop -module Ae1 . Then I2 D D.Ae2 / is a simple injective module in mod A and I1 D D.Ae1 / is an injective module in mod A such that soc.I1 / Š P1 and I1 = soc.I1 / Š I2 . In particular, we have in mod A a left minimal almost split homomorphism I1 ! I2 , by Lemma 7.7. Further, we have irrA .I1 ; I2 / D HomA .I1 ; I2 / Š e2 Ae1 and irrA .I2 ; I1 / D 0, because HomA .I2 ; I1 / D 0. We have also isomorphisms of R-algebras EndA .I1 / Š D.e1 Ae1 / Š R and EndA .I2 / Š D.e2 Ae2 / Š C, and so FI1 Š R and FI2 Š C. Applying Corollary 9.4, we conclude that dI1 I2 D dimFI2 irrA .I1 ; I2 / D dimC C D 1 and dI01 I2 D dimFI1 irrA .I1 ; I2 / D dimR C D 2. Hence, we have in A the valued arrow .1;2/

I1 ! I2 :

306


Moreover, since I2 is a simple injective nonprojective module in mod A, applying Theorem 8.4 (i) and Corollary 9.7 (ii), we conclude that there is in mod A an almost split sequence 0 ! A I2 ! I12 ! I2 ! 0; where dimR A I2 D 2 dimR I1 dimR I2 D 6 2 D 4. We claim that A1 P1 Š I1 and P2 Š A I2 . Observe that soc.P2 / D rad P2 Š S1 ˚ S1 , top.P2 / Š I2 , soc.I1 / D rad I1 Š P1 and top.I1 / D I1 = rad I1 D I1 = soc.I1 / Š I2 . In particular, we infer from the almost split sequence 0 ! P1 ! P2 ! A1 P1 ! 0 that `.A1 P1 / D `.P2 /`.P1 / D 31 D 2, and soc.A1 P1 / Š P1 , top.A1 P1 / Š I2 . Since soc.I1 / Š P1 , it follows from Theorem I.8.18 that there is in mod A a monomorphism u W A1 P1 ! I1 (injective envelope of A1 P1 ) which is an isomorphism, because dimR A1 P1 D 3 D dimR I1 . Since P2 is noninjective and I2 is nonprojective, applying Lemma 9.1 and Proposition 9.6, we conclude that A contains the valued arrows .2;1/

P2 ! A1 P1

and

.2;1/

A I2 ! I1 :

On the other hand, there is only one valued arrow with the target A1 P1 , because there is only one valued arrow in A with the source P1 . Since A1 P1 Š I1 , we obtain P2 Š A I2 . Summing up, we have proved that A admits a finite component C of the form > P2 AA ? I2 }} AA.2;1/ ~~ } ~ } AA ~ }} A ~~~ .1;2/ }} P1 I1 .1;2/

with P1 D A I1 and P2 D A I2 . Since A is an indecomposable R-algebra, applying Theorem 10.2, we conclude that C D A , A is of finite representation type, and P1 ; P2 ; I1 ; I2 form a complete set of pairwise nonisomorphic indecomposable modules in mod A. (b) Let K be a field and A D KQ=I the bound quiver algebra over K given by the quiver QW

o 1

˛ ˇ

/ o 2

/ o 3

/ ; 4

and the ideal I of KQ generated by ˛, , ˇ , , ˛ˇ , . Then A is a 14-dimensional K-algebra of Loewy length 3 (see Corollary I.5.20). Applying Proposition I.8.27, we conclude that the indecomposable projective A-modules are


307

injective and are of the forms P .1/ D I.1/ W

P .2/ D I.2/ W

P .3/ D I.3/ W

K2

10

10

0 1

o

0

P .4/ D I.4/ W

o

o

K

0 1

/Ko

/ K2 o

o

/K

0

o

0 1

/0

10

10

/0;

/Ko

/0;

0 1

o

/0o

/ K2

/K

0 1

o

o

10

10

/K;

0 1

/ K2 :

Moreover, we have top.P .i // D S.i / D soc.P .i // for any i 2 f1; 2; 3; 4g. We will show, applying Theorem 10.2, that A is of finite representation type and describe the Auslander–Reiten quiver A . Since P .1/, P .2/, P .3/, P .4/ are projective-injective modules, applying Proposition 8.6, we infer that there are in mod A the almost split sequences of the forms 0 ! rad P .1/ ! S.2/ ˚ P .1/ ! P .1/=S.1/ ! 0; 0 ! rad P .2/ ! S.1/ ˚ S.3/ ˚ P .2/ ! P .2/=S.2/ ! 0; 0 ! rad P .3/ ! S.2/ ˚ S.4/ ˚ P .3/ ! P .3/=S.3/ ! 0; 0 ! rad P .4/ ! S.3/ ˚ P .4/ ! P .4/=S.4/ ! 0; where the left terms and the right terms are of the forms rad P .1/ W

K

rad P .2/ W

K

rad P .3/ W

0

rad P .4/ W

0

P .1/=S.1/ W

o

1 0

o

/Ko /Ko

0 1

o

/Ko o

K

o

/0o 0 1

/Ko

1 0 0 1

/0o

/ 0,

/Ko

/ 0,

/Ko /Ko /0o

1 0 0 1

/K, /K, /0;

308


P .2/=S.2/ W

K

P .3/=S.3/ W

0

P .4/=S.4/ W

0

o

1 0

o

/Ko

1

/Ko o

/Ko

0

1 0

/0o

/ 0,

/Ko

0

/Ko

1

1

0

/K, /K.

In the next step, we describe the translations A S.i / and A1 S.i /, i 2 f1; 2; 3; 4g of the simple A-modules. Assume i D 1. The simple module S.1/ has in mod A a minimal projective presentation of the form p11

p01

P .2/ ! P .1/ ! S.1/ ! 0 and a minimal injective copresentation of the form i01

i11

0 ! S.1/ ! I.1/ ! I.2/: Hence, applying Lemma 5.1 and Proposition 5.3, we obtain in mod A exact sequences of the forms NA .p11 /

0 ! A S.1/ ! NA .P .2// ! NA .P .1//; with NA .P .1// D I.1/ D P .1/ and NA .P .2// D I.2/ D P .2/, and 1 .i / NA 11

NA1 .I.1// ! NA1 .I.2// ! A1 S.1/ ! 0; with NA1 .I.1// D P .1/ D I.1/ and NA1 .I.2// D P .2/ D I.2/. Observe also that HomA .P .2/; P .1// is the K-vector space generated by p11 and HomA .P .1/; P .2// D HomA .I.1/; I.2// is the K-vector space generated by i11 . Hence we conclude that o

/Ko

A S.1/ W

0

A1 S.1/ W

0

o

/Ko

1 0 0 1

/Ko /Ko

/ 0, / 0.

Similarly, we prove that A S.4/ W

0

o

/Ko

0 1

/Ko

/ 0,


A1 S.4/ W

0

o

/Ko

1 0

/Ko

309

/ 0.

As a consequence, we obtain A S.1/ D A1 S.4/ and A S.4/ D A1 S.1/. Assume i D 2. The simple module S.2/ has in mod A a minimal projective presentation of the form p12

p02

P .1/ ˚ P .3/ ! P .2/ ! S.2/ ! 0 and a minimal injective copresentation of the form i02

i12

0 ! S.2/ ! I.2/ ! I.1/ ˚ I.3/: Applying Lemma 5.1 and Proposition 5.3, we obtain then in mod A exact sequences of the forms NA .p12 /

0 ! A S.2/ ! NA .P .1/ ˚ P .3// ! NA .P .2//; with NA .P .1/ ˚ P .3// D I.1/ ˚ I.3/ D P .1/ ˚ P .3/ and NA .P .2// D I.2/ D P .2/, and 1 .i / NA 12

NA1 .I.2// ! NA1 .I.1/ ˚ I.3// ! A1 S.2/ ! 0; with NA1 .I.2// D P .2/ D I.2/ and NA1 .I.1/ ˚ I.3// D P .1/ ˚ P .3/ D I.1/ ˚ I.3/. Denote by u1 W P .1/ ! P .1/ ˚ P .3/, u3 W P .3/ ! P .1/ ˚ P .3/ the canonical monomorphisms, and by v1 W P .1/ ˚ P .3/ ! P .1/, v3 W P .1/ ˚ P .3/ ! P .3/ the canonical epimorphisms such that v1 u1 D 1P .1/ , v3 u3 D 1P .3/ , v3 u1 D 0, v1 u3 D 0. Then we easily conclude that HomA .P .1/˚P .3/; P .2// is the 2-dimensional Kvector space generated by p12 u1 v1 and p12 u3 v3 , and HomA .P .2/; P .1/ ˚ P .3// is the 2-dimensional K-vector space generated by u1 v1 i12 and u3 v3 i12 (observe that P .i/ D I.i / for i 2 f1; 2; 3g). Hence we conclude that A S.2/ W

K

A1 S.2/ W

K

o

1 0

o

0 1

/Ko /Ko

0 1 1 0

/Ko /Ko

1 0 0 1

/K,

/K.

Similarly, we show that A S.3/ W

K

o

0 1

/Ko

1 0

/Ko

0 1

/K,

310


A1 S.3/ W

K

o

1 0

/Ko

0 1

/Ko

1 0

/K.

As a consequence, we obtain that A S.2/ D A1 S.3/ and A S.3/ D A1 S.2/. We describe now the almost split sequences in mod A with simple left and right terms. Since we have in A the valued arrow rad P .2/ ! S.1/, there is in A the valued arrow A S.1/ ! rad P .2/, and rad P .2/ is a direct summand of the middle term of an almost split sequence in mod A with right term S.1/. Then the equalities dimK rad P .2/ D 3 D dimK A S.1/ C dimK S.1/ force that there is in mod A an almost split sequence 0 ! A S.1/ ! rad P .2/ ! S.1/ ! 0: Further, we have in A the valued arrow S.1/ ! P .2/=S.2/, and hence the valued arrow P .2/=S.2/ ! A1 S.1/. Hence P .2/=S.2/ is a direct summand of the middle term of an almost split sequence in mod A with the left term S.1/. Since dimK S.1/ C dimK A1 S.1/ D 3 D dimK P .2/=S.2/, we conclude that there is in mod A an almost split sequence of the form 0 ! S.1/ ! P .2/=S.2/ ! A1 S.1/ ! 0: Similarly, we show that there are in mod A almost split sequences of the forms 0 ! A S.4/ ! rad P .3/ ! S.4/ ! 0; 0 ! S.4/ ! P .3/=S.3/ ! A1 S.4/ ! 0: Observe now that we have in A the valued arrows rad P .1/ ! S.2/ and rad P .3/ ! S.2/, and hence also the valued arrows A S.2/ ! rad P .1/ and A S.2/ ! rad P .3/. Since dimK S.2/ C dimK A S.2/ D 5 D dimK rad P .1/ C dimK rad P .3/; we conclude, as above, that there is in mod A an almost split sequence of the form 0 ! A S.2/ ! rad P .1/ ˚ rad P .3/ ! S.2/ ! 0: Further, we have in A the valued arrows S.2/ ! P .1/=S.1/ and S.2/ ! P .3/=S.3/, and hence the valued arrows P .1/=S.1/ ! A1 S.2/ and P .3/=S.3/ ! A1 S.2/. Then the equalities dimK S.2/ C dimK A1 S.2/ D 5 D dimK P .1/=S.1/ C dimK P .3/=S.3/ force that there is in mod A an almost split sequence of the form 0 ! S.2/ ! P .1/=S.1/ ˚ P .3/=S.3/ ! A1 S.2/ ! 0:


311

Similarly, we prove that there are in mod A almost split sequences of the forms 0 ! A S.3/ ! rad P .2/ ˚ rad P .4/ ! S.3/ ! 0; 0 ! S.3/ ! P .2/=S.2/ ˚ P .4/=S.4/ ! A1 S.3/ ! 0: Recall that A S.1/ D A1 S.4/, A S.2/ D A1 S.3/, A S.3/ D A1 S.2/, A S.4/ D A1 S.1/. It follows from the above discussion that P .1/=S.1/ ! A S.3/ is the unique arrow in A with the source P .1/=S.1/ and A S.3/ ! rad P .4/ is the unique arrow in A with the target rad P .4/, and hence we conclude that P .1/=S.1/ D A rad P .4/. Similarly, P .4/=S.4/ ! A S.2/ is the unique arrow in A with the source P .4/=S.4/ and A S.2/ ! rad P .1/ is the unique arrow in A with the target rad P .1/, and then we infer that P .4/=S.4/ D A rad P .1/. Further, there are in A exactly two arrows P .3/=S.3/ ! A S.1/ and P .3/=S.3/ ! A S.3/ with the source P .3/=S.3/ and exactly two arrows A S.1/ ! rad P .2/ and A S.3/ ! rad P .2/ with the target rad P .2/, and consequently A rad P .2/ D P .3/=S.3/. Finally, there are in A exactly two arrows P .2/=S.2/ ! A S.2/ and P .2/=S.2/ ! A S.4/ with the source P .2/=S.2/ and exactly two arrows A S.2/ ! rad P .3/ and A S.4/ ! rad P .3/ with the target rad P .3/, and hence A rad P .3/ D P .2/=S.2/. Summing up, we conclude that A admits a finite component C of the form P .1/ P .4/ A @ A @ @@@ @@@ @@ @@ @ @ P .4/=S.4/ rad P .1/ P .1/=S.1/ rad P .4/ P .4/=S.4/ @@ A ~ @@ A A 0, G a finite group with jGj divisible by p, and KG the group algebra of G. Assume that B is a block of KG which is nonsimple and of finite representation type. Then B is Morita equivalent to a Brauer tree algebra A.TSm /.

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We also note that only exceptional Brauer tree algebras are Morita equivalent to blocks of the group algebras of finite groups, as it was shown by W. Feit in [Fe].

5 Simple algebras We will show in this section that all finite dimensional simple algebras, and then semisimple algebras, over a field are symmetric algebras. A prominent role in the proof of these facts will be played by the Noether–Skolem theorem on the finite dimensional central simple algebras over a field. We prove first several preliminary results. Let A be a finite dimensional K-algebra over a field K. Then the center of A is the K-subalgebra C.A/ D fa 2 A j ax D xa for all x 2 Ag of A. Lemma 5.1. Let A be a finite dimensional simple K-algebra over a field K. Then C.A/ is a finite field extension of K. Proof. Observe that C.A/ is a commutative K-subalgebra of A. Let c be a nonzero element in C.A/. Then cA D AcA D Ac is a nonzero two-sided ideal of A. Since A is a simple K-algebra, we conclude that cA D A, and so there exists an element d 2 A such that cd D 1A , and clearly also dc D cd D 1A . Moreover, for any element x 2 A, we have c.dx/ D .cd /x D 1A x D x D x1A D x.dc/ D .xd /c D c.xd /; because c 2 C.A/, and hence dx D xd . This shows that d 2 C.A/, and conse quently C.A/ is a field extension of K with dimK C.A/ dimK A < 1. Lemma 5.2. Let A and B be finite dimensional K-algebras over a field K. Then C.A ˝K B/ D C.A/ ˝K C.B/. Proof. Since C.A/ is a K-vector subspace of A, we have a decomposition A D A0 ˚ C.A/ of K-vector spaces. Then we have a decomposition A ˝K B D A0 ˝K B ˚ C.A/ ˝K B in mod K. We shall show first that C.A ˝K B/ C.A/ ˝K B. Take c 2 C.A˝K B/. Then c D x Cy for some x 2 A0 ˝K B and y 2 C.A/˝K B. Hence, for any a 2 A, we have c.a˝1B / D .a˝1B /c and y.a˝1B / D .a˝1B /y, and consequently we obtain that x.a ˝ 1B / D .a ˝ 1B /x. Further, choose a basis b1 ; b2P ; : : : ; bm of B over K. Then x 2 A0 ˝K B can be written in the form 0 x D m . But then x.a ˝ 1B / D iD1 xi ˝ bi for some elements Pm x1 ; : : : ; xm of PA m .a ˝ 1B /x leads to the equality iD1 axi ˝ bi D iD1 xi a ˝ bi , and hence to the equalities axi D xi a for all i 2 f1; : : : ; mg. Hence x1 ; : : : ; xm 2 C.A/, and

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so x1 D 0; : : : ; xm D 0. Thus x D 0, and c D y 2 C.A/ ˝K B. Therefore, C.A ˝K B/ C.A/ ˝K B. Similarly, we have a decomposition of K-vector spaces B D B 0 ˚ C.B/, and hence also C.A/ ˝K B D C.A/ ˝K B 0 ˚ C.A/ ˝K C.B/. Then, choosing a basis of C.A/ over K, we show as above that C.C.A/ ˝K B/ C.A/ ˝K C.B/. But then C.A ˝K B/ C.C.A ˝K B// C.C.A/ ˝K B/ C.A/ ˝K C.B/. Since the inclusion C.A/ ˝K C.B/ C.A ˝K B/ is obvious, we conclude that C.A ˝K B/ D C.A/ ˝K C.B/. Let A be a finite dimensional K-algebra over a field K and Aop the opposite algebra of A. For an element a in A, we denote by a0 the element a considered as an element of Aop . Then, for a; b 2 A, we have a0 b 0 D .ba/0 in Aop . Consider now the K-vector space Ae D Aop ˝K A together with the multiplication given by .a0 ˝ b/.c 0 ˝ d / D a0 c 0 ˝ bd D .ca/0 ˝ bd; for a; b; c; d 2 A. Then Ae is a K-algebra of dimension dimK Ae D .dimK A/2 and the identity 1Ae D 1Aop ˝ 1A D 1A0 ˝ 1A , called the enveloping algebra of A. We consider the category Bimod A of all A-bimodules, that is, the category whose objects are A-bimodules, the morphisms are homomorphisms of A-bimodules, and the composition of morphisms is the usual composition of maps. By a homomorphism f W M ! N of A-bimodules we mean a K-linear map f W M ! N such that f .amb/ D af .m/b for all a; b 2 A and m 2 M . Moreover, we have the full subcategory bimod A of Bimod A whose objects are the finite dimensional (over K) A-bimodules. Observe that bimod A contains the canonical A-bimodule A D A AA , called the regular A-bimodule. It follows from Proposition II.3.7 that there is a K-linear equivalence of categories F W Mod Ae ! Bimod A which restricts to a K-linear equivalence of categories F W mod Ae ! bimod A: We will identify Mod Ae with Bimod A and mod Ae with bimod A. Lemma 5.3. Let A be a finite dimensional K-algebra over a field K. Then there is a canonical isomorphism of K-algebras A W EndAe .A/ ! C.A/; which assigns to f 2 EndAe .A/ the element A .f / D f .1A /. Proof. Let f W A ! A be a homomorphism of right Ae -modules (A-bimodules). Then, for any a 2 A, we have af .1A / D f .a1A / D f .a/ D f .1A a/ D f .1A /a,

370


which shows that f .1A / 2 C.A/. Clearly, for any element c 2 C.A/, the K-linear map fc W A ! A such that f .a/ D ac D ca for any a 2 A is a homomorphism of Abimodules (right Ae -modules). Therefore, the map A is an isomorphism. Finally, observe that A .idA / D idA .1A / D 1A and A .fg/ D .fg/.1A / D f .g.1A // D f .1A g.1A // D f .1A /g.1A / D A .f / A .g/ for f; g 2 EndAe .A/. Thus A is an isomorphism of K-algebras. Proposition 5.4. Let A be a finite dimensional K-algebra over a field K. The following equivalences hold. (i) A is an indecomposable K-algebra if and only if A is an indecomposable module in mod Ae . (ii) A is a semisimple K-algebra if and only if A is a semisimple module in mod Ae . (iii) A is a simple K-algebra if and only if A is a simple module in mod Ae . Proof. Observe that a K-vector subspace I of A is a two-sided ideal of A if and only if I is a right Ae -submodule of A. Then the equivalence (i) follows from Proposition I.3.16. Moreover, it follows from Lemma I.6.11 that the radical rad A of A coincides with the intersection J.A/ of all maximal two-sided ideals of A. Hence, by the above remark, rad A D J.A/ is the intersection of all maximal right Ae -submodules of A. In particular, we obtain that rad A D 0 if and only if J.A/ D A rad.Ae / D 0. Therefore, the equivalence (ii) also holds. Obviously, then the equivalence (iii) also holds. A finite dimensional K-algebra A over a field K is called a central algebra if C.A/ D K. Corollary 5.5. Let A be a finite dimensional K-algebra over a field K. Then A is a central simple K-algebra if and only if A is a simple module in Ae with EndAe .A/ Š K. Lemma 5.6. Let n be a positive integer and F a finite dimensional division Kalgebra over a field K. Then C .Mn .F // D faIn j a 2 C.F /g D C.F /: In particular, Mn .F / is a central K-algebra if and only if F is a central K-algebra. Proof. For r; s 2 f1; : : : ; ng, let Ers be the elementary matrix in Mn .F / having 1F in the position .r; s/ and 0F elsewhere. Take a matrix Œaij 2 C.Mn .F //. Then Err Œaij D Œaij Err for any r 2 f1; : : : ; ng, which forces aij D 0 for all i ¤ j in f1; : : : ; ng. Moreover, we have E1r Œaij D Œaij E1r , which implies arr D a11 , for any r 2 f1; : : : ; ng. Hence Œaij D aIn for a D a11 2 F . Clearly, for any x 2 F , we have also .xIn /Œaij D Œaij .xIn /, which gives a 2 C.F /. Observe

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also that every matrix aIn , with a 2 C.F /, belongs to C.Mn .F //. Therefore, indeed C.Mn .F // D faIn j a 2 C.F /g D C.F /. Moreover, C.Mn .F // D K if and only if C.F / D K. We will establish now another characterization of central simple algebras. For a finite dimensional K-algebra over a field K, we consider the K-linear map T A W Ae ! EndK .A/ such that for a; b 2 A, T A .a0 ˝ b/ 2 EndK .A/ is given by T A .a0 ˝ b/.x/ D axb for any x 2 A. Observe that T A is a homomorphism of K-algebras. Proposition 5.7. Let A be a finite dimensional K-algebra over a field K. The following conditions are equivalent. (i) A is a central simple K-algebra. (ii) T A W Ae ! EndK .A/ is an isomorphism of K-algebras. (iii) Ae is a central simple K-algebra. Proof. Let n D dimK A. Observe that EndK .A/ Š Mn .K/ as K-algebras. Assume A is a central simple K-algebra. Then it follows from Corollary 5.5 that A is a simple module in mod Ae with EndAe .A/ Š K. Hence, we conclude that ƒA D EndAe .A/ Š K and .Ae /A Š Endƒop .A/ Š EndK .A/. Therefore, the A homomorphism of K-algebras T A W Ae ! EndK .A/ is the composition of the epimorphism of K-algebras rA W Ae ! .Ae /A , established in Corollary I.6.13, with ! EndK .A/. This shows that T A is an epimorphism. the isomorphism .Ae /A e Moreover, dimK A D .dimK A/2 D dimK EndK .A/. Thus T A is an isomorphism. Hence, (i) implies (ii). Assume now that T A W Ae ! EndK .A/ is an isomorphism of K-algebras. Then e A Š Mn .K/ as K-algebras, and consequently Ae is a central simple K-algebra, by Corollary I.6.5 and Lemma 5.6. Thus (ii) implies (iii). Finally, assume that Ae is a central simple K-algebra. Since dimK Ae D n2 , it follows from Corollary I.6.5 and Lemma 5.6 that Ae is isomorphic to the matrix algebra Mn .K/. But then every indecomposable right Ae -module X has dimK X D n and EndAe .A/ Š K. Moreover, every right Ae -module of dimension n is indecomposable. Hence, we infer that A is an indecomposable right Ae -module with EndAe .A/ Š K. Applying Proposition 5.4 and Corollary 5.5, we obtain that A is a central simple K-algebra. Therefore, (iii) implies (i). Proposition 5.8. Let A and B be finite dimensional K-algebras over a field K, and assume that B is a central simple K-algebra. Then every two-sided ideal of A ˝K B is of the form I ˝K B for some two-sided ideal of A.

372


Proof. Observe that, if I is a two-sided ideal of A, then I ˝K B is a two-sided ideal of A ˝K B. Let J be a two-sided ideal of A ˝K B. Consider I D fa 2 A j a ˝ 1B 2 J g, and observe that I is a two-sided ideal of A. We will show K. Then every element that J D I ˝K B. Let b1 ; : : : ; bm be a basis of B over P z 2 A ˝K B has a unique expression in the form z D m iD1 ai ˝ bi for some a1 ; : : : ; am 2 A. Since B is a central simple K-algebra, by Proposition 5.7, the Klinear homomorphism T B W B op ˝K B ! EndK .B/ such that T B .b 0 ˝c/.x/ D bxc for b; c 2 B is an isomorphism of K-algebras. Consider now the endomorphisms b1 ; : : : ; bm in EndK .B/ such that bk .bi / D ıik 1B for i; k 2 f1; : : : ; mg. Further, B we choose the elements u1 ; : : : ; um 2 B op ˝K B such that TP .uk / D bk for any m k 2 f1; : : : ; mg. Moreover, we have presentations uk D j D1 xj0 k ˝ bj with P xj k 2 B for i; k 2 f1; : : : ; mg. Take now an element m iD1 ai ˝ bi 2 A ˝K B. Then we have the equalities m X

1A ˝

j D1

xj0 k

m X

m X m X ai ˝ bi 1A ˝ bj D ai ˝ xj0 k bi bj

iD1

D

m X

ai ˝

j D1

iD1

D

m X

m X

0 xik bi bj

iD1

j D1

m X

D

ai ˝ T B .uk /.bi /

iD1

ai ˝ bk .bi / D ak ˝ 1B ;

iD1

P for any k 2 f1; : : : ; mg. Hence, if m iD1 ai ˝ bi 2 J , then ak ˝ 1B 2 J , and consequently ak 2 I , for any k 2 f1; : : : ; mg. This shows that J I ˝K B. Obviously, we have I ˝K B D .I ˝K 1B /.A˝K B/ J . Therefore, J D I ˝K B. As a direct consequence of Proposition 5.8 we obtain the following fact. Corollary 5.9. Let A and B be finite dimensional K-algebras over a field K, and assume that B is a central simple K-algebra. Then A ˝K B is a simple K-algebra if and only if A is a simple K-algebra. We note that the assumption on B to be a central simple K-algebra is essential for the validity of the above equivalence. For example, we have that C ˝R C Š C C as R-algebras (see Example 11.2). Corollary 5.10. Let A and B be finite dimensional K-algebras over a field K, and assume that B is a central simple K-algebra. Then rad.A ˝K B/ D .rad A/ ˝K B. Proof. It follows from Lemma I.6.11 that rad.A˝K B/ D J.A˝K B/ and rad A D J.A/ (intersections of maximal two-sided ideals in A ˝K B and A, respectively). Let Mi , i 2 I , be the family of all maximal two-sided ideals of A. Then, by

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373

Proposition 5.8, Mi ˝K B, i 2 I , is the family of all maximal two-sided ideals of A ˝K B, and we have \ J.A ˝K B/ D .Mi ˝K B/ D L ˝K B i2I

for a two-sided ideal L of M . Since L T ˝K B Mi ˝K B, we conclude that L Mi for any i 2 I , and hence L i2I Mi D J.A/. It follows that \ \ L ˝K B Mi ˝K B .Mi ˝K B/ D L ˝K B; i2I

i2I

so that J.A ˝A B/ D J.A/ ˝K B. Thus we have the equality rad.A ˝K B/ D .rad A/ ˝K B. The following corollary is an immediate consequence of Corollary 5.10. Corollary 5.11. Let A and B be finite dimensional K-algebras over a field K, and assume that B is a central simple K-algebra. Then A ˝K B is a semisimple K-algebra if and only if A is a semisimple K-algebra. We note that the assumption on B to be a central simple K-algebra is essential for the validity of the above equivalence (see Proposition 11.10 and Theorem 11.11). Corollary 5.12. Let A and B be finite dimensional K-algebras over a field K, and assume that B is a central simple K-algebra. Then A ˝K B is a central simple K-algebra if and only if A is a central simple K-algebra. Proof. It follows from Lemma 5.2 that C.A˝K B/ D C.A/˝K C.B/ D C.A/˝K K, because C.B/ D K. Hence, C.A ˝K B/ D K ˝K K D K1A˝B if and only if C.A/ D K D K1A . Moreover, by Corollary 5.9, A ˝K B is a simple K-algebra if and only if A is a simple K-algebra. The following important theorem is called the Noether–Skolem theorem for finite dimensional algebras. Theorem 5.13. Let A and B be finite dimensional K-algebras over a field K, and assume that A is a simple K-algebra and B is a central simple K-algebra. Moreover, let f and g be two K-algebra homomorphisms from A to B. Then there is an invertible element b 2 B such that g.a/ D bf .a/b 1 for all a 2 A. Proof. Consider the .A; B/-bimodules f B and g B such that f B D B and g B D B as K-vector spaces and the .A; B/-bimodule structure is given by axb D f .a/xb

and ayb D g.a/yb

374


for a 2 A, b 2 B, x 2 f B, y 2 g B. Then, by Proposition II.3.7, f B and g B are right .Aop ˝K B/-modules by x.a0 ˝ b/ D f .a/xb

and

y.a0 ˝ b/ D g.a/yb

for a; b 2 B, x 2 f B, y 2 g B. Since A is a simple K-algebra, Aop is also a simple K-algebra. Then it follows from Corollary 5.9 that Aop ˝K B is also a simple K-algebra. Since dimK f B D dimK B D dimK g B, applying Corollary I.6.8, we conclude that f B Š g B in mod Aop ˝K B. Let ' W f B ! g B be an isomorphism of right .Aop ˝K B/-modules, or equivalently, an isomorphism of .A; B/-bimodules. Then ' is an isomorphism of right B-modules, and consequently (see Lemma I.6.1) there is an invertible element b 2 B such that '.x/ D bx for any element x 2 B. Since ' W f B ! g B is also a homomorphism of left A-modules, we have '.f .a/x/ D g.a/'.x/ for any a 2 A and x 2 f B. Taking x D 1B , we obtain the equalities bf .a/ D '.f .a// D '.f .a/1B / D g.a/'.1B / D g.a/b; or equivalently, g.a/ D bf .a/b 1 , for any a 2 A.

Corollary 5.14. Let A be a finite dimensional central simple K-algebra over a field K. Then every K-algebra automorphism of A is an inner automorphism. Proof. Take A D B and two K-algebra homomorphisms f D idA and g D from A to A. Then it follows from Theorem 5.13 that there is an invertible element a 2 A such that .x/ D g.x/ D af .x/a1 D axa1 for any x 2 A. This shows that is an inner automorphism of A. In particular, we obtain the following fact. Corollary 5.15. Let n be a positive integer and K be a field. Then every K-algebra automorphism of Mn .K/ is an inner automorphism. Proposition 5.16. Let A be a finite dimensional simple K-algebra. Then A is a symmetric K-algebra. Proof. It follows from Corollary I.6.5 that there exists a positive integer n and a finite dimensional division K-algebra F such that A Š Mn .F / as K-algebras. Moreover, by Lemma II.6.13, the K-algebras Mn .F / and F are Morita equivalent. Therefore, by Corollary 4.3, it is enough to show that F is a symmetric K-algebra. Let L D C.F /. Then L is a finite field extension of K (Lemma 5.1) and F is a central simple L-algebra. Observe also that F is a finite dimensional basic selfinjective L-algebra, and hence a Frobenius L-algebra, by Proposition 3.9. Then it follows from Theorem 2.1 that there exists a nondegenerate associative L-bilinear form .; / W F F ! L. Let be the associated Nakayama automorphism of

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the L-algebra F , that is, ..x/; y/ D .y; x/ for all elements x; y 2 F . Since F is a central simple L-algebra, applying Corollary 5.14, we conclude that is an inner automorphism of F . Hence, by Corollary 3.4, F is a symmetric L-algebra. Then it follows from Theorem 2.2 that there exists a nondegenerate associative symmetric L-bilinear form .; / W F F ! L. Clearly, .; / is also a Kbilinear form. Applying Theorem 2.2 again, we conclude that F is a symmetric K-algebra. Summing up, we obtain that the K-algebra A, Morita equivalent to F , is a symmetric K-algebra. Corollary 5.17. Let A be a finite dimensional semisimple K-algebra over a field K. Then A is a symmetric K-algebra. Proof. It follows from Corollary I.6.6 that A is isomorphic to a product A1 Ar of finite dimensional simple K-algebras A1 ; : : : ; Ar . Then A1 ; : : : ; Ar are symmetric K-algebras, by Proposition 5.16. Applying Proposition 2.4, we obtain that A1 Ar , and hence also A, is a symmetric K-algebra.

6 The Nakayama theorems In this section we present four classical theorems of T. Nakayama that play a fundamental role in further considerations. Let A be a finite dimensional K-algebra over a field K. It follows from Corollary I.5.9 that there is a decomposition 1A D

nA mX A .i/ X

eij

iD1 j D1

of 1A into a sum of pairwise orthogonal primitive idempotents of A such that eij A Š eil A eij A © ekl A

for j; l 2 f1; : : : ; mA .i /g; i 2 f1; : : : ; nA g; for i; k 2 f1; : : : ; nA g with i ¤ k and all j 2 f1; : : : ; mA .i /g; l 2 f1; : : : ; mA .k/g:

We will call such a decomposition of 1A a canonical decomposition of 1A . The idempotents ei D ei1 , i 2 f1; : : : ; nA g, are said to be basic primitive idempotents of A. Then, applying Propositions I.8.2 and I.8.19, Corollary I.8.6 and Lemma I.8.22, we conclude that • Pi D ei A, i 2 f1; : : : ; nA g, is a complete set of pairwise nonisomorphic indecomposable projective modules in mod A; • Ii D D.Aei /, i 2 f1; : : : ; nA g, is a complete set of pairwise nonisomorphic indecomposable injective modules in mod A;

376


• Si D top.Pi / D ei A=ei rad A, i 2 f1; : : : ; nA g, is a complete set of pairwise nonisomorphic simple modules in mod A; • Si Š soc.Ii /, for any i 2 f1; : : : ; nA g. Observe that the algebra A is basic if and only if mA .i / D 1 for any i 2 f1; : : : ; nA g. The idempotent nA nA X X ei1 D ei eA D iD1

iD1

is said to be a basic idempotent of A. The following theorem is the first Nakayama theorem, proved in [Nak3], presented in this section. Theorem 6.1. Let A be a finite dimensional K-algebra over a field K. Then A is a selfinjective algebra if and only if there exists a permutation of the set f1; : : : ; nA g such that top.ei A/ Š soc.e.i/ A/ in mod A, for all i 2 f1; : : : ; nA g. Proof. Assume A is a selfinjective algebra. Then it follows from the definition of idempotents ei D ei1 , for i 2 f1; : : : ; nA g, that the set of indecomposable projective modules Pi D ei A, i 2 f1; : : : ; nA g, is a complete set of pairwise nonisomorphic indecomposable injective modules in mod A. Since the indecomposable projective modules in mod A are uniquely determined (up to isomorphism) by their simple tops (Corollary I.8.6) and the indecomposable injective modules in mod A are uniquely determined by their simple socles (Corollary I.8.21), we conclude, by Lemma I.8.22, that there is a permutation of the set f1; : : : ; nA g such that top.ei A/ Š soc.e.i/ A/ in mod A. Conversely, assume that there is a permutation of the set f1; : : : ; nA g such that top.ei A/ Š soc.e.i/ A/ in mod A, for all i 2 f1; : : : ; nA g. Then, for each i 2 f1; : : : ; nA g, the indecomposable projective right A-module Pi D ei A has a simple socle, isomorphic to S 1 .i/ D top.e 1 .i/ A/, and hence we have an injective envelope fi W Pi ! I 1 .i/ of Pi in mod A. Hence the homomorphisms fi , i 2 f1; : : : ; nA g, induce a monomorphism f W eA A D

nA M iD1

Pi !

nA M iD1

I 1 .i/ D

nA M

Ij D D.AeA /

j D1

of right A-modules. Then f induces a monomorphism g W eA AeA ! D.eA AeA / of right eA AeA -modules. Since dimK eA AeA D dimK D.eA AeA /, we conclude that g is an isomorphism of right eA AeA -modules. Therefore, for the basic algebra Ab D eA AeA of A, we have an isomorphism Ab ! D.Ab / of right Ab -modules,


377

and so Ab is a selfinjective algebra, because D.Ab / is injective in mod Ab . Since A is Morita equivalent to Ab , by Theorem II.6.16, applying Proposition 3.10, we obtain that A is a selfinjective algebra. For a finite dimensional K-algebra A, the permutation of the set f1; : : : ; nA g (in the notation introduced above) such that top.ei1 A/ Š soc.e.i/1 A/ in mod A, for all i 2 f1; : : : ; nA g, is said to be a Nakayama permutation of A. The second theorem of T. Nakayama, proved in [Nak2], gives a criterion for a selfinjective algebra to be a Frobenius algebra. Theorem 6.2. Let A be a finite dimensional selfinjective K-algebra over a field K, 1A D

nA mX A .i/ X

eij

iD1 j D1

be a canonical decomposition of 1A , and the associated Nakayama permutation of A. Then A is a Frobenius algebra if and only if mA .i / D mA ..i // for all i 2 f1; : : : ; nA g. Proof. Observe that we have two canonical direct sum decompositions AA D

nA m A .i/ M M

eij A in mod A

and

AA

D

iD1 j D1

nA m A .i/ M M

Aeij in mod Aop .

iD1 j D1

Then we have in mod A a decomposition of D.A/A of the form D.A/A D D.A A/ D

nA m A .i/ M M

D.Aeij /:

iD1 j D1

Moreover, by Lemma I.8.22, we have soc D.Aeij / Š eij A=eij rad A D top.eij A/ in mod A, for all i 2 f1; : : : ; nA g, j 2 f1; : : : ; mA .i /g. In particular, we conclude that D.Aeij / Š D.Aeil / D.Aeij / © D.Aekl /

for j; l 2 f1; : : : ; mA .i /g; i 2 f1; : : : ; nA g; for i; k 2 f1; : : : ; nA g with i ¤ k and all j 2 f1; : : : ; mA .i /g; l 2 f1; : : : ; mA .k/g:

On the other hand, it follows from the definition of the Nakayama permutation that top.ei1 A/ Š soc.e.1/1 A/ for i 2 f1; : : : ; nA g:

378


Further, by Theorem 2.1, A is a Frobenius algebra if and only if AA Š D.A/A as right A-modules. Then we obtain that AA Š D.A/A in mod A if and only if we have in mod A isomorphisms m A .i/ M j D1

D.Aeij / Š

mAM ..i//

e.i/l A

for i 2 f1; : : : ; nA g:

lD1

Therefore, we conclude that A is a Frobenius algebra if and only if mA .i / D mA ..i// for all i 2 f1; : : : ; nA g. A finite dimensional K-algebra A over a field K is said to be weakly symmetric if soc.P / Š top.P / for any indecomposable projective module P in mod A (see [NaNe]). Corollary 6.3. Let A be a weakly symmetric finite dimensional K-algebra over a field K. Then A is a Frobenius algebra. Proof. It follows from Theorem 6.1 that A is a selfinjective algebra with the identity Nakayama permutation. Then we conclude from Theorem 6.2 that A is a Frobenius algebra. Corollary 6.4. Let A be a symmetric algebra over a field K. Then A is a weakly symmetric algebra. Proof. Since A is a symmetric algebra, the identity automorphism idA of A is a Nakayama automorphism of A. Then it follows from Lemma I.8.22 and Corollary 3.14 that, for every primitive idempotent e of A, soc.eA/ Š soc.NA .eA// Š top.eA/, and consequently A is a weakly symmetric algebra. Corollary 6.5. Let A be a finite dimensional local K-algebra over a field K. Then A is a selfinjective algebra if and only if A is a weakly symmetric algebra. Proof. It is a direct consequence of Theorem 6.1 because, by Corollary I.6.15, the identity 1A is a unique primitive idempotent of A. Observe that the local K-algebras A , 2 K nf0; 1g, presented in Example 2.8, are weakly symmetric but nonsymmetric algebras (see also [NaNe], p. 665, for a more general class of such weakly symmetric algebras). Corollary 6.6. Let A be a finite dimensional K-algebra over a field K. Then A is a weakly symmetric algebra if and only if Aop is a weakly symmetric algebra. Proof. It is a direct consequence of Corollaries I.8.6 and I.8.21, Lemma I.8.22, and Theorem 6.1. The following proposition is an immediate consequence of Corollaries 5.17, 6.3 and 6.4.


379

Proposition 6.7. Let A be a finite dimensional semisimple K-algebra over a field K. Then A is a Frobenius K-algebra. We note also the following fact. Proposition 6.8. Let A be a nonsimple indecomposable finite dimensional selfinjective K-algebra over a field K. Then every simple module in mod A is nonprojective. Proof. Assume that S is a simple projective module in mod A. Then S is also an injective module in mod A, because A is selfinjective. Then we have a decomposition AA D P ˚ Q, where P is a direct sum of simple modules isomorphic to S and Q is a direct sum of indecomposable projective modules nonisomorphic to S. It follows from Theorem I.6.3, Corollary I.6.5 and the assumption on A that A is not a semisimple algebra, and hence Q ¤ 0. We claim that HomA .P; Q/ D 0 and HomA .Q; P / D 0. Assume HomA .P; Q/ ¤ 0. Then there is a nonzero homomorphism f W S ! Q, which is a section, because S is simple injective (Lemma I.8.13). Hence, applying Lemma I.4.2, we conclude that S is isomorphic to a direct summand of Q, a contradiction. Assume HomA .Q; P / ¤ 0. Then there is a nonzero homomorphism g W Q ! S , which is a retraction, because S is simple projective (Lemma I.8.1). Applying Lemma I.4.2 again, we infer that S is isomorphic to a direct summand of Q, a contradiction. Further, by Lemma I.5.7, there are nontrivial orthogonal idempotents e; f 2 A such that 1A D e C f and P D eA, Q D fA. Then, invoking Lemma I.8.7, we obtain that fAe Š HomA .P; Q/ D 0 and eAf Š HomA .Q; P / D 0. Therefore, we get a nontrivial decomposition A D eAe ˚ fAf into a product of K-algebras, which contradicts the indecomposability of A as K-algebra. The third theorem of T. Nakayama asserts that there is a Galois correspondence between the right and left ideals of a selfinjective algebra. We need a characterization of the annihilators of ideals of algebras. Proposition 6.9. Let A be a finite dimensional K-algebra over a field K, I a left ideal of A, and J a right ideal of A. Then there exist canonical isomorphisms (i) HomAop .A A=I; A A/ ! rA .I / of right A-modules; ! À .J / of left A-modules. (ii) HomA .AA =J; AA / Proof. (i) Recall that HomAop .A A; A A/ has the canonical structure of a right Amodule given by .f b/.a/ D f .a/b for f 2 HomAop .A A; A A/ and a; b 2 A. Then the canonical K-linear isomorphism ' W HomAop .A A; A A/A ! AA ; given by '.f / D f .1A / for f 2 HomAop .A A; A A/, is an isomorphism of right A-modules. Indeed, for b 2 A and f 2 HomAop .A A; A A/, we have '.f b/ D

380


.f b/.1A / D f .1A /b D '.f /b. Further, for f 2 HomAop .A A; A A/, we have f .I / D f .I1A / D If .1A / D I '.f /, and hence f .I / D 0 if and only if '.f / 2 rA .I /. Observe also that ff 2 HomAop .A A; A A/ j f .I / D 0g is a right A-submodule of HomAop .A A; A A/, because, for f 2 HomAop .A A; A A/ and b 2 A, we have .f b/.I / D f .I /b. Therefore, we have a commutative diagram in mod A, '

HomAop .A A; A A/A S j ff 2 HomAop .A A; A A/ j f .I / D 0g

/ AA S j '

/ rA .I / .

The canonical epimorphism W A A ! A A=I of left A-modules induces the canonical isomorphism of right A-modules W HomAop .A A=I; A A/ ! ff 2 HomAop .A A; A A/ j f .I / D 0g ; given by .g/ D g, for g 2 HomAop .A A=I; A A/. Clearly, .g/.I / D g.I / D 0 for any g 2 HomAop .A A=I; A A/. Conversely, if f 2 HomAop .A A; A A/ satisfies f .I / D 0, then f D .gf / for gf 2 HomAop .A A=I; A A/ defined by gf .a C I / D f .a/ for any a 2 A. Moreover, is a homomorphism of right A-modules. Indeed, for g 2 HomAop .A A=I; A A/ and a; b 2 A, we have .gb/.a/ D .gb/.a/ D .gb/.a C I / D g.a C I /b D .g/.a/b D .g/.a/b D . .g/b/.a/, and hence .gb/ D .g/b. Summing up, we have the canonical isomorphism of right A-modules ' W HomAop .A A=I; A A/ ! rA .I / such that .' /.g/ D .g/.1A / D g.1A C I /, for g 2 HomAop .A A=I; A A/. (ii) Similarly as above, it is easy to check that the canonical map '0

0

W HomA .AA =J; AA / ! À .J /

such that .' 0 0 /.g 0 / D .g 0 0 /.1A / D g 0 .1A CJ /, for g 0 2 HomA .AA =J; AA /, with 0 the canonical epimorphism AA ! AA =J of right A-modules, is an isomorphism of left A-modules. The following theorem is the announced third theorem of T. Nakayama, proved in [Nak2] and [Nak3]. Theorem 6.10. Let A be a finite dimensional selfinjective K-algebra over a field K. Then the annihilator operations À and rA induce mutually inverse Galois bijections ²

right ideals of A

³ o

À rA

/

²

left ideals of A

³ :

381


Proof. Note that, for every nonempty subset X of A, À .X / D fa 2 A j aX D 0g is a left ideal of A and rA .X / D fa 2 A j Xa D 0g is a right ideal of A. Let I be a left ideal of A and J be a right ideal of A. We will show that À rA .I / D I

rA À .J / D J:

and

Observe that I À rA .I / and J rA À .J / from the definition of the annihilator operations À and rA . We claim that dimK I D dimK À rA .I / and dimK J D dimK rA À .J / Since A A and AA are injective modules, by the Morita–Azumaya duality Theorem II.7.11, the exact functors DAop D HomAop .; A A/ W mod Aop ! mod A; DA D HomA .; AA / W mod A ! mod Aop ; define a duality between mod Aop and mod A. Consider the canonical exact sequence in mod Aop , 0 ! I ! A A ! A A=I ! 0: Applying the duality functor DAop to this sequence, we obtain, by Proposition 6.9 (i), a commutative diagram in mod A, 0

/ HomAop .A A=I; A A/

0

/ rA .I /

Š

/ HomAop .A A; A A/ Š

/ AA

/ HomAop .I; A A/

/0

Š

/ AA =rA .I /

/0

with exact rows and the vertical homomorphisms being isomorphisms. Then, applying Proposition 6.9 (ii), we obtain isomorphisms of left A-modules I Š DA .DAop .I // D HomA .HomAop .I; A A/; AA / Š HomA .AA =rA .I /; AA / Š À rA .I /; and hence dimK I D dimK À rA .I /, as claimed. Then I À rA .I / implies the equality I D À rA .I /. Similarly, consider the canonical exact sequence in mod A, 0 ! J ! AA ! AA =J ! 0: Applying the duality functor DA to this sequence, we obtain, by Proposition 6.9 (ii), a commutative diagram in mod Aop , 0

/ HomA .AA =J; AA /

0

/ À .J /

Š

/ HomA .AA ; AA / Š

/ AA

/ HomA .J; AA /

/0

Š

/ A A=À .J /

/0

382


with exact rows and the vertical homomorphisms being isomorphisms. Then, applying Proposition 6.9 (i), we obtain isomorphisms of right A-modules J Š DAop .DA .J // D HomAop .HomA .J; AA /; A A/ Š HomAop .A A=À .J /; A A/ Š rA À .J /; and hence dimK J D dimK rA À .J /. Then the inclusion J rA À .J / implies the equality J D rA À .J /. Finally, observe that, for nonempty subsets X Y of A we have the inclusions À .Y / À .X / and rA .Y / rA .X /. Therefore, the annihilator operations À and rA induce mutually inverse Galois bijections between the left ideals and right ideals of A. Corollary 6.11. Let A be a finite dimensional selfinjective K-algebra over a field K. Then the annihilator operations À and rA induce mutually inverse Galois bijections ² ³ ² ³ À two-sided two-sided / o : ideals of A ideals of A rA Proof. Observe that for a two-sided ideal I of A the annihilators À and rA are two-sided ideals of A. Indeed, we have the inclusions .À .I /A/I D À .I /.AI / À .I /I D 0; I.ArA .I // D .IA/rA .I / I rA .I / D 0; and hence À .I /A À .I / and ArA .I / rA .I /, so À .I / is a right ideal of A and rA .I / is a left ideal of A. Then the corollary follows from Theorem 6.10. Corollary 6.12. Let A be a finite dimensional selfinjective K-algebra over a field K, I a left ideal of A and J a right ideal of A. Then the following equivalences hold. (i) I ¤ A A if and only if rA .I / ¤ 0. (ii) J ¤ AA if and only if À .J / ¤ 0. Proof. It follows from Theorem 6.10 that (i) I ¤ A A if and only if rA .I / ¤ rA .A/ D 0; (ii) J ¤ AA if and only if À .J / ¤ À .A/ D 0.

The following fourth theorem of T. Nakayama, proved in [Nak3], will be important for our further considerations.


383

Theorem 6.13. Let A be a finite dimensional selfinjective K-algebra over a field K. Then soc.A A/ D soc.AA /. Proof. It follows from Proposition 6.9 that we have isomorphisms HomAop .A= rad A; A A/ Š rA .rad A/ of right A-modules and HomA .A= rad A; AA / Š À .rad A/ of left A-modules. Further, since A= rad A is semisimple as left A-module and right A-module, applying the duality functors HomAop .; A A/ W mod Aop ! mod A and HomA .; AA / W mod A ! mod Aop to A= rad A, we conclude that HomAop .A= rad A; A A/ is a semisimple right Amodule and HomA .A= rad A; AA / is a semisimple left A-module. Hence, by Proposition 6.9, rA .rad A) is a semisimple right A-module and À .rad A/ is a semisimple left A-module, and consequently rA .rad A/ soc.AA / and À .rad A/ soc.A A/. On the other hand, soc.A A/ is a semisimple left A-module and soc.AA / is a semisimple right A-module, so, applying Proposition I.5.13 and Corollary I.5.15, we conclude that rad.A/ soc.A A/ D rad.soc A A/ D 0 and soc.AA /.rad A/ D rad.soc AA / D 0. This shows that soc.A A/ rA .rad A/ and soc.AA / À .rad A/. Summing up, we have proved the inclusions soc.A A/ rA .rad A/ soc.AA /

and

soc.AA / À .rad A/ soc.A A/:

In particular, we obtain the equality soc.A A/ D soc.AA /.

For a finite dimensional selfinjective K-algebra A, we will denote soc A WD soc.A A/ D soc.AA / and call the socle of A. Since soc A is a two-sided ideal of A, we may consider the factor algebra A= soc A, which is usually not a selfinjective algebra. This allows us to introduce an important equivalence relation between finite dimensional selfinjective algebras. Two finite dimensional selfinjective Kalgebras A and ƒ are said to be socle equivalent if the factor algebras A= soc A and ƒ= soc ƒ are isomorphic as K-algebras. We also note the following direct consequence of the proof of Theorem 6.13 and Corollary 6.11. Corollary 6.14. Let A be a finite dimensional selfinjective K-algebra. Then the following equalities hold. (i) À .rad A/ D soc A D rA .rad A/. (ii) À .soc A/ D rad A D rA .soc A/.

384


We end this section with examples of Nakayama selfinjective algebras. Let K be a field, m and n positive integer, n the cyclic quiver ˛n

1 0

˛1

" 2

n A ˛n1

˛2

3,

n 1 Q ˛n2

d ˛i

q i

˛3

˛i 1

Im;n the admissible ideal of the path algebra K n generated by all compositions of m C 1 consecutive arrows in n , and Nnm .K/ D K n =Im;n the associated bound quiver algebra. Theorem 6.15. Let Q be a finite connected quiver with nonempty set of arrows, K a field, I an admissible ideal of the path algebra KQ, and A D KQ=I the associated bound quiver algebra. Then the following conditions are equivalent. (i) A is a selfinjective Nakayama algebra. (ii) A D Nnm .K/ for some positive integers m and n. Proof. Since Q has at least one arrow, it follows from Lemma I.6.16 that A is not a semisimple algebra. Moreover, by Proposition I.3.15, A is an indecomposable algebra. Finally, A is a basic algebra, by Lemma II.6.17. Assume that A is a selfinjective Nakayama algebra. We note first that, by Proposition 6.8, every simple module in mod A is not projective, equivalently not injective. Applying now Theorem I.10.3 we conclude that Q is a quiver of the form n , for a positive integer n. We claim that I D Im;n , for some positive integer m, and consequently A D Nnm .K/. Observe that I is generated by a finite number of paths in Q D n (see Lemma I.1.6). Let w be the shortest path of n which belongs to I , i be the source of w and j be the target of w. Denote by s the length of w, and observe that s 2, because the ideal I is admissible. We will show that all paths in n of length s belong to I , and consequently I D Im;n for m D s 1. For each vertex k 2 f1; : : : ; ng of n , denote by P .k/ the indecomposable projective right A-module at k and by I.k/ the indecomposable injective right A-module at k. The path w is of the form ˛i ˛iC1 : : : ˛j 1 . We show that ˛iC1 : : : ˛j 1 ˛j belongs to I . Suppose it is not the case. Then invoking the canonical equivalence of categories mod A ! repK .Q; I / (Theorem I.2.10) and the description of


385

indecomposable projective modules and indecomposable injective modules given in Proposition I.8.27, we conclude that top.I.j // is the simple module S.i C 1/ at the vertex i C 1 and dimK I.j / < dimK P .i C 1/. Hence there is a minimal epimorphism h W P .i C 1/ ! I.j / in mod A which is not an isomorphism. On the other hand, the injective module I.j / is projective, and so h is a retraction, by Lemma I.8.1. Then the indecomposability of P .i C 1/ and I.j / forces h to be an isomorphism, a contradiction. Repeating the above arguments we conclude that all compositions of s consecutive arrows in the cycle Q D n belong to I . This shows that (i) implies (ii). Assume A D Nnm .K/ for some positive integers m and n. Applying Theorem I.10.3, we infer that A is a Nakayama algebra. We claim that, for any vertex i 2 f1; : : : ; ng of n , the indecomposable projective A-module P .i / associated to i is injective. Fix i 2 f1; : : : ; ng. Then the ideal Im;n contains the path ˛i ˛iC1 : : : ˛iCm , but no proper subpath of this path. Applying Theorem I.2.10 and Proposition I.8.27 again, we conclude that P .i / D I.i C m 1/, and so P .i / is injective. Therefore, we obtain that all indecomposable projective modules in mod A are injective, and hence A is a selfinjective algebra. Corollary 6.16. Let Q be a finite connected quiver with nonempty set of arrows, K a field, I an admissible ideal of the path algebra KQ, and A D KQ=I the associated bound quiver algebra. Then the following conditions are equivalent. (i) A is a symmetric Nakayama algebra. (ii) A is a weakly symmetric Nakayama algebra. (iii) A D Nnm .K/ for some positive integers m and n, with n dividing m. Proof. The implication (i) ) (ii) follows from Corollary 6.4 and Theorem 6.15. Further, the implication (ii) ) (iii) follows from Theorem 6.15 and the description of the indecomposable projective modules over the Nakayama algebra Nnm .K/, invoking Proposition I.8.27. Assume now that A D Nnm .K/ and n divides m, say m D nr. Then Nnm .K/ is isomorphic to the Brauer tree algebra A.TSr / of the Brauer star T D TSr ? ?? 1 ??n ?? ?? 2 n1 3 S

: : :: :: :

i

::

:::

::: :::

: :::

::

: : :: ::

386


over K, with the exceptional vertex S of multiplicity r in the center of TSr (see Example 4.7), and consequently Nnm .K/ is a symmetric algebra, by Theorem 4.11. Hence the implication (iii) ) (i) also holds. The following example shows that the selfinjective Nakayama bound quiver algebras provide examples of selfinjective algebra with the Nakayama permutations of arbitrary large order. Example 6.17. Let K be a field, n 2 an integer, and A D Nnn1 .K/ D K n =In1;n . Since In1;n is the ideal of K n generated by all compositions of n consecutive arrows in n , we easily deduce that the Nakayama permutation of A is the cyclic permutation

1 2 3 ::: n 1 n ; n 1 2 ::: n 2 n 1 and hence has order n.

7 Non-Frobenius selfinjective algebras In this section we present a general form of non-Frobenius selfinjective finite dimensional K-algebras over a field K. We will give a construction of all such selfinjective algebras (up to isomorphism), proposed in [SY]. Let ƒ be a basic finite dimensional K-algebra over a field K, n D nƒ , and let ƒ D P1 ˚ P2 ˚ ˚ Pn be a decomposition of ƒ into a direct sum of indecomposable right ƒ-modules. Then P1 ; P2 ; : : : ; Pn are pairwise nonisomorphic indecomposable projective modules in mod ƒ. Let m.1/; m.2/; : : : ; m.n/ be an arbitrary sequence of positive integers. For each i 2 f1; : : : ; ng, we denote by Mi the direct sum of m.i / copies Pij , j 2 f1; : : : ; m.i /g, of Pi . Consider the right ƒ-module m.i/ n n M M M Mi D Pij M D iD1

iD1 j D1

and the endomorphism K-algebra ƒ.m.1/; m.2/; : : : ; m.n// WD Endƒ .M /: For each i 2 f1; : : : ; ng and j 2 f1; : : : ; m.i /g, we denote by eij the primitive idempotent of ƒ.m.1/; m.2/; : : : ; m.n// given by the composition ij

uij

M ! Pij ! M

7. Non-Frobenius selfinjective algebras

387

of the canonical projection ij W M ! Pij and the canonical inclusion uij W Pij ! M . Clearly, we have the decomposition 1ƒ.m.1/;:::;m.n// D

n m.i/ X X

eij

iD1 j D1

into a sum of pairwise orthogonal idempotents of ƒ.m.1/; : : : ; m.n//. We note that the idempotents eij are primitive because the right ƒ-modules Pij are indecomposable (see Lemma I.4.4). Moreover, since M is a progenerator of mod ƒ, it follows from Theorem II.6.7 that ƒ.m.1/; : : : ; m.n// is Morita equivalent to ƒ. Proposition 7.1. Let ƒ be a basic finite dimensional selfinjective K-algebra over a field K, n D nƒ , m.1/; : : : ; m.n/ be a sequence of positive integers, and A D ƒ.m.1/; : : : ; m.n//. Then the following statements hold. (i) A is a finite dimensional selfinjective K-algebra. (ii) ƒ is a basic algebra of A. (iii) The Nakayama permutations of ƒ and A coincide. Proof. Let be the Nakayama permutation of f1; : : : ; ng associated to a decomposition ƒ D P1 ˚ P2 ˚ ˚ Pn of ƒ into a direct sum of indecomposable right ƒ-submodules, that is, we have in mod ƒ isomorphisms top.Pi / Š soc.P.i/ /, for any i 2 f1; : : : ; ng. Moreover, in the notation introduced above, we have the decomposition n m.i/ X X eij 1A D iD1 j D1

of the identity of A D ƒ.m.1/; : : : ; m.n// into a sum of pairwise orthogonal primitive idempotents of A. Hence, for i 2 f1; : : : ; ng and j 2 f1; : : : ; m.i /g, we have that eij A D Homƒ .M; Pij / Homƒ .M; M / D A is a right A-submodule of A, where, for g 2 Homƒ .M; Pij / and f 2 Endƒ .M / D A, the right multiplication of g by f is the composition gf 2 Homƒ .M; Pij /. Since Pij D Pi1 D Pi for j 2 f1; : : : ; m.i /g, the identity map Homƒ .M; Pij / ! Homƒ .M; Pi1 / induces an isomorphism eij A ! ei1 A of right A-modules. Moreover, for i; k 2 f1; : : : ; ng with i ¤ k, the right A-modules ei1 A and ek1 A are not isomorphic, because the right ƒ-modules Pi and Pk are not isomorphic. Therefore, 1A D

n m.i/ X X iD1 j D1

eij

388


P is a canonical decomposition of 1A , and eA D niD1 ei1 is a basic idempotent of A. In particular, we obtain isomorphisms of K-algebras Ab D eA AeA Š Endƒ .Pi1 ˚ ˚ Pi n / D Endƒ .ƒ/ Š ƒ; and hence ƒ is a basic algebra of A. Since A and Ab are Morita equivalent (Theorem II.6.16), it follows from Proposition 3.10 that A is a selfinjective K-algebra. Therefore, the statements (i) and (ii) hold. In order to prove the statement (iii), we first show that Homƒ .M; soc.Pi1 //, considered as a right A-submodule of ei1 A D Homƒ .M; Pi1 /, is the socle soc.ei1 A/ of ei1 A. Obviously, Homƒ .M; soc.Pi1 // is a right A-submodule of ei1 A, by definition of the right A-module structure on A D Homƒ .M; M /. Observe also that the right A-module Homƒ .M; soc.Pi1 // is nonzero, because we have a nonzero composed epimorphism of right ƒ-modules M ! P 1 .i/1 D P 1 .i/ ! top P 1 .i/ ! soc.Pi / D soc.Pi1 /: On the other hand, since A is a selfinjective algebra and ei1 A is an indecomposable projective right A-module, soc.ei1 A/ is a simple right A-module. We claim that Homƒ .M; soc.Pi1 // is a semisimple right A-module. Indeed, it follows from Proposition I.5.16 and Lemma III.1.3 that rad A D rad Endƒ .M / D Homƒ .M; rad M /. For f 2 Homƒ .M; rad M / and g 2 Homƒ .M; soc.Pi1 // we then have .gf /.M / D g.f .M // g.rad M / D g.M rad ƒ/ D g.M / rad ƒ soc.Pi1 / rad ƒ D 0; by Corollary I.5.15. Hence, we obtain that Homƒ .M; soc.Pi1 //.rad A/ D 0. Applying Corollary I.5.15 again, we conclude that indeed Homƒ .M; soc.Pi1 // is a semisimple right A-module. Therefore, we have 0 ¤ Homƒ .M; soc.Pi1 // soc.ei1 A/, and consequently Homƒ .M; soc.Pi1 // D soc.ei1 A/ as right A-modules. In particular, for i 2 f1; : : : ; ng, we have soc.e.i/1 A/ei1 D Homƒ .M; soc.P.i/1 //ei1 : It is obvious that Homƒ .M; soc.P.i/1 //ei1 Š Homƒ .Pi1 ; soc.P.i/1 // as Kvector spaces and Homƒ .Pi1 ; soc.P.i/1 // ¤ 0, because Pi1 D Pi , P.i/1 D P.i/ and top.Pi / Š soc.P.i/ /. Therefore, we obtain, by Lemma I.8.7, that HomA .ei1 A; soc.e.i/1 A// Š soc.e.i/1 A/ei1 ¤ 0; and so top.ei1 A/ Š soc.e.i/1 A/, for any i 2 f1; : : : ; ng. Hence the Nakayama permutation of ƒ is the Nakayama permutation of A, for the chosen canonical decompositions of 1ƒ and 1A .


389

Corollary 7.2. Let ƒ be a basic finite dimensional selfinjective K-algebra, n D nƒ , the Nakayama permutation of f1; : : : ; ng, and m.1/; : : : ; m.n/ a sequence of positive integers. Then ƒ.m.1/; : : : ; m.n// is a Frobenius algebra if and only if m.i/ D m..i // for any i 2 f1; : : : ; ng. Proof. It is a direct consequence of Theorem 6.2 and Proposition 7.1.

The following theorem gives a general form of non-Frobenius selfinjective algebras. Theorem 7.3. Let A be a finite dimensional selfinjective K-algebra over a field K. Then the following conditions are equivalent. (i) A is not a Frobenius algebra. (ii) A is isomorphic to an algebra ƒ.m.1/; : : : ; m.n//, where ƒ is a basic finite dimensional selfinjective K-algebra with nƒ D nA and m.1/; : : : ; m.n/ is a sequence of positive integers such that m.i / ¤ m.j / and top.ei1 A/ Š soc.ej1 A/ for some i; j 2 f1; : : : ; ng. Proof. Let ƒ D Ab . We know from Theorem II.6.16 that A is Morita equivalent to ƒ. Then, applying Corollary II.6.22, we conclude that A is isomorphic to an algebra ƒ.m.1/; : : : ; m.n//, for some sequence m.1/; : : : ; m.n/ of positive integers. The equivalence of (i) and (ii) then follows from Corollary 7.2 and the definition of the Nakayama permutation. Corollary 7.4. Let ƒ be a basic indecomposable finite dimensional selfinjective K-algebra which is not weakly symmetric, and ƒ D P1 ˚ ˚Pn a decomposition of ƒ into a direct sum of indecomposable right ƒ-modules and top.Pi / © soc.Pi / for some i 2 f1; : : : ; ng. Take an integer r 2 and the sequence m.1/; : : : ; m.n/ of positive integers such that m.j / D 1 for j 2 f1; : : : ; ng n fi g and m.i / D r. Then the finite dimensional selfinjective K-algebra ƒ.m.1/; : : : ; m.n// is not a Frobenius algebra. Proof. Let be the Nakayama permutation of f1; : : : ; ng. As top.Pi / Š soc.P.i/ /, we have .i / ¤ i (by the choice of i ), and hence m..i // D 1 ¤ r D m.i /. Then the claim follows from Corollary 7.2. Example 7.5. Let K be a field, Q the quiver 1 o

˛ ˇ

/ ; 2

I the ideal of KQ generated by ˛ˇ and ˇ˛, and ƒ D KQ=I the associated bound quiver algebra. Then ƒ is a basic indecomposable selfinjective Nakayama K-algebra with .rad ƒ/2 D 0 (see Theorem I.10.3). In fact, letting "N1 D "1 C I ,

390


"N2 D "2 C I , ˛N D ˛ C I and ˇN D ˇ C I , where "1 and "2 are the trivial paths at 1 and 2, we have the decomposition ƒ D P1 ˚ P2 of ƒ into a direct sum of indecomposable projective ƒ-modules P1 D "N1 ƒ D K "N1 ˚ K ˛N

N and P2 D "N2 ƒ D K "N2 ˚ K ˇ:

Moreover, we have isomorphisms in mod ƒ, S1 WD top.P1 / D K "N1 ;

S2 WD top.P2 / D K "N2 ;

soc.P1 / D K ˛N Š K "N2 ;

soc.P2 / D K ˇN Š K "N1 ;

and hence top.P1 / Š soc.P2 / and top.P2 / Š soc.P1 /. This shows that the Nakayama permutation D ƒ of f1; nƒ g D f1; 2g is the transposition .1; 2/. In particular, ƒ is not a weakly symmetric K-algebra. Let m.1/ D 2, m.2/ D 1, M D P10 ˚ P11 ˚ P2 ; where P10 D P1 , P11 D P1 , and A D ƒ.2; 1/ D Endƒ .M /: Let e0 , e1 , e2 be the primitive idempotents of A corresponding to the direct summands P10 , P11 , P2 of ƒ, that is, the compositions e0 W M ! P10 ! M; e1 W M ! P11 ! M; e2 W M ! P2 ! M; of the canonical projections and injections. Moreover, let 1 u0 W

P10 ! P11

and

0 u1 W

P11 ! P10

be the identity homomorphisms idP1 W P1 ! P1 . Further, let 2 v0 W

P10 ! P2

and

2 v1 W

P11 ! P2

be the canonical composite homomorphisms P1 ! S1 ,! P2 , assigning to "N1 2 P1 the element ˇN D ˇN "N1 D "N2 ˇN of P2 , and 0 w2 W

P2 ! P10

and

1 w2 W

P2 ! P11


391

the canonical composite homomorphisms P2 ! S2 ,! P1 , assigning to "N2 2 P2 the element ˛N D ˛N "N2 D "N1 ˛N of P1 . Then we have the equalities e 0 D 0 u1 1 u0 ; 2 v0 0 w2

e1 D 1 u0 0 u1 ;

D e2 2 v0 e0 ;

2 v1

D e0 0 w2 e2 ;

1 w2

Then the algebra A D ƒ.2; 1/ D K-algebra of matrices 2 a0 0 b1 6 1 b0 a1 6 6 0 0 6 6 6 4 0

D e2 2 v1 e1 ; D e1 1 w2 e2 :

Endƒ .P10 ˚ P11 ˚ P2 / is isomorphic to the 3

0 1

0

1 2

a2 a2 0 0

2 0

a0 1 b0

7 7 7 7; 7 2 1 7 5 0 b1 a1

where a0 2 Ke0 , a1 2 Ke1 , a2 2 Ke2 , 0 b1 2 K 0 u1 , 1 b0 2 K 1 u0 , 0 2 2 K 0 w2 , 2 K 1 w2 , 2 0 2 K 2 v0 , 2 1 2 K 2 v1 . This is exactly the first selfinjective non-Frobenius K-algebra presented by T. Nakayama in 1939 in the paper [Nak2], p. 624.

1 2

We may visualize the hierarchy of finite dimensional selfinjective algebras over a field as follows.

division algebras simple algebras semisimple algebras symmetric algebras weakly symmetric algebras Frobenius algebras selfinjective algebras

392


8 The syzygy functors In this section we introduce the syzygy functors on the (stable) module categories of finite dimensional algebras, which play a prominent role in the modern representation theory of algebras. In particular, we obtain two new (mutually inverse) equivalences of the stable module categories of finite dimensional selfinjective algebras. Let A be a finite dimensional K-algebra over a field K. For a module M in mod A, we have two canonical exact sequences iM

hM

0 ! A .M / ! PA .M / ! M ! 0; uM

pM

0 ! M ! EA .M / ! A1 .M / ! 0 in mod A, where hM W PA .M / ! M is a projective cover of M and uM W M ! EA .M / is an injective envelope of M in mod A (see Theorems I.8.4 and I.8.18), iM the embedding of A .M / D Ker hM into PA .M /, and pM the canonical projection of EA .M / onto A1 .M / D Coker uM . Then A .M / is called the syzygy module of M and A1 .M / the cosyzygy module of M . We note that, by Theorems I.8.4 (ii) and I.8.18 (ii), A .M / and A1 .M / are uniquely determined by M , up to isomorphism. We collect now some general properties of the syzygy an cosyzygy modules. Proposition 8.1. Let A be a finite dimensional K-algebra over a field K and M; N modules in mod A. The following statements hold. (i) A .M / D 0 if and only if M is from proj A. (ii) A .M / is a module in modI A. (iii) A .M ˚ N / Š A .M / ˚ A .N /. (iv) If M Š N , then A .M / Š A .N /. i

h

(v) Let 0 ! U ! P ! M ! 0 be an exact sequence in mod A with P from proj A. Then U Š A .M / ˚ Q in mod A for a direct summand Q of P . (vi) A1 .M / D 0 if and only if M is from inj A. (vii) A1 .M / is a module in modP A. (viii) A1 .M ˚ N / Š A1 .M / ˚ A1 .N /. (ix) If M Š N , then A1 .M / Š A1 .N /. u

p

E ! V ! 0 be an exact sequence in mod A with E from (x) Let 0 ! N ! inj A. Then V Š A1 .N / ˚ F in mod A for a direct summand F of E.

8. The syzygy functors

393

Proof. Let hM W PA .M / ! M and hN W PA .N / ! N be the projective covers of M and N in mod A, respectively. (i) Observe that M is from proj A if and only if hM W PA .M / ! M is an isomorphism. Hence the statement (i) is obvious. (ii) Suppose A .M / admits a nonzero injective direct summand Q in mod A. Then Q is a right A-submodule of PA .M / and let f W Q ! PA .M / be the associated homomorphism. Since Q is injective, it follows from Lemma I.8.13 that f is a section in mod A, and consequently there exists a homomorphism g W PA .M / ! Q such that gf D idQ . Applying now Lemma I.4.2 we obtain that PA .M / D Im f ˚Ker g D Q˚X with X D Ker g. Then Q A .M / D Ker hM gives the equality PA .M / D Ker hM C X , with X ¤ PA .M /. This contradicts the fact that Ker hM is a superfluous submodule of PA .M /. Therefore, A .M / is a module in modI A. (iii) We have the canonical exact sequence h

iM 0 0 iN

i

hM 0 0 hN

0 ! A .M / ˚ A .M / ! PA .M / ˚ PA .M / ! M ˚ N ! 0 in mod A, where iM W A .M / ! PA .M / and iN W A .N / ! PA .N / are canonical embeddings. Further, by Proposition I.5.13 (ii), we have rad.PA .M / ˚ PA .N // D rad PA .M / ˚ rad PA .N /, and consequently top.PA .M / ˚ PA .N // D top.PA .M // ˚ top.PA .N //: Since hM and hN are minimal epimorphisms, the induced homomorphisms top.hM / W top.PA .M // ! top.M / and top.hN / W top.PA .N // ! top.N / are isomorphisms, and hence the homomorphism

0 top.hM / hM 0 D W top.PA .M /˚PA .N // ! top.M ˚N / top 0 hN 0 top.hN / is also an isomorphism. Applying now Theorem I.8.4, we infer that 0 hM W PA .M / ˚ PA .N / ! M ˚ N 0 hN is a projective cover of M ˚ N in mod A. Then for a chosen projective cover hM ˚N W PA .M ˚ N / ! M ˚ N of M ˚ N in mod A, there is an isomorphism i h 0 g W PA .M / ˚ PA .N / ! PA .M ˚ N / in mod A such that hM ˚N g D hM 0 hN (see Theorem I.8.4 (ii)). This leads to the isomorphisms 0 hM A .M ˚ N / D Ker hM ˚N Š hM ˚N g D Ker 0 hN D Ker hM ˚ Ker hN D A .M / ˚ A .N /:

394


(iv) Assume f W M ! N is an isomorphism in mod A. Then, since f hM is an epimorphism and Ker f hM D Ker f being superfluous in PA .M /, it follows that f hM W PA .M / ! N is also a projective cover of N in mod A. Hence, by Theorem I.8.4, there is an isomorphism g W PA .M / ! PA .N / in mod A such that hN g D f hM . Hence we obtain that g induces an isomorphism A .M / ! A .N /. (v) We have the commutative diagram in mod A, 0

/U

0

/ A .M /

i

/P

iM

/ PA .M /

h

/M

hM

/M

g

v

/0

idM

/ 0.

Since h D hM g is an epimorphism and hM is a minimal epimorphism, we conclude that g is also an epimorphism, by Lemma I.8.3. Then it follows from Lemma I.8.1 that g is a retraction in mod A, and so there exists a homomorphism f W PA .M / ! P in mod A such that gf D idPA .M / . Observe that then we have hf D hM gf D hM D idM hM . This implies that there is a homomorphism u W A .M / ! U in mod A such that i u D f iM , by the exactness of the rows of the above diagram. Moreover, we have iM vu D giu D gf iM D iM , and so vu D idA .M / , because iM is a monomorphism. Applying Lemma I.4.2, we obtain that P D Im f ˚ Ker g and U D Im u ˚ Ker v, where Im f Š PA .M / and Im u Š A .M /. We claim that Ker v Š Ker g. Indeed, from the equality gi D iM v, we conclude that the restriction of i to Ker v induces a homomorphism j W Ker v ! Ker g, which is clearly a monomorphism. On the other hand, we have the equalities dimK Ker g D dimK P dimK Im f D dimK P dimK PA .M / D .dimK M C dimK U / .dimK M C dimK A .M // D dimK U dimK A .M / D dimK Ker v: Hence j W Ker v ! Ker g is an isomorphism in mod A. But then we conclude that U Š A .M / ˚ Q for Q D Ker g being a direct summand of P . The proofs of the statements (vi)–(x) are similar and are left to the reader.

In general, the syzygy and cosyzygy operators do not reflect much of the properties of module categories. For example, if A is a hereditary finite dimensional K-algebra over a field K, then A .M / belongs to proj A and A1 .M / belongs to inj A for any module M in mod A, by Theorems I.9.1, I.9.2 and I.9.3. The following example exhibits a bad behaviour of the syzygy and cosyzygy operators.


395

Example 8.2. Let Q be the quiver 3 4 2 @@ ~ @@ ~ ~ @ˇ ~~ ˛ @@ ~~ 1 and KQ the path algebra of Q over a field K. Then A is a hereditary algebra, by Theorem I.9.5. We identify mod A with the category repK .Q/ of finite dimensional K-linear representations of Q (see Corollary I.2.11). Then the representations 0? 0 0 ?? ?? K S.1/ D P .1/ , KA 0 0 AA AA 1 K P .2/ , 0? K K ?? } ??1 }}} ~} 1 K L2 ,

K? 0 0 ?? ?? 0 S.2/ D I.2/ ,

0 ?? K 0 ?? ?? 0 S.3/ D I.3/ ,

0= 0 K == == 0 S.4/ D I.4/ ,

0? K 0 ?? ??1 K P .3/ ,

0? 0 K ?? }} ?? } ~}} 1 K P .4/ ,

KA K K AA } AA1 }}} 1 ~} 1 K I.1/ ,

KA 0 K AA }} AA } ~}} 1 1 K L3 ,

KA K 0 AA AA1 1 K L4 ,

K K B K BB 0 || | BB1 1 ! }|| 1 0 1 K2 M

form a complete family of pairwise nonisomorphic indecomposable representations in repK .Q/ (see Exercise I.12.7). Here, according to Section I.8, S.i /, P .i /, I.i / denote the simple, projective, injective representation of Q over K at the vertex i 2 f1; 2; 3; 4g. Then a simple checking shows that, if X is one of the representations S.2/, S.3/, S.4/, L2 , L3 , L4 , M , then A .X / Š S.1/, and A .I.1// Š S.1/ ˚ S.1/. Moreover, we have also isomorphisms A1 .S.1// Š S.2/ ˚ S.3/ ˚ S.4/ Š A1 .M /; A1 .P .3// Š S.2/ ˚ S.4/; A1 .L2 / Š S.2/;

A1 .P .2// Š S.3/ ˚ S.4/;

A1 .P .4// Š S.2/ ˚ S.3/;

A1 .L3 / Š S.3/;

A1 .L4 / Š S.4/:

This shows that the syzygy (respectively, cosyzygy) modules of indecomposable modules are not necessarily indecomposable, and nonisomorphic modules may have isomorphic syzygy (respectively, cosyzygy) modules. The following proposition shows that, for finite dimensional selfinjective algebras over a field, the syzygy and cosyzygy operators have good properties.

396


Proposition 8.3. Let A be a finite dimensional selfinjective K-algebra over a field K and M; N modules in modP A. Then the following statements hold. (i) M Š A1 .A .M // and M Š A .A1 .M //. (ii) M is indecomposable if and only if A .M / is indecomposable. (iii) M is indecomposable if and only if A1 .M / is indecomposable. (iv) M Š N if and only if A .M / Š A .N /. (v) M Š N if and only if A1 .M / Š A1 .N /. Proof. Since A is selfinjective, we have proj A D inj A, and hence modP A D modI A. In particular, we conclude from Proposition 8.1 that A .M /, A .N /, A1 .M /, A1 .N / belong to modP A. (i) We have the commutative diagram 0

/ A .M /

0

/ A .M /

uA .M /

idA .M /

/ EA .A .M //

pA .M /

v

iM

/ PA .M /

/ 1 .A .M // A

/0

w hM

/M

/0

in mod A with exact rows, as PA .M / is injective, A1 .A .M // D Coker uA .M / , and hM vuA .M / D hM iM idA .M / D 0. Further, since uA .M / is a minimal monomorphisms and vuA .M / D iM is a monomorphism, applying Lemma I.8.15, we conclude that v is a monomorphism. Then, by Lemma I.8.13, v is a section, because EA .A .M // is injective, and so v 0 v D idEA .A .M // for some homomorphisms v 0 W PA .M / ! EA .A .M // in mod A. In particular, we have PA .M / D Im v ˚ Ker v 0 , by Lemma I.4.2. Observe also that Ker hM D Im iM D Im vuA .M / Im v, and hence the restriction f W Ker v 0 ! M of hM to Ker v 0 is a monomorphism. Moreover, Ker v 0 belongs to proj A D inj A as a direct summand of PA .M /, by Lemma I.8.1. Applying Lemma I.8.13 again, we infer that f is a section, so gf D idKer v0 for some homomorphism g W M ! Ker v 0 in mod A. But then it follows from Lemma I.4.2 that M D Im f ˚ Ker g, where Im f Š Ker v 0 is an injective module. Then our assumption that M is in modP A D modI A implies that Im f D 0. Hence PA .M / D Im v. This shows that v is an isomorphism in mod A, because v is a monomorphism. Therefore, we obtain that w W A1 .A .M // ! M is an isomorphism (see Exercise I.12.17). The proof that M Š A .A1 .M // is similar. The remaining statements (ii)–(v) follow from (i) and the corresponding statements of Proposition 8.1.


397

Theorem 8.4. Let A be a finite dimensional selfinjective K-algebra over a field K. Then we have the mutually inverse equivalences of categories mod A o

A 1 A

/ mod A .

Proof. We first show that there are well-defined covariant functors A W mod A ! mod A and A1 W mod A ! mod A. Let f W M ! N be a homomorphism in mod A. Then there exists a commutative diagram in mod A, 0

/ A .M /

0

/ A .N /

iM

f1

/ PA .M /

hM

f0

iN

/ PA .N /

/M

/0

f hN

/N

/ 0.

Assume we have in mod A another commutative diagram of the form 0

/ A .M /

0

/ A .N /

iM

fQ1

/ PA .M /

hM

fQ0

iN

/ PA .N /

/M

/0

f hN

/N

/ 0.

Then hN .f0 fQ0 / D hN f0 hN fQ0 D f hM f hM D 0. Since A .N / D Ker hN , there exists a homomorphism s W PA .M / ! A .N / such that f0 fQ0 D iN s. Further, we have the equalities iN .f1 fQ1 / D iN f1 iN fQ1 D f0 iM fQ0 iM D .f0 fQ0 /iM D iN siM ; and hence f1 fQ1 D siM , because iN is a monomorphism. This shows that f1 fQ1 2 PA .A .M /; A .N //, and hence we have f1 C PA .A .M /; A .N // D fQ1 C PA .A .M /; A .N //. This allows us to assign to f 2 HomA .M; N / a welldefined element A .f / D f1 C PA .A .M /; A .N // D f1 in HomA .A .M /; A .N // D HomA .A .M /; A .N //=PA .A .M /; A .N //: We also note that, if f 0 2 HomA .M; N /, f00 2 HomA .PA .M /; PA .N // and f10 2 HomA .A .M /; A .N // satisfy the equalities hN f00 D f 0 hM and iN f10 D f00 iM , then hN .f0 C f00 / D .f C f 0 /hM and iN .f1 C f10 / D .f0 C f00 /iM ,

398


and consequently A .f C f 0 / D A .f / C A .f 0 /. Moreover, for 2 K, we have hN .f0 / D .f /hM and iN .f1 / D .f0 /iM , and so A .f / D A .f /. Therefore, we have a K-linear homomorphism A W HomA .M; N / ! HomA .A .M /; A .N //. We will show now that A vanishes on all homomorphisms from PA .M; N /. Let g 2 PA .M; N / and g D ˇ˛ for homomorphisms ˛ W M ! P and ˇ W P ! N in mod A with P from proj A. Since hN W PA .N / ! N is an epimorphism, there exists a homomorphism W P ! PA .N / in mod A such that ˇ D hN . Taking g0 D ˛hM and g1 D 0 in HomA .A .M /; A .N //, we obtain the commutative diagram in mod A, 0

/ A .M /

0

/ A .N /

iM

g1

/ PA .M /

hM

/M

g0

iN

/ PA .N /

/0

g

/N

hN

/ 0,

and consequently A .g/ D 0. As a consequence, we obtain that the syzygy operator induces a K-linear homomorphism A W HomA .M; N / ! HomA .A .M /; A .N // such that A .f / D A .f / for any f 2 HomA .M; N /. The above arguments show also that A .gf / D A .g/A .f / for any homomorphisms f 2 HomA .M; N / and g 2 HomA .N; U /, and obviously we have also A .idM / D idA .M / . Summing up, we have defined a covariant K-linear functor A W mod A ! mod A: Dually, for a homomorphism f W M ! N in mod A, we have a commutative diagram in mod A, 0

/M

0

/N

uM

/ EA .M /

pM

/ 1 .M / A

f0

f uN

/ EA .N /

pN

/0

f1

/ 1 .N / A

/ 0.

Assume we have in mod A another commutative diagram of the form 0

/M

uM

/N

pM

uN

/ EA .N /

/ 1 .M / A

fQ0

f

0

/ EA .M /

pN

/0

fQ1

/ 1 .N / A

/ 0.


399

Then .f 0 fQ0 /uM D f 0 uM fQ0 uM D uN f uN f D 0. Since A1 .M / D Coker uM , there exists a homomorphism t W A1 .M / ! EA .N / such that f 0 fQ0 D tpM . Further, we have the equalities .f 1 fQ1 /pM D f 1 pM fQ1 pM D pN f 0 pN fQ0 D pN .f 0 fQ0 / D pN tpM ; and hence f 1 fQ1 D pN t , because pM is an epimorphism. Since EA .N / belongs to inj A D proj A, we obtain that f 1 fQ1 2 PA .A1 .M /; A1 .N //. This allows us to assign to f 2 HomA .M; N / a well-defined element A1 .f / D f 1 C PA .A1 .M /; A1 .N // D f 1 in HomA .A1 .M /; A1 .N //. Applying arguments as above, we obtain that the cosyzygy operator induces a K-linear homomorphism A1 W HomA .M; N / ! HomA .A1 .M /; A1 .N // such that A1 .f / D A1 .f / for any f 2 HomA .M; N /. Moreover, we have A1 .gf / D A1 .g/A1 .f / for any homomorphisms f 2 HomA .M; N / and g 2 HomA .N; U /, and A1 .idM / D id1 .M / . Therefore, we obtain a covariant A K-linear functor A1 W mod A ! mod A: For a module M in mod A, there is a decomposition M D PM ˚ MP , where PM is a module from proj A and MP a module from modP A, both uniquely determined (up to isomorphism) by M . Then the canonical section u W MP ! M and retraction v W M ! MP with vu D idMP induce the mutually inverse isomorphisms u W MP ! M and v W M ! MP in mod A. Applying Proposition 8.3, we then infer that there exist natural isomorphisms of functors A1 B A Š 1mod A

and

A B A1 Š 1mod A :

Let A be a finite dimensional selfinjective K-algebra over a field K. Since mod A D mod A, we have the mutually inverse Auslander–Reiten translations mod A o

A 1 A

/ mod A

with A D D Tr and A1 D Tr D, and the mutually inverse Nakayama functors mod A o

NA 1 NA

/ mod A

with NA D D HomA .; A/ and NA1 D HomA .; A/D (see Sections III.4 and III.5). Moreover, we have the mutually inverse higher syzygy and cosyzygy functors mod A o

i A i A

/ mod A ,

400


where A0 D 1mod A and AiC1 D A B Ai , A.iC1/ D A1 B Ai for all nonnegative integers i . The following theorem relates the syzygy and cosyzygy functors with the Auslander–Reiten and Nakayama functors. Theorem 8.5. Let A be a finite dimensional selfinjective K-algebra over a field K. Then the following statements hold. (i) The functors A , A2 NA , and NA A2 from mod A to mod A are naturally isomorphic. (ii) The functors A1 , A2 NA1 , and NA1 A2 from mod A to mod A are naturally isomorphic. Proof. Since A is selfinjective, AA is an injective cogenerator in mod A with A Š EndA .AA /, and hence, by the Morita–Azumaya Theorem II.7.11, the functors HomA .; A/ W mod A ! mod Aop ; HomAop .; A/ W mod Aop ! mod A define a duality between mod A and mod Aop . Moreover we have the standard duality D D HomK .; K/ between mod A and mod Aop . Therefore, the Nakayama functors NA D D HomA .; A/ and NA1 D HomAop .; A/D give the mutually inverse equivalences of categories mod A o

NA 1 NA

/ mod A .

p1

p0

(i) Let M be a module from modP A and P1 ! P0 ! M ! 0 be a minimal projective presentation of M in mod A. Then A .M / Š Ker p0 D Im p1 , p1 W P1 ! Im p1 is a projective cover of Im p1 in mod A, and consequently we have A2 .M / D A .A .M // Š Ker p1 . This leads to an exact sequence p1

p0

0 ! A2 .M / ! P1 ! P0 ! M ! 0 in mod A. Applying the Nakayama functor NA W mod A ! mod A, we obtain then an exact sequence in mod A of the form NA .p1 /

NA .p0 /

0 ! NA .A2 .M // ! NA .P1 / ! NA .P0 / ! NA .M / ! 0: On the other hand, we have from Proposition III.5.3 an exact sequence of the form NA .p1 /

NA .p0 /

0 ! A M ! NA .P1 / ! NA .P0 / ! NA .M / ! 0:


401

! NA .A2 .M //. Passing now from Therefore, we obtain an isomorphism A M mod A to mod A, we obtain a natural isomorphism of functors A ! NA A2 from mod A to mod A. Further, since the Nakayama functor NA W mod A ! mod A is an equivap0 p1 lence of categories, the minimal projective presentation P1 ! P0 ! M ! 0 NA .p1 /

of M in mod A leads to the minimal projective presentation NA .P1 / ! NA .p0 /

NA .P0 / ! NA .M / ! 0 of NA .M / in mod A. Then we conclude that A .NA .M // Š Ker NA .p0 / D Im NA .p1 /, NA .p1 / W NA .P1 / ! Im NA .p1 / is a projective cover of Im NA .p1 / in mod A, and consequently we obtain that A2 .NA .M // D A .A .NA .M /// Š Ker NA .p1 /. This gives an exact sequence in mod A of the form NA .p1 /

NA .p0 /

0 ! A2 .NA .M // ! NA .P1 / ! NA .P0 / ! NA .M / ! 0: Comparing this with the exact sequence NA .p1 /

NA .p0 /

0 ! A M ! NA .P1 / ! NA .P0 / ! NA .M / ! 0; given by Proposition III.5.3, we obtain an isomorphism A M Š A2 .NA .M // in mod A. Passing from mod A to mod A, we conclude that there is a natural ! A2 NA from mod A to mod A. This finishes the isomorphism of functors A proof of (i). The equivalences in (ii) follow from the equivalences in (i) and the fact that A and A1 , A2 and A2 , NA and NA1 , are mutually inverse pairs of functors from mod A to mod A. Corollary 8.6. Let A be a finite dimensional symmetric K-algebra over a field K. Then the following statements hold. (i) NA Š 1mod A as functors from mod A to mod A. (ii) D Š HomA .; A/ as functors from mod A to mod Aop . (iii) A Š A2 and A1 Š A2 as functors from mod A to mod A. Proof. (i) Since A is a symmetric algebra, it follows from Theorem 2.2 that there is an isomorphism W A AA ! A D.A/A of A-bimodules. Further, by Proposition III.5.2, we know that the functors NA and ˝A D.A/ from mod A to mod A are naturally isomorphic. Then we obtain natural isomorphisms of functors from mod A to mod A NA Š ˝A D.A/ Š ˝A A Š 1mod A (see also Lemma II.3.5).

402


(ii) For the standard duality functor D D HomK .; K/, we have a natural isomorphism of functors DD Š 1mod A . Hence an equivalence of functors NA D D HomA .; A/ Š 1mod A established in (i), leads to a natural isomorphism of functors HomA .; A/ Š DD HomA .; A/ D DNA Š D form mod A to mod Aop . (iii) The existence of natural isomorphisms of functors A Š A2 and A1 Š 2 A follow from (i) and Theorem 8.5.

9 The higher extension spaces The aim of this section is to introduce the higher extension spaces of finite dimensional modules over finite dimensional algebras. Furthermore, the extension algebra of a finite dimensional module over a finite dimensional selfinjective algebra will be defined. Let A be a finite dimensional K-algebra over a field K and N , L be modules in mod A. Consider a minimal projective resolution dnC1

d1

dn

d0

! PnC1 ! Pn ! Pn1 ! ! P1 ! P0 ! N ! 0 of N in mod A (see Proposition I.8.30). Fix a positive integer n. Applying the contravariant functor HomA .; L/ W mod A ! mod K, we obtain the chain of K-vector spaces HomA .dnC1 ;L/

HomA .dn ;L/

HomA .Pn1 ; L/ ! HomA .Pn ; L/ ! HomA .PnC1 ; L/ with HomA .dnC1 ; L/ HomA .dn ; L/ D HomA .dn dnC1 ; N / D 0. This allows us to define the K-vector space ExtAn .N; L/ D Ker HomA .dnC1 ; L/= Im HomA .dn ; L/: We note that, by Proposition I.8.30 and Lemma I.8.31, the space ExtAn .N; L/ is well defined (does not depend on the choice of minimal projective resolution). The K-vector space ExtAn .N; L/ is called the n-th extension space of L by N . Let u W L ! U be a homomorphism in mod A. Then we have the commutative diagram in mod K, HomA .Pn1 ; L/

HomA .dn ;L/

HomA .Pn1 ;u/

/ HomA .Pn ; L/

HomA .dnC1 ;L/

HomA .Pn ;u/

/ HomA .PnC1 ; L/ HomA .PnC1 ;u/

Hom .d ;U / HomA .dn ;U / / HomA .Pn ; U / A nC1 / HomA .PnC1 ; U / , HomA .Pn1 ; U /

9. The higher extension spaces

403

from which we infer that HomA .Pn ; u/.Ker HomA .dnC1 ; L// Ker HomA .dnC1 ; U /; HomA .Pn ; u/.Im HomA .dn ; L// Im HomA .dn ; U /: Hence, we may define the K-linear homomorphism ExtAn .N; u/ W ExtAn .N; L/ ! ExtAn .N; U / by ExtAn .N; u/.' C Im HomA .dn ; L// D u' C Im HomA .dn ; U / for ' 2 HomA .Pn ; L/ with 'dnC1 D HomA .dnC1 ; L/.'/ D 0. Observe that u' D HomA .Pn ; u/.'/. Moreover, ExtAn .N; t u/ D ExtAn .N; t / ExtAn .N; u/ for u 2 HomA .L; U / and t 2 HomA .U; T /, and clearly ExtAn .N; idL / D idExtAn .N;L/ . Therefore, for any module N in mod A, we have the covariant functor ExtAn .N; / W mod A ! mod K: Let v W V ! N be a homomorphism in mod A. Consider also a minimal projective resolution dnC1

d1

dn

d0

! PnC1 ! Pn ! Pn1 ! ! P1 ! P0 ! V ! 0

of V in mod A. Then invoking the projectivity of the modules Pi , i 0, and the exactness of the projective resolution of N , we conclude that there exists a commutative diagram in mod A of the form :::

/ P nC1

dnC1

/ P n

vnC1

:::

/ PnC1

dn

/ P n1

/ Pn

/ P 1

/ :::

/ P1

vn1

vn

dnC1

/ :::

dn

/ Pn1

d1

v1

/ P 0

d0

/V

v0 d1

/ P0

/0 v

d0

/N

/ 0.

Then we obtain the commutative diagram in mod K, HomA .Pn1 ; L/

HomA .dn ;L/

HomA .vn1 ;L/

/ HomA .Pn ; L/

HomA .dnC1 ;L/

HomA .vn ;L/

/ HomA .PnC1 ; L/ HomA .vnC1 ;L/

HomA .dnC1 ;L/ HomA .dn ;L/ / HomA .P ; L/ / HomA .P ; L/ , HomA .Pn1 ; L/ nC1 n

from which we infer that HomA .vn ; L/.Ker HomA .dnC1 ; L// Ker HomA .dnC1 ; L/;

404


HomA .vn ; L/.Im HomA .dn ; L// Im HomA .dn ; L/: We may then define the K-linear homomorphism ExtAn .v; L/ W ExtAn .N; L/ ! ExtAn .V; L/ by

ExtAn .v; L/.' C Im HomA .dn ; L// D 'vn C Im HomA .dn ; L/

for ' 2 HomA .Pn ; L/ with 'dnC1 D HomA .dnC1 ; L/.'/ D 0. Observe that 'vn D HomA .vn ; L/.'/. Unfortunately, the homomorphisms vn W Pn ! Pn , n 0, occurring in the above commutative diagram of projective resolutions, are not uniquely determined by the homomorphism v W V ! N . Assume vn0 2 HomA .Pn ; Pn /, n 0, is another family of homomorphisms such that d0 v00 D vd0 0 and dn vn0 D vn1 dn for n 1. Then, invoking the projectivity of the modules Pi , i 0, and the exactness of the minimal projective resolutions of N and V , we conclude that there exist homomorphisms sn 2 HomA .Pn ; PnC1 /, n 0, such that v0 v00 D d1 s0 and vn vn0 D dnC1 sn C sn1 dn , for n 0. Then, for any homomorphism ' 2 Ker HomA .dnC1 ; L/, we obtain that 'vn 'vn0 D '.vn vn0 / D 'dnC1 sn C 'sn1 dn D 'sn1 dn D HomA .dn ; L/.'sn1 / with 'sn1 2 HomA .Pn1 ; L/. Therefore, we have 'vn C Im HomA .dn ; L/ D 'vn0 C Im HomA .dn ; L/: As a consequence, the homomorphism ExtAn .v; L/ does not depend on the choice of homomorphisms vn W Pn ! Pn , n 0. Moreover, we have ExtAn .vw; L/ D ExtAn .w; L/ ExtAn .v; L/ for v 2 HomA .V; N / and w 2 HomA .W; V /, and clearly ExtAn .idN ; L/ D idExtAn .N;L/ . Therefore, for any module L in mod A, we have the contravariant functor ExtAn .; L/ W mod A ! mod K: We note the following facts. Lemma 9.1. Let A be a finite dimensional K-algebra over a field K, N and L modules in mod A, and n a positive integer. Then the following statements hold. (i) If N is in proj A, then ExtAn .N; L/ D 0. (ii) If L is in inj A, then ExtAn .N; L/ D 0. Proof. Let dnC1

dn

d1

d0

! PnC1 ! Pn ! Pn1 ! ! P1 ! P0 ! N ! 0 be a minimal projective resolution of N in mod A.


405

(i) Assume N is in proj A. Then Pi D 0 for i 1, and we get ExtAn .N; L/ D 0. (ii) Assume L is in inj A. Then it follows from Proposition II.2.6 (ii) that the contravariant functor HomA .; L/ W mod A ! mod K is exact. In particular, we obtain the exact sequence of K-vector spaces HomA .dn ;L/

HomA .dnC1 ;L/

HomA .Pn1 ; L/ ! HomA .Pn ; L/ ! HomA .PnC1 ; L/; and consequently ExtAn .N; L/ D 0.

Corollary 9.2. Let A be a finite dimensional K-algebra over a field K, N and L modules in mod A, and n a positive integer. Then we have the covariant functor ExtAn .N; / W mod A ! mod K and the contravariant functor ExtAn .; L/ W mod A ! mod K: Proof. By Lemma 9.1, ExtAn .N; u/ D 0 for any u 2 IA .L; U / and ExtAn .v; L/ D 0 for any v 2 PA .V; N /. Then the covariant functor ExtAn .N; / W mod A ! mod K induces the covariant functor ExtAn .N; / W mod A ! mod K and the contravariant functor ExtAn .; L/ W mod A ! mod K induces the contravariant functor ExtAn .; L/ W mod A ! mod K. Consider now a minimal injective resolution d0

d1

dn

d nC1

0 ! L ! I0 ! I1 ! ! In1 ! In ! InC1 ! of L in mod A (see Proposition I.8.30). Fix a positive integer n. Applying the covariant functor HomA .N; / W mod A ! mod K, we obtain the chain of Kvector spaces HomA .N;d n /

HomA .N;d nC1 /

HomA .N; In1 / ! HomA .N; In / ! HomA .N; InC1 / with HomA .N; d nC1 / HomA .N; d n / D HomA .N; d nC1 d n / D 0. This allows us to define the K-vector space

e

ExtAn .N; L/ D Ker HomA .N; d nC1 /= Im HomA .N; d n /:

e

We note that, by Proposition I.8.30 and Lemma I.8.32, the space ExtAn .N; L/ is well defined (does not depend on the choice of minimal injective resolution). Let u W L ! U be a homomorphism in mod A. Consider also a minimal injective resolution Q0

Q1

Qn

Q nC1

d d d d 0 ! U ! IQ0 ! IQ1 ! ! IQn1 ! IQn ! IQnC1 !

406


of U in mod A. Then invoking the injectivity of the modules IQi , i 0, and the exactness of the injective resolution of L, we conclude that there exists in mod A a commutative diagram of the form 0

/L

0

/U

d0

d1

/ I0 u0

u dQ 0

/ IQ 0

/ I1

/ :::

u1

dQ 1

/ IQ 1

/ :::

/ In1

dn

un1

/ IQ n1

dQ n

/ In

d nC1/

InC1

un

/ IQ n

dQ nC1/

/ :::

unC1

IQnC1

/ ::::

Then we obtain the commutative diagram in mod K, HomA .N; In1 /

HomA .N;d n /

HomA .N;un1 /

HomA .N; IQn1 /

HomA .N;dQ n /

/ HomA .N; In /HomA .N;d

nC1 /

/ HomA .N; InC1 / HomA .N;unC1 /

HomA .N;un /

Q nC1 / / Hom .N; IQ /HomA .N;d / Hom .N; IQ / , n nC1 A A

from which we infer that HomA .N; un /.Ker HomA .N; d nC1 // Ker HomA .N; dQ nC1 /; HomA .N; un /.Im HomA .N; d n // Im HomA .N; dQ n /: We may then define the K-linear homomorphism

e

e

e

ExtAn .N; u/ W ExtAn .N; L/ ! ExtAn .N; U / by

e

ExtAn .N; u/.' C Im HomA .N; d n // D un ' C Im HomA .N; dQ n /

for ' 2 HomA .N; In / with d nC1 ' D HomA .N; d nC1 /.'/ D 0. Observe that un ' D HomA .N; un /.'/. Unfortunately, the homomorphisms un W In ! IQn , n 0, occurring in the above commutative diagram of injective resolutions, are not uniquely determined by the homomorphism u W L ! U . Assume u0n 2 HomA .In ; IQn /, n 0, is another family of homomorphisms such that u00 d 0 D dQ 0 u and u0n d n D dQ n u0n1 for n 1. Invoking the injectivity of the modules IQi , i 0, and the exactness of the minimal injective resolutions of L and U , we conclude that there exist homomorphisms tn 2 HomA .InC1 ; IQn /, n 0, such that u0 u00 D t0 d 1 and un u0n D tn d nC1 C dQ n tn1 for n 1. Then, for any homomorphism ' 2 Ker HomA .N; d nC1 /, we obtain that un ' u0n ' D .un u0n /' D tn d nC1 ' C dQ n tn1 ' D dQ n tn1 ' D HomA .N; dQ n /.tn1 '/ with tn1 ' 2 HomA .N; IQn1 /. Therefore, we have un ' C Im HomA .N; dQ n / D u0n ' C Im HomA .N; dQ n /:


e

407

As a consequence, the homomorphism ExtAn .N; u/ does not depend on the choice of homomorphisms un W In ! IQn , n 0. Moreover, we have ExtAn .N; t u/ D ExtAn .N; t/ExtAn .N; u/ for u 2 HomA .L; U / and t 2 HomA .U; T /, and clearly ExtAn .N; idL / D idExt e An .N;L/ . Therefore, for any module N in mod A, we have the covariant functor ExtAn .N; / W mod A ! mod K:

e e

e

e

e

Let v W V ! N be a homomorphism in mod A. Then we have the commutative diagram in mod K, HomA .N; In1 /

HomA .N;d n /

HomA .v;In1 /

HomA .V; In1 /

/ HomA .N; In /HomA .N;d

nC1 /

/ HomA .N; InC1 / HomA .v;InC1 /

HomA .v;In /

HomA

.V;d n /

nC1 / / HomA .V; In / HomA .V;d / HomA .V; InC1 / ,

from which we infer that HomA .v; In /.Ker HomA .N; d nC1 // Ker HomA .V; d nC1 /; HomA .v; In /.Im HomA .N; d n // Im HomA .V; d n /: Hence, we may define the K-linear homomorphism

e

e

e

ExtAn .v; L/ W ExtAn .N; L/ ! ExtAn .V; L/ by

e

ExtAn .v; L/.' C Im HomA .N; d n // D 'v C Im HomA .V; d n / for ' 2 HomA .N; In / with d nC1 ' D HomA .N; d nC1 /.'/ D 0. Observe that 'v D HomA .v; L/.'/. Moreover, we have ExtAn .vw; L/ D ExtAn .w; L/ExtAn .v; L/ for v 2 HomA .V; N /, w 2 HomA .W; V /, and clearly ExtAn .idN ; L/ D idExt e An .N;L/ . Therefore, for any module L in mod A, we have the contravariant functor

e

e

e

e

e

ExtAn .; L/ W mod A ! mod K: We note the following facts. Lemma 9.3. Let A be a finite dimensional K-algebra over a field K, N and L modules in mod A, and n a positive integer. Then the following statements hold.

e ext .N; L/ D 0. (ii) If L is in inj A, then E

(i) If N is in proj A, then ExtAn .N; L/ D 0. n A

408


Proof. Let d0

d1

dn

d nC1

0 ! L ! I0 ! I1 ! ! In1 ! In ! InC1 ! be a minimal injective resolution of L in mod A. (i) Assume N is in proj A. Then it follows from Proposition II.2.6 (i) that the covariant functor HomA .N; / W mod A ! mod K is exact. In particular, we obtain the exact sequence of K-vector spaces HomA .N;d n /

HomA .N;d nC1 /

HomA .N; In1 / ! HomA .N; In / ! HomA .N; InC1 /;

e

and consequently ExtAn .N; L/ D 0. (ii) Assume L is in inj A. Then Ii D 0 for i 1, and we obtain ExtAn .N; L/ D 0.

e

Corollary 9.4. Let A be a finite dimensional K-algebra over a field K, N and L modules in mod A, and n a positive integer. Then we have the covariant functor

e

ExtAn .N; / W mod A ! mod K and the contravariant functor

e ext .N; u/ D 0 for any u 2 I .L; U / and Eext .v; L/ D Proof. By Lemma 9.3, E ext .N; / W mod A ! 0 for any v 2 P .V; N /. Then the covariant functor E e mod K induces the covariant functor Ext .N; / W mod A ! mod K and the conext .; L/ W mod A ! mod K induces the contravariant functor travariant functor E ext .; L/ W mod A ! mod K. E ExtAn .; L/ W mod A ! mod K: n A

n A

A

n A

A

n A

n A

n A

The following result extends Corollary III.3.11 to the higher extension functors. Theorem 9.5. Let A be a finite dimensional K-algebra over a field K, L and N modules in mod A, and n a positive integer. The following statements hold.

e

(i) The covariant functors ExtAn .N; / and ExtAn .N; / from mod A to mod K are naturally isomorphic.

e

(ii) The contravariant functors ExtAn .; L/ and ExtAn .; L/ from mod A to mod K are naturally isomorphic. The usual proof of the above theorem is tedious and requires consideration of bicomplexes involving simultaneously projective resolutions of modules on the contravariant place and injective resolutions of modules on the covariant place of


409

the hom functors (for a proof we refer to the book [CE]). For selfinjective algebras, there is a much simpler proof invoking the syzygy functors, which we will present below. Let A be a finite dimensional selfinjective K-algebra over a field K. Then proj A D inj A and mod A D mod A. Then, for any modules N and L in mod A, and a positive integer n, we have, by Corollaries 9.2 and 9.4, the covariant functors

e

ExtAn .N; /; ExtAn .N; / W mod A ! mod K and the contravariant functors

e

ExtAn .; L/; ExtAn .; L/ W mod A ! mod K: Theorem 9.6. Let A be a finite dimensional selfinjective K-algebra over a field K and n be a positive integer. For any modules N and L in mod A, there exist K-linear isomorphisms N;L W HomA An .N /; L ! ExtAn .N; L/ such that the following statements hold. (i) For any homomorphism u W L ! U in mod A the diagram in mod K, HomA An .N /; L

N;L

n .N /;u HomA .A / n N;U HomA A .N /; U

/ Ext n .N; L/ A n .N;u/ ExtA

/ Ext n .N; U / , A

is commutative. (ii) For any homomorphism v W V ! N in mod A the diagram in mod K, HomA An .N /; L

N;L

A

n .v/;L HomA .A /

HomA An .V /; L

/ Ext n .N; L/

V;L

n .v;L/ ExtA

/ Ext n .V; L/ , A

is commutative. Proof. Let L and N be modules in mod A. Consider a minimal projective resolution dnC1

dn

dn1

d1

d0

! PnC1 ! Pn ! Pn1 ! ! P1 ! P0 ! N ! 0

410


of N in mod A. Then we have Im dn D Ker dn1 D An .N / and Im dn Š Pn = Ker dn D Pn = Im dnC1 D Coker dnC1 . Let n W Pn ! Im dn be the canonical epimorphism induced by dn and in W An .N / ! Pn1 the canonical embedding. Thus dn D in n . Furthermore, in n dnC1 D dn dnC1 D 0, and hence n dnC1 D 0, because in is a monomorphism. Take a homomorphism f 2 HomA .An .N /; L/. Then 'f D f n 2 HomA .Pn ; L/ and HomA .dnC1 ; L/.'f / D 'f dnC1 D f n dnC1 D 0. Hence we may assign to f the element 'f C Im HomA .dn ; L/ 2 ExtAn .N; L/. Suppose now that g 2 HomA .An .N /; L/ and f g 2 PA .An .N /; L/, say f g D t r for some r 2 HomA .An .N /; P / and t 2 HomA .P; L/ with P in proj A. Then we obtain a commutative diagram in mod A, in

/ Pn1 || || | r || || s | ~|| P,

An .N /

because P is in proj A D inj A. Then we obtain that 'f 'g D f n gn D .f g/n D t rn D t sin n D t sdn D HomA .dn ; L/.t s/ with ts 2 HomA .Pn1 ; L/. This shows that 'f 'g 2 Im HomA .dn ; L/, and so 'f C Im HomA .dn ; L/ D 'g C Im HomA .dn ; L/. Therefore, we obtain the K-linear homomorphism N;L W HomA An .N /; L ! ExtAn .N; L/ such that

N;L f C PA .An .N /; L D 'f C Im HomA .dn ; L/

for any f 2 HomA .An .N /; L/. We claim that N;L is an isomorphism. Assume f 2 HomA .An .N /; L/ and 'f 2 Im HomA .dn ; L/. Then 'f D HomA .dn ; L/. / D dn for some 2 HomA .Pn1 ; L/. But then we have f n D 'f D dn D in n , which implies f D in , because n is an epimorphism. Therefore, f 2 PA .An .N /; L/. This shows that N;L is a monomorphism. Assume now that a homomorphism ' 2 HomA .Pn ; L/ has the property 'dnC1 D HomA .dnC1 ; L/.'/ D 0. Since n induces an isomorphism Pn = Im dnC1 ! An .N /, we infer that there exists a homomorphism f 2 HomA .An .N /; L/ such that ' D f n D 'f . Hence ' C Im HomA .dn ; L/ D 'f C Im HomA .dn ; L/. This shows that N;L is also an epimorphism. (i) Let u W L ! U be a homomorphism in mod A. Then we have, for any


411

f 2 HomA .An .N /; L/, the equalities n ExtA .N; u/N;L f D ExtAn .N; u/ 'f C Im HomA .dn ; L/ D u'f C Im HomA .dn ; L/ D uf n C Im HomA .dn ; L/ D 'uf C Im HomA .dn ; L/ D N;U uf D N;U HomA .An .N /; u/ f ; where f D f C PA .An .N /; L/, u D u C PA .L; U /, uf D uf C PA .An .N /; L/. Therefore, the required equality ExtAn .N; u/N;L D N;U HomA An .N /; u holds. (ii) Let v W V ! N be a homomorphism in mod A. Let dnC1

dn1

dn

d1

d0

! PnC1 ! Pn ! Pn1 ! ! P1 ! P0 ! V ! 0

be a minimal projective resolution of V in mod A. Then there exists a commutative diagram in mod A of the form :::

/ P nC1

dnC1

vnC1

:::

/ PnC1

dnC1

/ P n

dn

/ P n1

/ :::

/ P 1

/ Pn1 dn1 / : : :

/ P1

vn1

vn

/ Pn

dn1

dn

d1

v1

/ P 0

d0

v0 d1

/ P0

/V

/0 v

d0

/N

/ 0.

Hence we obtain also the commutative diagram of the form Pn1 FF : v FF n in vv FF v FF vv v F# vv vn1 vn An .V /

Pn

wn Pn G P n1 : GG u GGn in uuu GG u u GG u # uu n A .N / , where Im dn D Ker dn1 D An .V /, n W Pn ! An .V / is the canonical epi is the canonical embedding, and morphism induced by dn , in W An .V / ! Pn1

412


wn D wn C PA .An .V /; An .N // D An .v C PA .V; N // D An .v/. Clearly, we have dn D in n . Then we obtain, for any f 2 HomA .An .N /; L/, the equalities n ExtA .v; L/N;L f D ExtAn .v; L/ 'f C Im HomA .dn ; L/ D 'f vn C Im HomA .dn ; L/ D f n vn C Im HomA .dn ; L/ D f wn n C Im HomA .dn ; L/ D V;L f wn D V;L f wn D V;L f An .v/ D V;L HomA .An .v/; L/ f : Therefore, the required equality

ExtAn .v; L/N;L D V;L HomA An .v/; L

holds. Dually, we have the following theorem, and we leave its proof to the reader.

Theorem 9.7. Let A be a finite dimensional selfinjective K-algebra over a field K and n be a positive integer. For any modules N and L in mod A there exist K-linear isomorphisms QN;L W HomA N; An .L/ ! ExtAn .N; L/

e

such that the following statements hold. (i) For any homomorphism u W L ! U in mod A the diagram in mod K, HomA N; An .L/

QN;L

A

n .u/ HomA N;A

.

HomA N; An .U /

e

/ Ext n .N; L/

/

QN;U

e An .N;u/ Ext / Ext n .N; U / , A

e

is commutative. (ii) For any homomorphism v W V ! N in mod A the diagram in mod K, HomA N; An .L/

QN;L

n .L/ HomA .v;A / QV;L HomA V; An .L/

is commutative.

e

/ Ext n .N; L/ A

e An .v;L/ Ext / Ext n .V; L/ , A

e


413

Theorem 9.8. Let A be a finite dimensional selfinjective K-algebra over a field K and n be a positive integer. For any modules N and L in mod A, the n-th syzygy functor An W mod A ! mod A induces isomorphisms of K-vector spaces !N;L W HomA N; An .L/ ! HomA An .N /; L : Moreover, the following statements hold. (i) For any homomorphism u W L ! U in mod A the diagram in mod K, HomA N; An .L/

!N;L

n .u/ HomA .N;A / !N;U HomA N; An .U /

/ Hom n .N /; L A A n .N /;u HomA .A / / Hom n .N /; U , A A

is commutative. (ii) For any homomorphism v W V ! N in mod A the diagram in mod K, HomA N; An .L/

!N;L

/ Hom n .N /; L A A

n .L/ HomA .v;A /

HomA V; An .L/

!V;L

n .v/;L HomA .A /

/ Hom

A

n A .V /; L ,

is commutative. Proof. It follows from Theorem 8.4 that An and An are mutually inverse equivalences from mod A to mod A. In particular, there is a natural isomorphism of functors ' n W An B An ! 1mod A : Then for modules N and L in mod A, the required isomorphism of K-vector spaces !N;L W HomA N; An .L/ ! HomA An .N /; L is given for f 2 HomA N; An .L/ by !N;L f D 'Ln An f : A simple checking shows that the diagrams in (i) and (ii) are commutative.

Combining Theorems 9.6, 9.7 and 9.8, we obtain the following theorem. Theorem 9.9. Let A be a finite dimensional selfinjective K-algebra over a field K, L and N be modules in mod A, and n a positive integer. The following statements hold.

414


e

(i) The covariant functors ExtAn .N; /, ExtAn .N; / and HomA An .N /; from mod A to mod K are naturally isomorphic. (ii) The contravariant functors ExtAn .; L/, ExtAn .; L/ and HomA ; An .L/ from mod A to mod K are naturally isomorphic.

e

Let A be a finite dimensional selfinjective K-algebra over a field K and M be a module in mod A. The Ext-algebra of M is the K-algebra ExtA .M; M / such that ExtA .M; M / D

1 M

HomA Ar .M /; M

rD0

as K-vector space and the multiplication of f 2 HomA Ar .M /; M and g 2 s HomA A .M /; M is given by f g D f B Ar .g/ 2 HomA ArCs .M /; M : Hence ExtA .M; M / is a graded K-algebra, usually of infinite dimension over K.

10 Periodic modules In this section we introduce as well as describe properties of periodic modules with respect to actions of the syzygy and Auslander–Reiten operators. Let A be a finite dimensional K-algebra over a field K. A module M in mod A is said to be A -periodic (briefly, periodic) if An .M / Š M in mod A for some n 1. The smallest positive integer d with Ad .M / Š M is called the period of M . More generally, the module category mod A is said to be periodic if all modules in modP A are periodic. Observe that mod A is periodic if and only if every indecomposable nonprojective module in mod A is periodic (see Proposition 8.1 (iii)). We will show in Section 12 (Corollary 12.3) that the periodicity of all simple modules in mod A forces A to be a selfinjective algebra. In particular, we obtain that if a module category mod A is periodic then A is a selfinjective algebra. The following proposition exhibits a large class of algebras with periodic module categories. Proposition 10.1. Let A be a finite dimensional selfinjective K-algebra of finite representation type over a field K. Then the module category mod A is periodic. Proof. Let M be an indecomposable nonprojective module in mod A. Applying Proposition 8.3, we obtain the family An .M /, n 1, of indecomposable nonprojective modules in mod A. Since A is of finite representation type, there exists positive integers r > s such that Ar .M / Š As .M / in mod A. Applying Proposition 8.3 again, we conclude that Ars .M / Š M , and hence M is a periodic module. This shows that mod A is periodic.

10. Periodic modules

415

Let A be a finite dimensional K-algebra over a field K. A module M in mod A is said to be A -periodic if An .M / Š M for some n 1. More generally, the module category mod A is said to be A -periodic if all modules in modP A are A periodic. We note that mod A is A -periodic if and only if every indecomposable nonprojective module in mod A is A -periodic (see Corollary III.4.9). Proposition 10.2. Let A be a finite dimensional selfinjective K-algebra of finite representation type over a field K. Then the module category mod A is A -periodic. Proof. Let M be an indecomposable nonprojective module in mod A. As proj A D inj A, applying Corollary III.4.9, we obtain the family An M , n 1, of indecomposable nonprojective modules in mod A. Then the assumption that A is of finite representation type forces that Ar M Š As M for some positive integers r > s. Applying Corollary III.4.9 again, we conclude that Ars M Š M , and hence M is A -periodic. This shows that mod A is A -periodic. We have also the following fact. Proposition 10.3. Let A be a finite dimensional K-algebra over a field K such that mod A is A -periodic. Then A is a selfinjective algebra. Proof. Assume that A is not a selfinjective algebra. Since the number of pairwise nonisomorphic indecomposable projective modules in mod A is the same as the number of pairwise nonisomorphic indecomposable injective modules in mod A (Corollaries I.8.6 and I.8.21), we conclude that there exists in mod A an indecomposable injective but not projective module I . Then I belongs to modP A, and so is A -periodic, say I Š Am I for a positive integer m. But then I Š A .Am1 I /, which gives a contradiction, because A .Am1 I / is an indecomposable noninjective module in mod A, by Corollary III.4.9. Therefore, indeed A is a selfinjective algebra. We obtain the following direct consequence of Propositions 10.2 and 10.3. Corollary 10.4. Let A be a finite dimensional K-algebra of finite representation type over a field K. The following conditions are equivalent. (i) A is a selfinjective algebra. (ii) The module category mod A is A -periodic. Let A be a finite dimensional selfinjective K-algebra over a field K. Then it follows from the Morita–Azumaya duality Theorem II.7.11 that the Nakayama functor NA D D HomA .; A/ W mod A ! mod A is a selfequivalence of the category mod A. We say that NA has finite order if there is a positive integer m such that the functors NAm and 1mod A from mod A to mod A are naturally isomorphic.

416


Proposition 10.5. Let A be a finite dimensional selfinjective K-algebra over a field K such that the Nakayama functor NA W mod A ! mod A has finite order, and M be a module in mod A. Then the following equivalences hold. (i) M is periodic if and only if M is A -periodic. (ii) mod A is periodic if and only if mod A is A -periodic. Proof. Let m be a positive integer such that the functors NAm and 1mod A are naturally isomorphic. Then the functors NAm and 1mod A from mod A to mod A are also naturally isomorphic. Further, it follows from Theorem 8.5 that the functors A , A2 NA and NA A2 from mod A to mod A are naturally isomorphic. Then we infer that the functors Am and A2m from mod A to mod A are naturally isomorphic. (i) Observe that if M is periodic (respectively, A -periodic) then M belongs to modP A. Assume Ad .M / Š M in mod A for some positive integer d . Then Ad .M / Š M in mod A, and hence we obtain that Amd M Š A2md .M / Š M in mod A. Since Amd M and M are in modP A D modI A, we conclude, by Lemma III.4.3, that Amd M Š M in mod A. In a similar way, we show that if Ar M Š M in mod A, for a positive integer r, then A2mr .M / Š M in mod A. Therefore, the equivalence (i) holds. Clearly, the equivalence (ii) follows from (i). As a direct consequence of Corollary 8.6 and Proposition 10.5 we obtain the following facts. Corollary 10.6. Let A be a finite dimensional symmetric K-algebra over a field K, and M be a module in mod A. Then the following equivalences hold. (i) M is periodic if and only if M is A -periodic. (ii) mod A is periodic if and only if mod A is A -periodic. Let A be a finite dimensional Frobenius K-algebra over a field K and A a Nakayama automorphism of A, associated to a nondegenerated associative K-bilinear form .; /A W A A ! K. It follows from Proposition 3.13 that the functors NA D D HomA .; A/ and NA0 D ./ 1 from mod A to mod A are A naturally isomorphic. In particular, we obtain that if A is of finite order (as an automorphism of A) then the Nakayama functor NA has finite order. We will show in Chapter VI that all finite dimensional Hopf algebras over a field K are Frobenius algebras with Nakayama automorphisms of finite order (see Theorem VI.5.2). On the other hand, the following example shows that there are finite dimensional Frobenius algebras A over a field for which the Nakayama automorphism A (respectively, the Nakayama functor NA ) are of infinite order, and the A -periodicity and A -periodicity of modules does not coincide.


417

Example 10.7. Let K be a field and 2 K n f1; 0; 1g. Consider the local Kalgebra A D KhX; Y i=.X 2 ; Y 2 ; X Y YX /; and denote by x and y the cosets of X and Y in A . We know from Example 2.8 that A is a 4-dimensional nonsymmetric Frobenius K-algebra and 1 D 1A , x, y, xy D yx form a basis of A over K. For elements a; b 2 K n f0g, consider the right A -modules M.a; b/ D A =.ax C by/A ; where .ax C by/A is the right ideal of A generated by ax C by. Observe that dimK .ax C by/A D 2 and the elements ax C by and xy D yx form a basis of .axCby/A over K, because .axCby/x D byx and .axCby/y D axy D ayx. Then M.a; b/ is an indecomposable right A -module with dimK M.a; b/ D 2, simple top.M.a; b// and simple soc.M.a; b//. Clearly, every simple module in mod A is isomorphic to A = rad A , which is the K-vector space K with the zero actions of x and y. Then we have in mod A the exact sequence 0 ! A .M.a; b//!PA .M.a; b//!M.a; b/ ! 0; where PA .M.a; b// D A and A .M.a; b// D .ax C by/A . Moreover, .axCby/A Š M.a; b/ in mod A , because we have .axCby/.axCby/ D 0. Therefore, we obtain A .M.a; b// Š M.a; b/ in mod A . Hence, for any positive integer m, we have that Am .M.a; b// Š M ../m a; b/ : Moreover, for a; b; c; d 2 K n f0g, we have M.a; b/ Š M.c; d / in mod A if and only if there is a nonzero element 2 K such that c D a and d D b, or equivalently .a; b/ and .c; d / give the same point .a W b/ D .c W d / on the projective line P 1 .K/. Therefore, we conclude that a module M.a; b/ is periodic (A -periodic) if and only if 2m D 1 in K. Summing up, we conclude that the following conditions are equivalent: (1) is an n-th root of unity in K, for some n 1. (2) M.a; b/ is periodic in mod A , for some a; b 2 K n f0g. (3) M.a; b/ is periodic in mod A , for any pair a; b 2 K n f0g. We will show now that, for any 2 K n f1; 0; 1g and pair a; b 2 K n f0g, we have M.a; b/ Š A M.a; b/

418


in mod A , and hence M.a; b/ is A -periodic in mod A . Fix elements a; b 2 K n f0g. It follows from the above discussion that M.a; b/ admits a minimal projective presentation in mod A of the form p1

p0

P1 ! P0 ! M.a; b/ ! 0; where P1 D P0 D A , p0 is the canonical epimorphism of right A -modules A ! A =.ax Cby/A and p1 is the composed epimorphism of right A -modules p q A .ax C by/A , with q an isomorphism. Applying ! A =.ax C by/A ! the Nakayama functor NA D D HomA .; A / W mod A ! mod A , we obtain, by Proposition III.5.3, an exact sequence in mod A of the form NA .p1 /

NA .p0 /

0 ! A M.a; b/ ! NA .P1 / ! NA .P0 / ! NA .M.a; b// ! 0; and so A .M.a; b// Š Ker NA .p1 / in mod A . Since NA .p1 / D NA .qp/ D NA .q/NA .p/ with NA .q/ an isomorphism, we conclude that A M.a; b/ Š Ker NA .p1 / Š Ker NA .p/, where p W A ! M.a; b/ is the canonical projective cover of M.a; b/ in mod A . We will show that Ker NA .p/ Š M.a; b/ in mod A , and consequently A M.a; b/ Š M.a; b/. We calculate first the left A -module HomA .M.a; b/; A /. Observe that we have in mod A the exact sequence p

u

A ! A ! M.a; b/ ! 0 such that u.1A / D ax C by. This leads to the following commutative diagram op in mod A with exact rows: 0

/ Ker v

i

/ A

v

'

0

/ A '

Hom .p; A / Hom .u; A / / HomA .M.a; b/; A /A / HomA .A ; A / A / HomA .A ; A / ,

where ' W A ! HomA .A ; A / is an isomorphism of left A -modules such that '.c/.d / D cd for c; d 2 A (see Lemma I.6.1), and i W Ker v ! A is the canonical embedding. Observe that, for c 2 A , we have v.c/ D '.v.c// 1A D HomA .u; A /'.c/ 1A D .'.c/u/ 1A D c.ax C by/: Since .ax C by/.ax C by/ D 0, we conclude that Ker v D A .ax C by/. Moreover, is an isomorphism of left A -modules, and so we have an isomorphism HomA .M.a; b/; A / Š A .ax C by/


419

op

in mod A . We calculate now the right A -module D.A .ax C by//. Consider the canonical embedding w W A .ax C by/ ! A of left A -modules. This gives the epimorphism D.w/ W D.A / ! D.A .ax C by// of right A -modules. Let ' W A ! K be the K-linear homomorphism such that ' .1A / D 0, ' .x/ D 0, ' .y/ D 0 and ' .xy/ D 1. We know from Example 2.8 that Ker ' does not contain nonzero right ideals of A . Then it follows from the proof of Theorem 2.1 that the K-linear map W A ! D.A / such that .m/.n/ D ' .mn/ for m; n 2 A is an isomorphism of right A -modules. Hence, D D.w/ W A ! D .A .ax C by// is an epimorphism of right A -modules. Observe that dimK Ker

D dimK A dimK D.A .ax C by// D 4 2 D 2:

Moreover, .m/ D D.w/. .m// D .m/w for any m 2 A . Then we obtain the equalities .ax

by/ .A .ax C by// D .ax by/w .A .ax C by// D ' ..ax by/A .ax C by// D ' ..ax by/K.ax C by// D K' ..ax by/.ax C by// D 0;

and hence ax by 2 Ker . This shows that Ker D .ax by/A . This leads to an isomorphism M.a; b/ D A =.ax by/A Š D.A .ax C by// in mod A . Therefore, we obtain a commutative diagram in mod A with exact rows 0

/ A .M.a; b//

0

/ Ker NA .p/

j

/ A

h

/ M.a; b/

NA .p/

/ NA .M.a; b//

%

/ NA .A /

/0

with NA .A / D D.A / and the vertical homomorphisms being isomorphisms. Summing up, we have in mod A isomorphisms A M.a; b/ Š Ker NA .p/ Š A .M.a; b// Š M.a; b/ D M.a; b/: In particular, for any pair a; b 2 K n f0g, M.a; b/ is an indecomposable A -periodic module in mod A of A -period 1. Take now an element 2 K n f1; 0; 1g which is not n-th root of unity, for any natural number n 2. Observe that for K D C we have many such elements. Then, from earlier observation, for a; b 2 K n f0g, M.a; b/ Š A .M../m a; b// © Am .M.a; b// for any positive integer m, that is, M.a; b/ is not A -periodic. Since A is a Frobenius algebra, there exists a nondegenerate associative K-bilinear form .; / W A A ! K, and the associated Nakayama automorphism D A of A . We claim that is an automorphism of A of infinite order. Indeed,

420


by Proposition 3.13, the functors NA and NA0 D ./ 1 from mod A to mod A are naturally isomorphic. Hence, if is an automorphism of finite order, then the Nakayama functor NA W mod A ! mod A has finite order, and consequently, by Proposition 10.5, an indecomposable module M in mod A is A -periodic if and only if M is A -periodic. Since the modules M.a; b/, a; b 2 K n f0g, are A -periodic but not A -periodic, we conclude that the Nakayama automorphism D A of A is of infinite order. The following characterization of selfinjective Nakayama algebras shows that sometimes we may recover a finite dimensional selfinjective algebra A and its module category mod A from the action of the syzygy operator A (respectively, the Auslander–Reiten translation A ) on the simple modules in mod A. Theorem 10.8. Let A be an indecomposable finite dimensional selfinjective Kalgebra over a field K. The following conditions are equivalent. (i) A2 .S/ is simple for any simple module S in mod A. (ii) A S is simple for any simple module S in mod A. (iii) A is a Nakayama algebra with ``.A/ 2. Proof. Since A is a selfinjective algebra, we conclude that the Nakayama functors NA D D HomA .; A/ and NA1 D HomAop .; A/D from mod A to mod A are exact. Hence it follows that, for any simple module S in mod A, the modules NA .S/ and NA1 .S / are also simple. We also note that if one of the conditions (i), (ii), or (iii) holds, then every simple module in mod A is not projective (equivalently, not injective). Observe first that the implication (iii) ) (ii) follows from the description of almost split sequences over Nakayama algebras given in Theorem III.8.7. Indeed, for a simple module S and its projective cover P ! S in mod A, we have S D P = rad P and an almost split sequence of the form 0 ! rad P = rad2 P ! P = rad2 P ! P = rad P ! 0: Hence A S D rad P = rad2 P is simple, because P is a uniserial module. Assume (ii) holds. Let S be a simple module in mod A. Then T D NA1 .S / is a simple module in mod A and S Š NA .T /. Further, by (ii), A T is a simple module in mod A. Applying now Theorem 8.5, we conclude that A2 .S / Š A2 .NT .T // Š A T in mod A. Since the modules A2 .S / and A T belong to modP A, using Lemma III.4.3, we infer that A2 .S / Š A T in mod A. Therefore, A2 .S / is a simple module in mod A. This shows that (ii) implies (i). Assume (i) holds. Let S be a simple module in mod A. Then we have an exact sequence in mod A of the form i

g

h

0 ! A2 .S / ! PA .rad P / ! P ! S ! 0;


421

h

S is a projective cover of S in mod A, rad P D Ker h D Im g D where P ! A .S/. It follows from (i) that A2 .S / is a simple module in mod A. Further, by Proposition 8.3, we have A2 .A2 .S // Š S and i W A2 .S / ! PA .rad P / is an injective envelope of A2 .S / in mod A. Since A2 .S / is simple, we conclude that PA .rad P / is an indecomposable projective module, and consequently rad P = rad2 P D top.rad P / is a simple module. This show that the radical rad P of any indecomposable projective module P in mod A has simple top. Take now an arbitrary nonsimple module M in mod A with top.M / a simple module. Then we obtain a commutative diagram in mod A of the form 0

/ rad PA .M /

0

/ rad M

i

/ PA .M /

j

/M

g

u

/ top.PA .M //

v

/ top.M /

/0

top.h/

h

/0

with rad M ¤ 0, where h W PA .M / ! M is a projective cover of M in mod A. Moreover, top.h/ is an isomorphism and PA .M / is an indecomposable projective module, because top.M / is simple. We claim that g is an epimorphism. Take an element x 2 rad.M /. Since h is an epimorphism, there exists an element y 2 PA .M / such that j.x/ D h.y/. Then we obtain that top.h/u.y/ D vh.y/ D vj.x/ D 0. Hence u.y/ D 0, because top.h/ is an isomorphism. Thus Ker u D Im i implies that y D i.z/ for some element z 2 rad PA .M /. But then we obtain that j.x/ D h.y/ D h.i.z// D j.g.z//, and hence x D g.z/, since j is a monomorphism. Therefore, indeed g is an epimorphism. Applying now Lemma I.5.18, we obtain that top.g/ W top.rad PA .M // ! top.rad M / is an epimorphism. Since, by the first part of the proof, top.rad PA .M // is simple, we conclude that top.rad M / is also simple. Summing up, we infer that, for any indecomposable projective module P in mod A, the radical series P rad P rad2 P radm1 P radm P D 0 of P is a composition series of P . This shows that P is a uniserial right A-module, by Proposition I.10.1. Since proj A D inj A, it follows from Proposition I.8.16, that every indecomposable projective left A-module Q is also uniserial. This shows that A is a Nakayama algebra. Moreover, by the assumption (i), we have ``.A/ 2. Therefore, (i) implies (iii). Corollary 10.9. Let A be an indecomposable finite dimensional selfinjective Kalgebra over a field K. Then the following conditions are equivalent. (i) A .S/ is simple for any simple module S in mod A. (ii) A is a Nakayama algebra with ``.A/ D 2.

422


Proof. Assume (i) holds. Then A2 .S / D A .A .S // is simple for any simple module S in mod A. Hence it follows from Theorem 10.8 that A is a Nakayama algebra with ``.A/ 2. On the other hand, for any indecomposable projective module P in mod A we have the canonical exact sequence 0 ! rad P ! P ! P = rad P ! 0: Since top.P / D P = rad P is simple and rad P D A .top.P //, we conclude that rad P is a simple module. Clearly, then .rad P /.rad A/ D rad2 P D 0. This shows that .rad A/2 D 0, and hence ``.A/ D 2, because rad A ¤ 0. Thus, (i) implies (ii). Assume (ii) holds. Then, for any indecomposable projective module P in mod A, the radical series of P is a unique composition series of P , and is of the form P rad P rad2 P D 0. In particular, we have rad P D soc.P / and is simple, because P is an indecomposable injective module in mod A. Then, for any simple module S in mod A, A .S / D rad.PA .S // is a simple module in mod A, because PA .S / is an indecomposable projective module. This shows that (ii) implies (i). The following result proved by F. G. Carlson in [Car], called the Carlson theorem, provides an interesting property of indecomposable periodic modules over finite dimensional selfinjective algebras. Theorem 10.10. Let A be a finite dimensional selfinjective K-algebra over a field K and M be an indecomposable periodic module in mod A of period d . Moreover, let N.M / be the ideal of the algebra ExtA .M; M / generated by all nilpotent homogeneous elements. Then there is an isomorphism of graded K-algebras ExtA .M; M /=N.M / Š F Œx; where F Œx is the graded K-algebra of polynomials in one variable x of degree d over the finite dimensional division K-algebra F D EndA .M /= rad EndA .M /. Proof. We identify (see Theorems 9.6 and 9.8) HomA .Ai .M /; M / D ExtAi .M; M / D HomA .M; Ai .M //; for any i 1. Let f 2 HomA .As .M /; M / be a homogeneous nilpotent element of ExtA .M; M / and g 2 HomA .Am .M /; M / an arbitrary homogeneous element of ExtA .M; M /. We claim that f g D f As .g/ 2 HomA .AmCs .M /; M / is again a nilpotent element of ExtA .M; M /. Choose r such that r.m C s/ D qd , for some q 1, and consider the element h D .f As .g//r in ExtA .M; M /. Then h 2 HomA .Aqd .M /; M / Š HomA .M; M /;


423

because Aqd .M / Š M . Suppose h is an isomorphism. Then f W As .M / ! M is a retraction in mod A, and hence f f 0 D idM for some f 0 2 HomA .M; As .M //. But then .f /n .f 0 /n D idM for any n 1, and hence f is not nilpotent in ExtA .M; M /, a contradiction. Therefore, h belongs to the radical of the local algebra EndA .M /, and hence h is nilpotent. Then Aid .h/ 2 EndA .M / are nilpotent elements for all i 0, and hence belong to the radical of EndA .M /. Since .rad EndA .M //l D 0 for some l 1, we get hl D 0 in ExtA .M; M /. But then f g D f As .g/ is a nilpotent element in ExtA .M; M /. Similarly, using ExtAi .M; M / D HomA .M; Ai .M //; i 1; we prove that g f is nilpotent in ExtA .M; M /. Let s ¤ pd , for all p 1. We show that any element f 2 HomA .As .M /; M / is a nilpotent element of ExtA .M; M /. Choose r 1 such that rs D qd , for some q 1, and take h D f r in ExtA .M; M /. Since d is the period of M and s is not divisible by d , we conclude that f is not an isomorphism. Then h is not an isomorphism, hence h 2 EndA .M / is nilpotent. Therefore, h is a nilpotent element in ExtA .M; M /, and so f is nilpotent in ExtA .M; M /. Let x 2 HomA .Ad .M /; M / Š HomA .M; M / correspond to the residue class idM of the identity map idM from M to M . Observe that x is not nilpotent in ExtA .M; M /. We claim thatPx n … N.M / for any n 1. Suppose that x t 2 N.M / for some t 1. Then x t D gi fi hi , where fi are homogeneous nilpotent elements of ExtA .M; M / and gi , hi are elements of ExtA .M; M /. We may assume that the elements gi , hi are also homogeneous. It follows from the first part of the proof that gi fi hi D .gi fi /hi arePnilpotent elements in ExtA .M; M /, and hence are nilpotent in EndA .M /. But then gi fi hi is nilpotent in EndA .M /, and hence in ExtA .M; M /. This implies that x t , and hence x, is nilpotent A .M; M /, a L1 in Ext p contradiction. We conclude that ExtA .M; M /=N.M / Š pD0 F x D F Œx as graded K-algebras, with x of degree d , and F D EndA .M /= rad EndA .M /. We will present now a method allowing us to construct periodic modules over trivial extension algebras T.A/ D A Ë D.A/ of algebras A which admit A -periodic modules of projective dimension one. Let A be a finite dimensional K-algebra over a field K. We denote by T.mod A/ D mod A Ë . ˝ D.A// the category whose objects are pairs .X; ˛/, where X is a module in mod A and ˛ W X ˝A D.A/ ! X is a homomorphism in mod A with ˛.˛ ˝D.A// D 0. Moreover, if .X; ˛/ and .Y; ˇ/ are objects of T.mod A/ then a morphism u W .X; ˛/ ! .Y; ˇ/ in T.mod A/ is a homomorphism u W X ! Y in mod A such that the diagram

424


of homomorphisms in mod A, X ˝A D.A/

˛

/X

ˇ

/Y,

u

u˝D.A/

Y ˝A D.A/

is commutative. Recall also (Example 2.7) that the trivial extension algebra T.A/ D AËD.A/ of A by the A-bimodule D.A/ D HomK .A; K/ is a symmetric K-algebra whose underlying K-vector space is A ˚ D.A/ and the multiplication in T.A/ is given by .a; f /.b; g/ D .ab; ag C f b/ for a; b 2 A and f; g 2 D.A/. Lemma 10.11. Let A be a finite dimensional K-algebra over a field K. Then there is a canonical equivalence of categories mod T.A/ ! T.mod A/:

Proof. Let X be a module in mod T.A/. Since T.A/ D A ˚ D.A/, then X is a module in mod A by xa D x.a; 0/ for any x 2 X and a 2 A. Moreover, we have the A-bilinear map X D.A/ ! X given by xf D x.0; f / for any x 2 X and f 2 D.A/. Then it follows from Proposition II.3.1 that we have the induced K-linear homomorphism ˛ D ˛X W X ˝A D.A/ ! X , which is clearly a homomorphism in mod A. Further, for x 2 X and f; g 2 D.A/, we have ˛ .˛ ˝ D.A// .x ˝ f ˝ g/ D ˛.˛.x ˝ f / ˝ g/ D .x.0; f //.0; g/ D x..0; f /.0; g// D x.0; 0/ D 0: Hence ˛.˛ ˝ D.A// D 0. Observe that then x.a; f / D x..a; 0/ C .0; f // D x.a; 0/ C x.0; f / D xa C ˛.x ˝ f / for x 2 X , a 2 A, f 2 D.A/. Conversely, if .X; ˛/ is an object of T.mod A/ then we have the canonical right T.A/-module structure given for x 2 X , a 2 A, f 2 D.A/ by x.a; f / D xa C ˛.x ˝ f /; which is well defined because ˛.˛ ˝ D.A// D 0. Finally, we conclude that u W .X; ˛X / ! .Y; ˛Y / is a homomorphism in mod T.A/ if and only if u W X ! Y is a homomorphism in mod A, with the induced (restricted) A-module structures on X and Y , and ˛Y .u ˝ D.A// D u˛X . The canonical equivalence of categories established above allows us to identify mod T.A/ and T.mod A/. Observe that A is the quotient algebra of T.A/ by the two-sided ideal 0 ˚ D.A/, and modules X from mod A are identified with the pairs


425

.X; 0/, where 0 W X ˝A D.A/ ! X is the zero homomorphism in mod A. We also note that if e1 ; : : : ; en is a set of pairwise orthogonal primitive idempotents of T.A/ with 1A D e1 C C en , then eQ1 D .e1 ; 0/; : : : ; eQn D .en ; 0/ is a set of pairwise orthogonal primitive idempotents of T.A/ with 1T.A/ D eQ1 C C eQn . Then it follows from Proposition I.8.2 that every nonzero projective module P in mod T.A/ is a direct sum P D P1 ˚ ˚ Pm , where each Pj , j 2 f1; : : : ; mg, is isomorphic to a module eQi T.A/ D .ei ; 0/ T.A/ with i 2 f1; : : : ; mg. Observe that, under identification mod T.A/ D T.mod A/, eQi T.A/ D .ei A ˚ D.Aei /; ˛eQi T.A/ /, with ˛eQi T.A/ W .ei A ˚ D.Aei // ˝A D.A/ ! ei A ˚ D.Aei / the homomorphism in mod A given by ˛eQi T.A/ ..ai ; fi / ˝ g/ D .0; ai g/ for ai 2 ei A, fi 2 D.Aei / and g 2 D.A/. For modules X , Y in mod A and a homomorphism ' W X ˝A D.A/ ! Y of right A-modules, we denote by .X; Y; '/ the induced module .X ˚ Y; '/ Q in mod T.A/ D T.mod A/ with 'Q the composed homomorphism h

0 0 ' 0

i

! .X ˝A D.A// ˚ .Y ˝A D.A// ! X ˚ Y: .X ˚ Y / ˝A D.A/

Observe that then we have Pz D .P; idP ˝A D.A/ / D .P; P ˝A D.A/; idP ˝A D.A/ /: Moreover, X D .X; 0/ D .X; 0; 0/ D .0; X; 0/ for any module X in mod A. Proposition 10.12. Let A be a finite dimensional K-algebra over a field K and M an indecomposable module in mod A with pdA M D 1. Then T.A/ M Š A M in mod T.A/. Proof. Since pdA M D 1, X admits a minimal projective resolution in mod A of the form p1 p0 0 ! P1 ! P0 ! M ! 0: Then, applying Proposition III.5.3, we conclude that there is in mod A an exact sequence of the form NA .p1 /

NA .p0 /

0 ! A M ! NA .P1 / ! NA .P0 / ! NA .M / ! 0; where NA D D HomA .; A/ W mod A ! mod A is the Nakayama functor. Further, by Proposition III.5.2, the functors NA and ˝A D.A/ from mod A to mod A are naturally isomorphic. Therefore, we obtain an exact sequence in mod A of the form p1 ˝A D.A/

p0 ˝A D.A/

! P1 ˝A D.A/ ! P0 ˝A D.A/ ! M ˝A D.A/ ! 0: 0 ! A M On the other hand, from the description of projective modules in mod T.A/ D T.mod A/ it follows that M admits a projective cover of the form .p ;0/

0 Pz0 D .P0 ; P0 ˝A D.A/; idP0 ˝A D.A/ / ! .M; 0; 0/:

426


Then Ker.p0 ; 0/ D .P1 ; P0 ˝A D.A/; p1 ˝ D.A//. Thus Ker.p0 ; 0/ admits a projective cover in mod T.A/ D T.mod A/ of the form .idP ;p1 ˝D.A//

1 Pz1 D .P1 ;P1 ˝A D.A/; idP1 ˝A D.A/ / ! .P1 ;P0 ˝A D.A/; p1 ˝ D.A//:

Hence we conclude that there are in mod T.A/ isomorphisms 2T.A/ .M / Š Ker.idP1 ; p1 ˝ D.A// D .0; Ker.p1 ˝ D.A//; 0/ D .0; A M; 0/ D A M: Since T.A/ is a symmetric algebra, applying Corollary 8.6, we obtain that T.A/ M Š 2T.A/ .M / in mod T.A/. Therefore, T.A/ M Š A M in mod T.A/. Corollary 10.13. Let A be a finite dimensional K-algebra over a field K and M be an indecomposable A -periodic module with pdA M D 1. Then M is an indecomposable T.A/ -periodic and T.A/ -periodic module in mod T.A/. Example 10.14. Let r be a positive integer, Q.r/ the quiver 0 Z5o 55 5 ˇ1 55

˛

o 1

ˇ2

o 2

::: o

r

ˇ

r

o r 2 ˇr1 r 1

and A.r/ D KQ.r/ the path algebra of Q.r/ over a field K (see Exercise III.12.32). It follows from Theorem I.9.5 that A.r/ is a hereditary algebra, and so pdA.r/ X D 1 for any indecomposable nonprojective module X in mod A.r/. Moreover, the simple right A.r/-modules S.1/; S.2/; : : : ; S.r 1/, associated to the vertices 1; 2; : : : ; r 1 of Q.r/, and the right A.r/-module E K Z5o 55 55 5

1

0o

0o

::: o

0o

0

K

lie on the mouth of a stable tube T of A.r/ of rank r, and we have A S.i / D S.i 1/, for i 2 f2; : : : ; r 1g, A S.1/ D E and A E D S.r 1/. In particular, all indecomposable modules in T are A -periodic of period r. Applying now Proposition 10.12, we conclude that T is also a stable tube of rank r in T.A/ , and consequently all indecomposable right A-modules in T are T.A/ -periodic indecomposable right T.A/-modules of period r. Moreover, since T.A/ is a symmetric algebra, we have T.A/ M Š 2T.A/ .M / for any indecomposable nonprojective module M in mod T.A/. Therefore, we obtain that all modules in T are T.A/ -periodic indecomposable right T.A/-modules of period dividing 2r. In fact, a direct checking

11. Periodic algebras

427

shows that the modules T.A/ .S.1//; T.A/ .S.2//; : : : ; T.A/ .S.r 1//; T.A/ .E/ are not isomorphic to any of the modules S.1/; S.2/; : : : ; S.r 1/; E. Clearly, we have 2T.A/ .S.i // Š T.A/ .S.i // D S.i 1/ for i 2 f2; : : : ; r 1g, 2T.A/ .S.1// Š T.A/ .S.1// D E and 2T.A/ .E/ Š T.A/ E D S.r 1/. Hence, we conclude that all modules in the stable tube T are T.A/ -periodic indecomposable right T.A/modules of period 2r.

11 Periodic algebras In this section we introduce as well as describe properties of periodic finite dimensional algebras. Moreover, we show that the periodic algebras have periodic module categories. Proposition 11.1. Let A be a finite dimensional K-algebra over a field K and e1 ; : : : ; en be a set of pairwise orthogonal primitive idempotents of A with 1A D e1 C C en . The following statements hold. (i) fij D ei0 ˝ ej , i; j 2 f1; : : : ; ng, is a set of pairwise orthogonal idempotents of Ae with X fij : 1Ae D 1 i;j n

(ii) There is a decomposition M

Ae D

fij Ae

1 i;j n

of the enveloping algebra Ae of A into a direct sum of projective right Ae submodules. (iii) Every indecomposable projective module P in mod Ae is isomorphic to a direct summand of a projective right Ae -module fij Ae , for some i; j 2 f1; : : : ; ng. Proof. (i) For i; j; k; l 2 f1; : : : ; ng, we have the equalities fij fkl D .ei0 ˝ ej /.ek0 ˝ el / D ei0 ek0 ˝ ej el D .ek ei /0 ˝ ej el D ıik ıj l fij : Moreover, we have also the equalities 1Ae D 1Aop ˝ 1A D

n n X X ei0 ˝ ej D iD1

j D1

X 1 i;j n

ei0 ˝ ej D

X 1 i;j n

(ii) It is a direct consequence of Lemma I.5.7, Proposition I.8.2 and (i).

fij :

428


(iii) Since Ae is a finite dimensional K-algebra, it follows from Corollary I.5.9 that there is a decomposition Ae D f1 Ae ˚ ˚ fm Ae of Ae in mod Ae into a direct sum of indecomposable right ideals, where f1 ; : : : ; fm are pairwise orthogonal idempotents of Ae such that 1Ae D f1 C C fm . Further, by Proposition I.8.2, P is isomorphic as right Ae -module to a module fk Ae for some k 2 f1; : : : ; mg. On the other hand, by (ii), we have a decomposition M Ae D fij Ae 1 i;j n

in mod Ae . Taking now decompositions of all modules fij Ae , i; j 2 f1; : : : ; ng, into direct sums of indecomposable right Ae -modules, we conclude, using Theorem I.4.6, that fk Ae , and hence P , is isomorphic in mod Ae to a direct summand of a projective module of the form fij Ae , for some i; j 2 f1; : : : ; ng. The following example shows that the projective right Ae -modules (A-bimodules) fij Ae D Aei0 ˝K ej A, for primitive idempotents ei and ej of A, are not necessarily indecomposable, and hence the idempotents fij D ei0 ˝ ej are not necessarily primitive. Example 11.2. Consider the 2-dimensional R-algebra C of complex numbers. Then the identity 1C of C is a unique nonzero idempotent of C, and hence 1C is a primitive idempotent of the R-algebra C. We have a canonical isomorphism of R-algebras C Š RŒx=.x 2 C 1/. This leads to the following isomorphisms of R-algebras Ce Š Cop ˝R C D C ˝R C Š C ˝R RŒx=.x 2 C 1/ Š CŒx=.x 2 C 1/ D CŒx=.x C i /.x i / Š CŒx=.x C i / CŒx=.x i / Š C C; which shows that Ce admits two orthogonal central idempotents f1 and f2 with 1Ce D f1 C f2 , corresponding to the central idempotents e1 D .1; 0/ and e2 D .0; 1/ of C C. In particular, we obtain that 1Ce D 1C ˝ 1C is not a primitive idempotent of the R-algebra Ce . We also note that the above example shows also that the tensor product of two fields (respectively, division algebras, simple algebras) is not necessarily a simple algebra. On the other hand, the next proposition and corollary show that for a wide class of finite dimensional K-algebras, containing all bound quiver K-algebras KQ=I , the situation is very nice.


429

Proposition 11.3. Let A be a finite dimensional K-algebra over a field K, B D A= rad A, and e1 ; : : : ; en be a set of pairwise orthogonal primitive idempotents of A with 1A D e1 C Cen . Assume that B Š Mn1 .K/ Mnr .K/ as K-algebras, for some positive integers n1 ; : : : ; nr . The following statements hold. (i) fij D ei0 ˝ ej , i; j 2 f1; : : : ; ng, is a set of pairwise orthogonal primitive idempotents of Ae with X 1Ae D fij : 1 i;j n

(ii) There is a decomposition Ae D

M

fij Ae

1 i;j n

of Ae into a direct sum of indecomposable projective right Ae -submodules. (iii) Every indecomposable projective module P in mod Ae is isomorphic to a projective right Ae -module fij Ae , for some i; j 2 f1; : : : ; ng. Proof. Let N D rad Aop ˝K A C Aop ˝K rad A. Then N is a nilpotent two-sided ideal of Ae , since rad Aop and rad A are nilpotent two-sided ideals of rad Aop and A, respectively (Corollary I.3.4). Consider the canonical K-algebra homomorphism ' W Ae =N ! B e given by '.a0 ˝ b C N / D .a0 C rad Aop / ˝ .b C rad A/ for a; b 2 A. Applying Proposition II.3.1, we conclude that we have also the well-defined K-algebra homomorphism W B e ! Ae =N such that ..a0 C rad Aop / ˝ .b C rad A// D a0 ˝ b C N . Then ' and are mutually inverse homomorphisms of K-algebras, and consequently Ae =N Š B e as K-algebras. Since N is a nilpotent two-sided ideal of Ae , we conclude from Lemma I.3.12 that fij D ei0 ˝ ej , i; j 2 f1; : : : ; ng, is a set of pairwise orthogonal primitive idempotents of Ae D Aop ˝K A whose sum is the identity 1Ae of Ae if and only if fNij D fij C N , i; j 2 f1; : : : ; ng, is a set of pairwise orthogonal primitive idempotents of Ae =N whose sum is the identity 1Ae =N of Ae =N . The latter is equivalent to the statement: eNi0 ˝ eNj D .ei0 C rad Aop / ˝ .ej C rad A/, i; j 2 f1; : : : ; ng, is a set of pairwise orthogonal primitive idempotents of B e D B op ˝K B D .Aop = rad Aop / ˝K .A= rad A/ whose sum is the identity 1B e D 1B op ˝ 1B of B e , because Ae =N is isomorphic to B e . Clearly, the idempotents eNi0 ˝ eNj , i; j 2 f1; : : : ; ng, are pairwise orthogonal and their sum is the identity 1B e of B e . Fix i; j 2 f1; : : : ; ng. We claim that eNi0 ˝ eNj is a primitive idempotent of B e .

430


By Corollary I.8.9, it is enough to prove that .eNi0 ˝ eNj /B e .eNi0 ˝ eNj / D .eNi B eNi /op ˝K .eNj B eNj / is a local K-algebra. It follows from Lemma I.4.4, Proposition I.5.16 and Corollaries I.8.8 and I.8.9, that eNi D ei C rad A and eNj D ej C rad A are primitive idempotents of B. Invoking now our assumption B Š Mn1 .K/ Mnr .K/ as K-algebras, we infer that eNi B eNi Š K and eNj B eNj Š K. Then we obtain that .eNi B eNi /op ˝K .eNj B eNj / Š K ˝K K Š K as K-algebras. This ensures that .eNi0 ˝ eNj /B e .eNi0 ˝ eNj / is a local K-algebra. Summing up, we have proved that fij D ei0 ˝ ej , i; j 2 f1; : : : ; ng, is a set of pairwise orthogonal primitive idempotents of Ae with X fij ; 1Ae D 1 i;j n

so the statement (i) holds. The statements (ii) and (iii) are now consequences of (i) and Proposition I.8.2. Corollary 11.4. Let A be a finite dimensional K-algebra such that A= rad A Š K K as K-algebras. Then rad Ae D rad Aop ˝ A C Aop ˝ rad A. Proof. We know that N D rad Aop ˝K A C Aop ˝K rad A is a nilpotent two-sided ideal of Ae D Aop ˝K A such that Ae =N Š .A= rad A/op ˝K .A= rad A/ as Kalgebras. Now it follows from our assumption that both algebras are isomorphic to a product K K of a finite number of copies of K. Applying Lemma I.3.5, we conclude that N D rad Ae . We exhibit now some properties of the enveloping algebras of finite dimensional algebras. Proposition 11.5. Let A be a finite dimensional K-algebra over a field K. Then A is a selfinjective algebra if and only if Ae is a selfinjective algebra. Proof. We will show first that it is enough to prove the required equivalence for A a basic algebra. Consider the canonical decomposition 1Ae D

nA mX A .i/ X

eij ;

iD1 j D1

of 1A into sum of pairwise orthogonal primitive idempotents of A such that eij A Š eil A eij A © ekl A

for j; l 2 f1; : : : ; mA .i /g; i 2 f1; : : : ; nA g; for i; k 2 f1; : : : ; nA g; with i ¤ k and all j 2 f1; : : : ; mA .i /g; l 2 f1; : : : ; mA .k/g;

in mod A, the associated primitive idempotents ei D ei1 , i 2 f1; : : : ; nA g, and the basic idempotent nA nA X X ei1 D ei eA D iD1

iD1


431

of A. Then Ab D eA AeA is the associated basic algebra of A. Moreover, by Theorem II.6.16, A is Morita equivalent to Ab . In particular, we conclude from Proposition 3.10 that A is selfinjective if and only if Ab is selfinjective. It follows from Proposition 11.1 that fij;kl D eij0 ˝ ekl , for i; k 2 f1; : : : ; nA g, j 2 f1; : : : ; mA .i /g, l 2 f1; : : : ; mA .k/g, are pairwise orthogonal idempotents of the enveloping algebra Ae D Aop ˝K A such that X X X 1Ae D fij;kl 1 i;k nA 1 j mA .i/ 1 l mA .k/

and there is a decomposition M Ae D

M

M

fij;kl Ae

1 i;k nA 1 j mA .i/ 1 l mA .k/

of Ae into a direct sum of projective right Ae -submodules. Applying Lemma I.8.12, we obtain isomorphisms in mod Aop D A-mod Aeij Š Aeil

for j; l 2 f1; : : : ; mA .i /g; i 2 f1; : : : ; nA g:

Therefore, we obtain isomorphisms in mod Ae D bimod A fij;kl Ae D Aeij ˝K ekl A Š Aeir ˝K eks A D fir;ks Ae for i; k 2 f1; : : : ; nA g, j; r 2 f1; : : : ; mA .i /g, l; s 2 f1; : : : ; mA .k/g. Consider now the idempotent Ae of the form X X 0 fA D ei0 ˝ ek D ei1 ˝ ek1 D eA0 ˝ eA : 1 i;j n

1 i;j n

Observe that every right Ae -submodule of Ae of the form fij;kl Ae for i; k 2 f1; : : : ; nA g, j 2 f1; : : : ; mA .i /g, l 2 f1; : : : ; mA .k/g is isomorphic in mod Ae to a direct summand of the module fA Ae . Hence, there are a positive integer m and an epimorphism .fA Ae /m ! Ae of right Ae -modules which shows that fA Ae is a projective generator (progenerator) in mod Ae (see Proposition II.5.3). Applying now the Morita equivalence Theorem II.6.7, we conclude that Ae is Morita equivalent to EndAe .fA Ae /. Invoking Corollary I.8.8, we obtain isomorphisms of K-algebras EndAe .fA Ae / Š fA Ae fA D .eA0 ˝ eA /.Aop ˝K A/.eA0 ˝ eA / D eA0 Aop eA0 ˝K eA AeA D .eA AeA /op ˝K eA AeA D .Ab /op ˝K Ab D .Ab /e : Hence, we conclude that Ae is Morita equivalent to .Ab /e . Moreover, it follows from Proposition 3.10 that Ae is a selfinjective if and only if .Ab /e is a selfinjective.

432


Now let A be a basic algebra, that is, A Š Ab . Assume A is a selfinjective algebra. Then A is a Frobenius algebra, by Proposition 3.9. Then it follows from Theorem 2.1 that there are isomorphisms AA Š D.A/A in mod A and A A Š A D.A/ in mod Aop . We obtain then isomorphisms of right Ae -modules (A-bimodules) Ae D Aop ˝K A Š D.Aop / ˝K D.A/ Š D.Aop ˝K A/ D D.Ae /: This shows that Ae is a Frobenius algebra, and consequently a selfinjective algebra, by Theorem 2.1 and Proposition 3.8. Conversely, assume that Ae is a selfinjective algebra. Then proj Ae D inj Ae . It follows also from Proposition I.8.2 (ii) that every indecomposable projective right Ae -module is isomorphic to a direct summand of the right Ae -module Ae . Since proj Ae D inj Ae , we obtain that every indecomposable projective right Ae -module is injective and isomorphic to a direct summand of the right Ae -module D.Ae /. Hence, Ae is a direct summand of a module D.Ae /t in mod Ae D bimod A, for some positive integer t . In particular, we obtain that Ae is a direct summand of D.Ae /t in mod A. Observe now that there are isomorphisms of right A-modules op /

Ae D Aop ˝K A Š AdimK .A and

op

D.Ae / D D.Aop ˝K A/ Š D.Aop / ˝K D.A/ Š D.A/dimK D.A / : op

op

Thus .AA /dimK .A / is isomorphic to a direct summand of D.A/t dimK D.A / in mod A. Clearly, this implies that AA is an injective module in mod A. Therefore, A is a selfinjective algebra. A finite dimensional K-algebra A over a field K is said to be separable if A is a projective module in mod Ae . Our next aim is to show that every separable K-algebra is a semisimple K-algebra. We need some preliminary facts. Let A be a finite dimensional K-algebra over a field K. For a bimodule M in bimod A, consider the K-vector space M .A/ D fm 2 M j am D ma for all a 2 Ag : Observe that, if ' W M ! N is a homomorphism in bimod A, then '.M .A/ / N .A/ , because a'.m/ D '.am/ D '.ma/ D '.m/a for all m 2 M .A/ and a; b 2 A. Therefore, we have the covariant K-linear functor ./.A/ W bimod A ! mod K: Lemma 11.6. Let A be a finite dimensional K-algebra over a field K. Then the functors HomAe .A; / and ./.A/ from bimod A to mod K are naturally isomorphic.


433

Proof. For a bimodule M in bimod A, consider the K-linear map M W HomAe .A; M / ! M .A/ given by M .'/ D '.1A / for any ' 2 HomAe .A; M /. Observe that '.1A / belongs to M .A/ , because, for a 2 A, we have a'.1A / D '.a1A / D '.1A a/ D '.1A /a. Clearly, M is a monomorphism of K-vector spaces. Moreover, for any m 2 M .A/ , the map 'm W A ! M such that 'm .a/ D am D ma for a 2 A belongs to HomAe .A; M /, and M .'m / D 'm .1A / D 1A m D m. This shows that M is an isomorphism of K-vector spaces. Further, for a homomorphism f W M ! N in bimod A D mod Ae , the following diagram in mod K, HomAe .A; M /

M

/ M .A/

HomAe .A;f /

HomAe .A; N /

N

f .A/ Df jM .A/

/ N .A/ ,

is commutative. Therefore, the family of K-linear maps M , for M bimodules in bimod A, defines a natural isomorphism W HomAe .A; / ! ./.A/ of functors from bimod A to mod K. Corollary 11.7. Let A be a finite dimensional K-algebra over a field K. Then the following statements hold. (i) The functor ./.A/ W bimod A ! mod K is left exact. (ii) The functor ./.A/ W bimod A ! mod K is exact if and only if A is a separable K-algebra. Proof. It follows from Lemma II.2.5 and Proposition II.2.6 that the functor HomAe .A; / W bimod A ! mod K is left exact, and is exact if and only if A is a projective module in bimod A D mod Ae . Then the required statements are consequences of Lemma 11.6. Let A be a finite dimensional K-algebra over a field K and M; N modules in mod A. Then the K-vector space HomK .M; N / has a natural structure of Abimodule such that the left multiplication au and the right multiplication ua of u 2 HomK .M; N / by a 2 A are given by .au/.m/ D u.ma/

and .ua/.m/ D u.m/a

for m 2 M . Observe that, for a; b 2 A, u 2 HomK .M; N /, and m 2 M , we have ..au/b/.m/ D .au/.m/b D u.ma/b D .ub/.ma/ D .a.ub//.m/; and hence .au/b D a.ub/. Moreover, we have HomA .M; N / D HomK .M; N /.A/ :

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Proposition 11.8. Let A be a finite dimensional separable K-algebra over a field K. Then A is a semisimple K-algebra. Proof. In order to prove that A is a semisimple K-algebra, it is enough to show that every module in mod A is projective (see Corollary I.8.24). Let M be a module in mod A and f W X ! Y be an epimorphism in mod A. Then f W X ! Y is an epimorphism in mod K, and hence the induced homomorphism of K-vector spaces HomK .M; f / W HomK .M; X / ! HomK .M; Y /, which assigns to ' 2 HomK .M; X/ the homomorphism HomK .M; f /.'/ D f ', is an epimorphism. We claim that HomK .M; f / is a homomorphism of A-bimodules. Indeed, for a 2 A, ' 2 HomK .M; X /, and m 2 M , we have the equalities .HomK .M; f /.a'// .m/ D .f .a'//.m/ D f .'.ma// D .f '/.ma/ D .a.f '//.m/ D .a HomK .M; f /.'// .m/ and .HomK .M; f /.'a// .m/ D .f .'a//.m/ D f .'.m/a/ D f .'.m//a D ..f '/a/.m/ D .HomK .M; f /.'/a/ .m/; and hence HomK .M; f /.a'/ D a HomK .M; f /.'/ and HomK .M; f /.'a/ D HomK .M; f /.'/a. Moreover, we have from Lemma 11.6 the commutative diagram of K-vector spaces HomAe .A; HomK .M; X //

HomK .M;X /

HomAe .A;HomK .M;f //

HomAe .A; HomK .M; Y //

HomK .M;Y /

/ Hom .M; X /.A/ K .HomK .M;f //.A/

/ Hom .M; Y /.A/ , K

where the horizontal homomorphisms are isomorphism. Further, since A is a projective module in mod Ae , the functor HomAe .A; / W bimod A ! mod K is exact. Then HomAe .A; HomK .M; f // is an epimorphism, because HomK .M; f / is an epimorphism in bimod A. This implies that HomA .M; f / D .HomK .M; f //.A/ is also an epimorphism. Therefore, M is a projective module in mod Ae D bimod A. As a direct consequence of Propositions 5.4 and 11.8, we obtain the following fact. Corollary 11.9. Let A be a finite dimensional separable K-algebra over a field K. Then A is a semisimple module in mod Ae .


435

We will now explain (without proofs) that in general the class of finite dimensional separable K-algebras over a field K is a proper subclass of the class of all finite dimensional semisimple K-algebras over K. Let K L be a finite field extension. Then K L is called a separable extension if the minimal irreducible polynomial fa .x/ 2 KŒx of any element a 2 L has no multiple roots. Further, a field K is called perfect if every finite field extension of K is separable. We note that the class of perfect fields contains: all algebraically closed fields, all fields of characteristic 0, and all finite fields. For a finite dimensional division K-algebra F , the center C.F / D fa 2 F j ab D ba for all b 2 F g is a finite field extension of K. Proposition 11.10. Let K L be a finite field extension. The following conditions are equivalent. (i) K L is separable extension. (ii) Le D L ˝K L is a semisimple K-algebra. The following general result gives a description of finite dimensional separable algebras over arbitrary fields. Theorem 11.11. Let A be a finite dimensional K-algebra over a field K. The following conditions are equivalent. (i) A is a separable K-algebra. (ii) Ae D Aop ˝K A is a semisimple K-algebra. (iii) There exist positive integers n1 ; : : : ; nr and finite dimensional division Kalgebras F1 ; : : : ; Fr whose centers C.F1 /; : : : ; C.Fr / are separable field extensions of K and A Š Mn1 .F1 / Mnr .Fr / as K-algebras. (iv) There exist a separable finite field extension L of K and positive integers m1 ; : : : ; ms such that A ˝K L Š Mm1 .L/ Mms .L/ as K-algebras. (v) For any finite field extension L of K, A ˝K L is a semisimple K-algebra. For proofs of Proposition 11.10 and Theorem 11.11 we refer to Chapters 5 and 6 of [DK] and Chapter 10 of [P].

436


Corollary 11.12. Let A be a finite dimensional K-algebra over a perfect field K. Then A is a separable K-algebra if and only if A is a semisimple K-algebra. Corollary 11.13. Let A be a finite dimensional K-algebra over a field K. Then A is a separable K-algebra if and only if every finite dimensional A-bimodule in semisimple. Lemma 11.14. Let A be a finite dimensional K-algebra over a field K and P be a projective module in mod Ae D bimod A. Then P is a projective left A-module and a projective right A-module. Proof. Let e1 ; : : : ; en be a set of pairwise orthogonal primitive idempotents of A with 1A D e1 C C en . It follows from Propositions I.8.2 and 11.1 that P is isomorphic to a direct sum of indecomposable projective right Ae -modules (A-bimodules) which are isomorphic to direct summands of projective right Ae modules of the form fij Ae D Aei ˝K ej A, for i; j 2 f1; : : : ; ng. Observe that, for fixed i; j 2 f1; : : : ; ng, there are isomorphisms fij Ae ! .Aei /dimK ej A

of left A-modules, and

fij Ae ! .ej A/dimK Aei

of right A-modules, and consequently fij Ae is a projective left A-module and a projective right A-module. Therefore, P is also a projective left A-module and a projective right A-module. Lemma 11.15. Let A be a finite dimensional K-algebra over a field K and P a projective module in mod Ae . Then the following statements hold. (i) For any module M in mod A, M ˝A P is a projective module in mod A. (ii) For any module N in mod Aop , P ˝A N is a projective module in mod Aop . Proof. (i) Since the functor M ˝A W mod Aop ! mod A commutes with direct sums (see Exercise II.8.2), we may assume that P is an indecomposable right Ae -module. Then, by Proposition 11.1, P is isomorphic to an indecomposable direct summand of a right Ae -module fij Ae D Aei ˝K ej A for some primitive idempotents ei and ej of A. Hence, M ˝A P is isomorphic in mod A to a direct summand of M ˝A fij Ae . Applying Lemma II.3.2, we obtain isomorphisms of right A-modules M ˝A fij Ae D M ˝A .Aei ˝K ej A/ Š .M ˝A Aei /˝K ej A Š .ej A/dimK .M ˝A Aei /; and hence M ˝A P is a projective module in mod A, by Lemma I.8.1. The proof of (ii) is similar.


437

Lemma 11.16. Let A be a finite dimensional K-algebra over a field K. Then, for any positive integer i, Ai e .A/ is a projective left A-module and a projective right A-module. Proof. Consider a minimal projective resolution di C1

di

di 1

d1

d0

! PiC1 ! Pi ! Pi1 ! ! P1 ! P0 ! A ! 0 of A in mod Ae . Then, for any positive integer i, we have in mod Ae an exact sequence of the form vi

ui

i 0 ! AiC1 e .A/ ! Pi ! Ae .A/ ! 0:

Since mod Ae D bimod A, it is also an exact sequence in mod Aop D A-mod and in mod A. Moreover, we have in mod Ae the exact sequence v0

u0

0 ! Ae .A/ ! P0 ! A ! 0; with v0 D d0 , being also an exact sequence in mod Aop and mod A. Since A is in proj Aop and proj A, we conclude by Lemma I.8.1, that d0 is a retraction in mod Aop and in mod A. Then, by Lemma III.3.1, we obtain that u0 is a section in mod Aop and in mod A. Applying now Lemma I.4.2, we conclude that there are isomorphisms P0 Š Ae .A/ ˚ A in mod Aop and in mod A. Therefore, Ae .A/ belongs to proj Aop and proj A, by Lemma I.8.1. Then, by induction on i 1, we conclude that vi is a retraction and ui a section in mod Aop and mod A, and hence all syzygy right Ae -modules (A-bimodules) Ai e .A/ are projective left A-modules and projective right A-modules. Proposition 11.17. Let A be a finite dimensional K-algebra over a field K and M a module in mod A. Then, for each i 0, we have an isomorphism Ai .M / Š M ˝A Ai e .A/ in mod A. Proof. Observe that, for i D 0, we have A0 .M / D M Š M ˝A A D M ˝A A0 e .A/, in mod A (see Lemma II.3.5), and hence in mod A. Consider now a minimal projective resolution di C1

di

di 1

d1

d0

! PiC1 ! Pi ! Pi1 ! ! P1 ! P0 ! A ! 0 of A in mod Ae . It follows from Lemma 11.16 (and its proof) that we have in mod Ae , hence in mod Aop and mod A, exact sequences ui

vi

i 0 ! AiC1 e .A/ ! Pi ! Ae .A/ ! 0

438


with ui sections and vi retractions in mod Aop and mod A, for all i 0. Since the tensor functor M ˝A W mod Aop ! mod K is right exact (see Lemma II.3.3), we obtain exact sequences of K-vector spaces idM ˝ui

idM ˝vi

! M ˝A Pi ! M ˝A Ai e .A/ ! 0; M ˝A AiC1 e .A/ and therefore in mod A, for all i 0. Furthermore, since the homomorphisms op ui W AiC1 and mod A, we conclude that the maps e .A/ ! Pi are sections in mod A idM ˝ui W M ˝A AiC1 e .A/ ! M ˝A Pi ;

i 0;

are also sections, hence monomorphisms, in mod A. Further, by Lemma 11.15 (i), M ˝A Pi , i 0, are projective modules in mod A. Summing up, we conclude that / M ˝A PiC1

:::

GF @A ````0 : : :

idM ˝di C1

/ M ˝A P0

/ M ˝A Pi idM ˝d0

idM ˝di

/ M ˝A Pi1

/ M ˝A A

ED BC /0

is a projective resolution of M ˝A A Š M in mod A. Applying now Proposition 8.1 (v), we conclude that, for any positive integer i , we have in mod A an isomorphism M ˝A Ai e .A/ Š Ai .M / ˚ P .i / for some module P .i / in proj A, and hence a required isomorphism M ˝A Ai e .A/ Š Ai .M /

in mod A.

Let A be a finite dimensional K-algebra over a field K and a K-algebra automorphism of A. For an A-bimodule M , we denote by M the A-bimodule with M D M as K-vector space and amb D am .b/ for all a; b 2 A and m 2 M. Proposition 11.18. Let A be a finite dimensional K-algebra over a field K. Assume there exists a positive integer d and a K-algebra automorphism of A such that Ad e .A/ Š A in mod Ae . Then A is a selfinjective algebra. Proof. If follows from Lemma II.3.5 that the K-linear homomorphism ' W D.A/ ˝A A ! D.A/ given by '.f ˝ a/ D f a, for f 2 D.A/ and a 2 A , is an isomorphism of A-bimodules. Consider a minimal projective resolution di C1

di

d1

d0

! PiC1 ! Pi ! Pi1 ! ! P1 ! P0 ! A ! 0


439

of A in mod Ae . Then we obtain in mod Ae an exact sequence ud 1

vd 1

0 ! Ad e .A/ ! Pd 1 ! Ad e1 .A/ ! 0 such that ud 1 is a section and vd 1 is a retraction simultaneously in mod Aop and in mod A. Letting ˛ D idD.A/ ˝ud 1 and ˇ D idD.A/ ˝vd 1 , this leads to the exact sequence in mod A of the form ˛

ˇ

0 ! D.A/ ˝A Ad e .A/ ! D.A/ ˝A Pd 1 ! D.A/ ˝A Ad e1 .A/ ! 0: Invoking now the assumption Ad e .A/ Š A in mod Ae , we conclude that there is a composed monomorphism in mod A: D.A/ ! D.A/ ˝A A ! D.A/ ˝A Ad e .A/ ! D.A/ ˝A Pd 1 :

On the other hand, the automorphism induces an isomorphism A 1 ! A of A-bimodules, and consequently the A-bimodules D.A/ D D. A/ and 1 D.A/ D D.A 1 / are isomorphic. Therefore, we obtain that there is a monomorphism of right A-modules D.A/A D 1 D.A/A ! D.A/ ˝A Pd 1 , which is a section because D.A/ is in inj A (Lemma I.8.13). In particular, by Lemma I.4.2, D.A/ is a direct summand of D.A/˝A Pd 1 in mod A. Since, by Lemma 11.15, D.A/˝A Pd 1 is in proj A, we infer that D.A/ is also in proj A. Applying now Proposition I.8.19, we obtain that inj A D proj A, equivalently A is a selfinjective algebra. A finite dimensional K-algebra over a field K is said to be a periodic algebra if A is a periodic module in mod Ae D bimod A, that is, Ame .A/ Š A in mod Ae for some positive integer m. The smallest positive integer d with Ad e .A/ Š A in mod Ae is called the period of A. Observe that, by Proposition 11.18, every periodic algebra is a selfinjective algebra. Theorem 11.19. Let A be a finite dimensional periodic K-algebra over a field K and d the period of A. Then the following statements hold. (i) The module category mod A is periodic. (ii) For any module M in modP A, the period of M in mod A divides d . Proof. Let M be a module in modP A. Since Ad e .A/ Š A in mod A, it follows from Propositions 11.17 and 11.18 that A is a selfinjective algebra and we have in mod A isomorphisms Ad .M / Š M ˝A Ad e .A/ Š M ˝A A Š M: Moreover, by Proposition 8.1, Ad .M / is in modP A D modI A. Applying now Lemma III.4.3, we infer that Ad .M / Š M in mod A. Therefore, M is a periodic module and its period divides d . This shows that (i) and (ii) hold.

440


Example 11.20. Let n 2 be a natural number and A D KŒX =.X n / the quotient polynomial algebra over a field K, and denote by x the coset of X in A. Then 1 D 1A ; x; : : : ; x n1 is a basis of A over K. The enveloping algebra Ae D Aop ˝K A D A ˝K A is a commutative K-algebra and the elements x i ˝ x j , i; j 2 f0; 1; : : : ; n 1g, form a basis of Ae over K. In particular, every element in Ae has a unique expression in the form n1 X

ij x i ˝ x j

i;j D0

with ij 2 K, for i; j 2 f0; 1; : : : ; n 1g. Moreover, since A is a local K-algebra with A= rad A Š K, we conclude, by Proposition 11.3, that 1Ae D 1A ˝1A D 1˝1 is a unique nonzero idempotent of Ae . In particular, Ae is a unique indecomposable projective right Ae -module (up to isomorphism), by Proposition I.8.2. Consider now the augmentation map W Ae ! A such that .a ˝ b/ D ab for a; b 2 A. Clearly, is an epimorphism of right Ae -modules. Further, a direct checking shows that Ker is the right Ae -module .x ˝ 1 1 ˝ x/Ae generated by x ˝ 1 1 ˝ x. Let ! W Ae ! Ae be the homomorphism of right Ae -modules such that !.1 ˝ 1/PD x ˝ 1 1 ˝ x. Then Ker ! is the right Ae -module yAe generated n1i by y D n1 ˝ x i (see Excercise 16.46). Therefore, we have a minimal iD0 x projective presentation ! Ae ! Ae ! A ! 0 of A in mod Ae with Ae .A/ D Ker D .x ˝ 1 1 ˝ x/Ae and A2 e .A/ D Ker ! D yAe . Observe that dimK Ae .A/ D dimK Ae dimK A D .dimK A/2 dimK A: Hence, we obtain dimK A2 e .A/ D dimK Ae dimK Ae .A/ D dimK A: Then the canonical epimorphism of right Ae -modules % W A ! yAe D A2 e .A/ given by %.1/ D y is an isomorphism, and consequently A Š A2 e .A/ in mod Ae . This shows that A is a periodic algebra. Moreover, for n 3, we have dimK Ae .A/ D .dimK A/2 dimK A D n2 n ¤ n D dimK A; so A is of period 2. On the other hand, for n D 2, we have dimK Ae .A/ D dimK A, and so the canonical epimorphism of right Ae -modules W A ! .x˝11˝x/Ae D Ae .A/ given by .1/ D x ˝ 1 1 ˝ x is an isomorphism, and consequently A is of period 1. We present now a general result proved byA. Dugas [Du], extending the previous partial results by K. Erdmann and T. Holm [EH] and S. Brenner, M. C. R. Butler and


441

A. D. King [BKK] to the general case, whose proof uses the classification of Morita equivalence classes of selfinjective algebras of finite representation type over an arbitrary algebraically closed field. Theorem 11.21. Let A be a nonsimple, indecomposable, finite dimensional selfinjective K-algebra of finite representation type over an algebraically closed field K. Then A is a periodic algebra. We end this section with a result describing the periodic blocks of group algebras over algebraically closed fields. Let K be an algebraically closed field, G a finite group, and KG the group algebra of G over K. We know from Maschke’s Theorem I.6.18 that KG is a semisimple algebra if and only if the characteristic of K does not divide the order of G. In general, the group algebra KG has a unique decomposition KG D B1 ˚ B2 ˚ ˚ Bs into a direct sum of indecomposable two-sided ideals B1 D e1 .KG/; B2 D e2 .KG/; : : : ; Bs D es .KG/; for central idempotents e1 ; e2 ; : : : ; es of KG with 1KG D e1 C e2 C C es , called the blocks of KG (see Proposition I.3.16). We know also from Theorem 4.12 that every nonsimple block B of KG of finite representation type is Morita equivalent to a Brauer tree algebra A.TSm /. Moreover, such a block B is a periodic algebra, by a result of K. Erdmann and T. Holm [EH], and also by Theorem 11.21. The following result proved by K. Erdmann and A. Skowroński [ES] completes the picture. Theorem 11.22. Let K be an algebraically closed field of positive characteristic p, G a finite group whose order is divisible by p, and B a block of KG of infinite representation type. The following conditions are equivalent. (i) The module category mod B is periodic. (ii) B is a periodic algebra. 4 e (iii) B e .B/ Š B in mod B .

We note that if the block B in the above theorem is periodic then p D 2 and B belongs to one of 12 families of algebras of quaternion type described by K. Erdmann in [Er].

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12 The Green–Snashall–Solberg theorems In this section we discuss the properties of finite dimensional algebras all of whose simple modules are periodic. The following lemma will be useful. Lemma 12.1. Let A be a finite dimensional K-algebra over a field K and M be a finite dimensional A-bimodule such that A M is projective in mod Aop and MA is projective in mod A. Then M.rad A/ D .rad A/M . Proof. We will show only the inclusion .rad A/M M.rad A/, because the converse inclusion will follow in a similar way. Consider the canonical homomorphism of K-algebras L W A ! EndA .MA / given for a 2 A and m 2 M by L.a/.m/ D am. Take a 2 rad A L and set f D L.a/. Since MA is a projective module in mod A, we may write MA D m iD1 Pi for some indecomposable projective right A-modules P1 ; : : : ; Pm . Then f is of the form f D .fij / W

m M

Pj !

j D1

m M

Pi ;

iD1

with fij W Pj ! Pi homomorphisms in mod A. Since a 2 rad A, it follows from Lemmas III.1.3 and III.1.4 that fij W Pj ! Pi is a nonisomorphism, or equivalently not an epimorphism, for all i; j 2 f1; : : : ; mg. Hence we have Im fij rad Pi D Pi .rad A/ for all i; j 2 f1; : : : ; mg, because rad Pi is the unique P maximal right A-submodule of Pi , by Proposition I.5.16. This implies that jmD1 Im fij Pi .rad A/, for any i 2 f1; : : : ; mg. Therefore, we obtain the inclusions Im f

m X m M iD1

j D1

m m M M .Pi rad A/ D Im fij Pi .rad A/ : iD1

iD1

Hence, aM M.rad A/ for any a 2 rad A, and consequently .rad A/M M.rad A/, as desired. The following important result has been proved by E. L. Green, N. Snashall and Ø. Solberg in [GSS]. Theorem 12.2. Let A be a finite dimensional K-algebra over a field K. The following conditions are equivalent. (i) Every simple module in mod A is periodic. (ii) There exists a natural number d and a K-algebra automorphism of A such that Ad e .A/ Š A in mod Ae , and .e/A Š eA in mod A for any primitive idempotent e of A.

12. The Green–Snashall–Solberg theorems

443

Proof. We prove first that (i) implies (ii). Assume that every simple module in mod A is periodic. Let d be the minimal natural number such that Ad .S / Š S for any simple right A-module S . Let B D Ad e .A/. We know from Lemma 11.16 that Ad e .A/ is a projective left A-module and a projective right A-module. Then we have the exact functors ˝A B W mod A ! mod A

and

B ˝A W mod Aop ! mod Aop

(see Lemma II.3.3 and Exercise II.8.8). Let S be a simple module in mod A. We will show that S ˝A B Š S in mod A. Observe first that S ˝A B is a semisimple module in mod A. Indeed, applying Lemma 12.1 to the A-bimodule B, we obtain .S ˝A B/ rad A D S ˝A .B.rad A// D S ˝A ..rad A/B/ D .S rad A/ ˝A B D 0; which implies that S ˝A B is a semisimple right A-module, by Proposition I.5.13 and Corollary I.5.15. As a consequence, we obtain also that S ˝A B belongs to modP A, because all simple modules in mod A are periodic. Further, applying Proposition 11.17, we obtain isomorphisms S Š Ad .S / Š S ˝A Ad e .A/ D S ˝A B in mod A. Then, by Lemma III.4.3, we have also S Š S ˝A B in mod A, because S and S ˝A B are in modP A. Clearly, then `.S ˝A B/ D `.S / D 1. We claim that `.M ˝A B/ D `.M / for any module M in mod A. Indeed, let M be a nonzero module in mod A. Then there is in mod A an exact sequence 0 ! L ! M ! S ! 0 with L a right A-submodule of M and S D M=L a simple right A-module (see Proposition I.7.4). Applying the exact functor ˝A B W mod A ! mod A, we obtain in mod A an exact sequence 0 ! L ˝A B ! M ˝A B ! S ˝A B ! 0: Then, by Lemma I.7.6, we have `.L/ D `.M /`.S / D `.M /1 and `.L˝A B/ D `.M ˝A B/ `.S ˝A B/ D `.M ˝A B/ 1. Hence, by induction on the length of a module in mod A, we conclude that `.M ˝A B/ D `.M /. Our next aim is to show that B Š AA in mod A. Let P be an indecomposable projective right A-module. We have the canonical exact sequence in mod A i

h

0 ! rad P ! P ! top.P / ! 0; and hence an exact sequence of the form i˝idB

h˝idB

0 ! rad P ˝A B ! P ˝A B ! top.P / ˝A B ! 0:

444


Since P is isomorphic to a direct summand eA of AA (see Proposition I.8.2), we conclude that P ˝A B is isomorphic to a direct summand eA ˝A B of A ˝A B D B in mod A. Further, by Lemma 11.16, B is a projective module in mod A, and so (Lemma I.8.1) P ˝A B is also a projective module in mod A. It follows from the first part of the proof that top.P / ˝A B is a simple right A-module isomorphic to the simple right A-module top.P /. Moreover, by Lemma I.5.18, top.h ˝ idB / W top.P ˝A B/ ! top.top.P / ˝A B/ D top.P / ˝A B is an epimorphism of semisimple right A-modules. Consequently, top.P ˝A B/ Š Ker.top.h˝ idB // ˚ .top.P / ˝A B/ Š Ker.top.h ˝ idB // ˚ top.P / (see Lemma I.5.4). Applying Theorem I.8.4 (and its proof) we conclude that P is isomorphic to a direct summand of P ˝A B, because P ˝A B and P are projective covers of top.P ˝A B/ and top.P / in mod A, respectively. On the other hand, `.P ˝A B/ D `.P /, by the first part of the proof. This shows that P ˝A B Š P in mod A. Therefore, B Š AA in mod A, because A ˝A B Š B in mod A and AA is a direct sum of indecomposable projective right A-modules. We will show now that B Š A A in mod Aop . By the arguments from the previous part of the proof, it is enough to show that B ˝A T Š T in mod Aop for any simple left A-module T . Let T be a simple module in mod Aop . It follows from the first part of the proof and Lemma II.3.2 that S ˝A .B ˝A T / Š .S ˝A B/˝A T Š S ˝A T in mod K for any simple right A-module S . Observe also that the canonical exact sequence of right A-modules 0 ! rad A!A!A= rad A ! 0: induces the exact sequence of K-vector spaces .rad A/ ˝A T !A ˝A T !.A= rad A/ ˝A T ! 0; with .rad A/ ˝A T D .A rad A/ ˝A T D A ˝A .rad A/T D 0, and hence we get that .A= rad A/ ˝A T Š A ˝A T Š T in mod Aop . Therefore, we obtain that .A= rad A/ ˝A .B ˝A T / Š T in mod Aop . On the other hand, we have in mod Aop the commutative diagram with exact rows rad A ˝A .B ˝A T / Š

.rad A/.B ˝A T /

/ A ˝A .B ˝A T / Š

/ B ˝A T

/ .A= rad A/ ˝A .B ˝A T /

/0

Š

/ .B ˝A T /=.rad A/.B ˝A T /

/ 0,

and hence .B ˝A T /=.rad A/.B ˝A T / Š T in mod Aop . Moreover, since B Š AA in mod A, we have isomorphisms of K-vector spaces B ˝A T Š A ˝A T Š T . This shows that .rad A/.B ˝A T / D 0, and consequently B ˝A T Š T in mod Aop . Summing up, we have proved that A B Š A A in mod Aop and BA Š AA in mod A.

12. The Green–Snashall–Solberg theorems

445

Let W A ! B be an isomorphism of left A-modules, and b D .1A /. Then .a/ D ab, for a 2 A, and Ab D B. Define the map W A ! A by .a/ D 1 .ba/, for a 2 A. Then, for a 2 A, we have ba D

.

1

.ba// D

. .a// D

..a/1A / D .a/ .1A / D .a/b:

Next we show that is a homomorphism of K-algebras. Obviously, is K-linear and .1A / D 1 .b/ D 1A . Moreover, for a; a0 2 A, we have .aa0 /b D b.aa0 / D .ba/a0 D . .a/b/a0 D .a/.ba0 / D .a/. .a0 /b/ D . .a/ .a0 //b: Hence, we obtain ..aa0 // D . .aa0 /1A / D .aa0 / .1A / D .aa0 /b D . .a/ .a0 //b D . .a/ .a0 // .1A / D . .a/ .a0 //; and so .aa0 / D .a/ .a0 /, because is an isomorphism. Therefore, is a homomorphism of K-algebras. We claim that is in fact an automorphism. It is enough to show that Ker D 0. Let a 2 Ker . Then 0 D .a/b D ba and hence Ba D .Ab/a D A.ba/ D 0. Since B is isomorphic to A as a right A-module, we obtain Aa D 0, and hence a D 0. Therefore, indeed Ker D 0. Finally, observe that the isomorphism W A ! B of left A-modules is an isomorphism W A ! B of A-bimodules. Indeed, for x; a 2 A, we have .x.a// D .x .a//b D x. .a/b/ D x.ba/ D .xb/a D

.x/a:

Therefore, we obtain Ad e .A/ Š A in mod Ae . Let e be a primitive idempotent of A. Then .e/ is a primitive idempotent of A and .e/A=.e/ rad A is a simple right A-module. Further, using Proposition 11.17, we obtain in mod A isomorphisms .e/A= .e/ rad A ! Ad .e/A= .e/ rad A ! .e/A= .e/ rad A ˝A Ad e .A/ ! .e/A= .e/ rad A ˝A A ! .e/A= .e/ rad A ! eA=e rad A:

Since the simple right A-modules are nonprojective, it follows from Lemma III.4.3 that .e/A=.e/ rad A Š eA=e rad A in mod A. Passing to the projective covers of these simple modules we obtain that .e/A Š eA in mod A. This completes the proof of implication (i) ) (ii).

446


Assume now that Ad e .A/ Š A in mod Ae for a positive natural number d and a K-algebra automorphism of A such that .e/A Š eA in mod A for any primitive idempotent e of A. Then it follows from Propositions 11.5 and 11.18 that A and Ae are selfinjective algebras. Moreover, since Ad e .A/ D A , we conclude from Proposition 6.8 and Corollary 11.9 that all simple modules in mod A are nonprojective. Let S be a simple module in mod A. Then there is a primitive idempotent of A such that S Š eA=e rad A in mod A (see Corollary I.8.6). Further, applying Proposition 11.17, we obtain isomorphisms in mod A, Ad .S / Š S ˝A Ad e .A/ Š S ˝A A Š S ; where the right A-module S is simple. Since Ad .S / and S are in modP A, it follows from Lemma III.4.3 that Ad .S / Š S in mod A. On the other hand, since eA Š .e/A in mod A, the automorphism induces isomorphisms of right A-modules eA ! .eA/ , e rad A ! .e rad A/ , and hence an isomorphism of simple right A-modules S ! S . Therefore, we obtain that Ad .S / Š S in mod A. This shows that (ii) implies (i). As a direct consequence of Proposition 11.18 and Theorem 12.2 we obtain the announced fact. Corollary 12.3. Let A be a finite dimensional K-algebra over a field K such that every simple module in mod A is periodic. Then A is a selfinjective algebra. We present now another result proved by E. L. Green, N. Snashall and Ø. Solberg in [GSS]. Theorem 12.4. Let A be an indecomposable finite dimensional K-algebra over a field K. Assume that An e .A/ Š A in mod Ae for a positive integer n and an algebra automorphism of A. Moreover, let N.A/ be the ideal of the algebra ExtA e .A; A/ generated by all nilpotent homogeneous elements. Then there are isomorphisms of graded K-algebras ´ F if A is nonperiodic, ExtA e .A; A/=N.A/ Š F Œx if A is periodic, where F is the finite dimensional division K-algebra EndAe .A/= rad EndAe .A/. Proof. Since An e .A/ Š A , it follows from Propositions 11.5 and 11.18 that A and Ae are selfinjective algebras. Recall also that the Ext-algebra of A in mod Ae is of the form 1 M HomAe .Ar e .A/; A/: ExtA e .A; A/ D rD0

If Ame .A/ Š A in mod Ae for some positive integer m, then, by Carlson’s Theorem 10.10, we have an isomorphism of graded K-algebras ExtA e .A; A/=N.A/ Š

13. Dynkin and Euclidean graphs

447

F Œx, where F Œx is the graded polynomial algebra over the division K-algebra F D EndAe .A/= rad EndAe .A/ in variable x of degree d , being the period of A in mod Ae . Observe that this situation occurs if the automorphism is of finite order. Assume now that Ame .A/ © A in mod Ae for any positive integer m. Then is of infinite order. Let s 1 and 2 HomAe .As e .A/; A/. We claim that is nilpotent in ExtA e .A; A/. Assume first that s D np, for some p 1. Then, for any e i 1, we have that Ai np e .A/ Š A ip is an indecomposable right A -module and e the induced homomorphism of right A -modules .i1/np A.i1/np . / W Ai np .A/ e e .A/ ! Ae

is not an isomorphism. Further, our assumption An e .A/ Š A implies that the indecomposable right Ae -modules Ai np e .A/, i 1, have bounded length (dimension). Then, applying the Harada–Sai Lemma III.2.1, we conclude that there exists a natural number t such that 2np np

t D Atnp e . / : : : Ae . /Ae . / D 0

in the algebra ExtA e .A; A/. Hence, is nilpotent. Now consider an arbitrary n with An e .A/ Š A in mod Ae and take positive integers r and q such that rs D nq. Then r 2 HomAe .Anqe .A/; A/, and hence (by the above argument) r is nilpotent, and consequently is nilpotent. Observe also that, since A is an indecomposable module in mod Ae , the radical rad EndAe .A/ of the local K-algebra EndAe .A/ is the set of all nilpotent elements of EndAe .A/, and F D EndAe .A/= rad EndAe .A/ is a finite dimensional division K-algebra (see Lemma I.3.8 and Corollary I.3.9). Therefore, ExtA e .A; A/ Š F as K-algebras

13 Dynkin and Euclidean graphs In this section we introduce Dynkin graphs and Euclidean graphs and present their combinatorial characterizations through the associated Cartan matrices. Let I be a finite set. A Cartan matrix on I is a matrix C D Cij i;j 2I with integral coefficients satisfying the following conditions: (1) Ci i D 2 for all i 2 I ; (2) Cij 0 for all i ¤ j in I ; (3) Cij D 0 if and only if Cj i D 0, for i; j 2 I . A Cartan matrix C on I is completely determined by the associated valued graph .C /. The set I is the set of vertices of .C /. For i ¤ j in I there is an edge in

448


.C / connecting i and j if and only if Cij ¤ 0, and this edge has the valuation .jCij j;jCj i j/

i

: j .1;1/

instead of . In order to simplify the notation, we will write A Cartan matrix C on I is said to be indecomposable if for all proper partitions I D I 0 [ I 00 of I there exist i 2 I 0 and j 2 I 00 with Cij ¤ 0. Clearly, a Cartan matrix C on I is indecomposable if and only if the associated valued graph .C / is connected. We also note that the transpose C T of a Cartan matrix C D ŒCij i;j 2I on I is also a Cartan matrix on I and its valued graph .C T / is obtained from the valued graph .C / by changing the valuations of all edges i

.jCij j;jCj i j/

j

i

on

.jCj i j;jCij j/

: j

For a valued graph , without loops and multiple edges, we denote by C. / the Cartan matrixon the set I D 0 of vertices of such that D .C. //. Let C D Cij i;j 2I be a Cartan matrix on a finite set I . A function d W I ! P NC D f1; 2; 3; : : : g is said to be an additive function for C if i2I di Cij D 0 for P all j 2 I , and a subadditive function for C if i2I di Cij 0 for all j 2 I , where we write di D d.i / for all i 2 I . We will show that the existence of a subadditive or additive function for a Cartan matrix C on a finite set I forces a distinguished shape of the associated valued graph .C /. Consider the following valued graphs, called Dynkin graphs, ::: (n vertices), n 1 An W Bn W

.1;2/

:::

(n vertices), n 2

Cn W MMM MMM Dn W qqq qqq

:::

(n vertices), n 3

:::

(n vertices), n 4

.2;1/

E6 W

E7 W


449

E8 W

F4 W

.1;2/

.1;3/

, G2 W and the following valued graphs, called Euclidean graphs, z 11 W A

.1;4/

z 12 W A

.2;2/

zn W A

UUUUU iiii UUUU iiii UUUU i i i i UUUU i i U iii i : : :

zn W B

.1;2/

znW C

.2;1/

:::

:::

fnW BC

.1;2/

:::

fnW BD

.1;2/

:::

.2;1/

:::

e

CDn W

MMM MMM zn W D qqq q q q z6 W E

:::

(n C 1 vertices), n 4

.2;1/


.1;2/


.1;2/

qqq qqq MMM MMM qqq qqq MMM MMM qqq qqq MMM MMM

(n C 1 vertices), n 2 (n C 1 vertices), n 3



z7 W E z8 W E

450


Fz41 W Fz42 W z 21 W G z 22 W G

.1;2/

.2;1/

.1;3/

.3;1/

.

Lemma 13.1. Let C D C. / be the Cartan matrix associated to a Euclidean graph . Then the following statements hold. (i) There is an additive function for C . (ii) Every subadditive function for C is additive. Proof. (i) Examples of additive functions for the Cartan matrices of the Euclidean graphs are listed below. z 11 W 2 1 A z 12 W 1 1 A 1 1 1 1

1 1 1

zn W 1 1 1 B

1 1 1

znW 1 2 2 C

2 2 1

zn W A

fnW 2 2 2 BC

2 2 2

fnW 2 2 2 BD

2 2

e

2 2

CDn W 1 2 2 zn W D

1 1

2 2

2 2

1 1 1 1 1 1

z6 W E

1 2 1 2 3 2 1

z7 W E

2 1 2 3 4 3 2 1

z8 W E

3 2 4 6 5 4 3 2 1


451

Fz41 W 1 2 3 2 1 Fz42 W 1 2 3 4 2 z 21 W 1 2 1 G z 22 W 1 2 3 G (ii) Let I be the set of vertices of and d W I ! NC be a subadditive function for the Cartan matrix C . Take the additive function e for the transpose Cartan in the matrix C T , associated to the Euclidean graph op to , listed opposite above table. Then we obtain that e.dC /T D e C T d T D eC T d T D 0, where d and e are considered as the corresponding one row matrices indexed by I . Since all coefficients of the matrix e are strictly positive and all coefficients of the matrix .dC /T are nonnegative, we conclude that .dC /T D 0, and hence dC D 0. Therefore, d is an additive function for C . Let C 0 and C be indecomposable Cartan matrices on finite sets I 0 and I , respectively. We say that C 0 is strictly smaller than C if there is an injection W I 0 ! I such that jCij0 j jC .i/ .j / j for all i; j 2 I 0 , and either .I 0 / ¤ I or jCij0 j < jC.i/ .j / j for some i ¤ j in I 0 . Lemma 13.2. Let C 0 and C be indecomposable Cartan matrices on finite sets I 0 and I , respectively. Assume W I 0 ! I is an injection such that C 0 is strictly smaller than C . Then for any subadditive function d W I ! NC for C the composition d 0 D d W I 0 ! NC is a subadditive but not additive function on C 0 . Proof. Let d W I ! NC be a subadditive function for C . Then for any j 2 I 0 we have the inequalities X X X di0 Cij0 d .i/ C .i/ .j / dl Cl.j / 0; i2I 0

i2I 0

l2I

and hence d 0 is a subadditive function for C 0 . In order to prove that d 0 is not an additive function for C 0 , we have two cases to consider. Assume first that .I 0 / ¤ I . Since the Cartan matrices C 0 and C are indecomposable, we may choose k 2 I n .I 0 / and j 2 I 0 such that Ck.j / ¤ 0. Then we obtain the inequalities X X X di0 Cij0 d .i/ C .i/ .j / > d.i/ C.i/.j / C dk Ck.j / i2I 0

i2I 0

X l2I

and so d 0 is not additive.

i2I 0

dl Cl.j / 0;

452


Assume jCi00 j0 j < jC0 .i0 /.j0 / j for some i0 ; j0 2 I 0 . Then we have the inequalities X X X di0 Cij0 0 > d .i/ C .i/ .j0 / dl Cl.j0 / 0; i2I 0

i2I 0

l2I

0

which shows that d is not additive.

The following characterizations of Dynkin and Euclidean graphs are due to E. B. Vinberg [Vin] and S. Berman, R. Moody and M. Wonenburger [BMW] (see also [HPR]). Theorem 13.3. Let C be the Cartan matrix on a finite set I and .C / the associated valued graph. Assume that d is a subadditive function for C . Then the following statements hold. (i) .C / is either a Dynkin graph or a Euclidean graph. (ii) If d is not additive, then .C / is a Dynkin graph. Proof. (i) Suppose .C / is neither a Dynkin graph nor a Euclidean graph. Then there is a subset I 0 of I and a Cartan matrix C 0 on I 0 such that C 0 is strictly smaller than C , via the inclusion map W I 0 ! I , and .C 0 / is a Euclidean graph. Then d 0 D d W I 0 ! NC is, by Lemma 13.2, a subadditive but not additive function for C 0 . On the other hand, by Lemma 13.1 (ii), every subadditive function for C 0 is additive, since .C 0 / is a Euclidean graph. This gives a contradiction. Therefore, .C / is either a Dynkin graph or a Euclidean graph. The statement (ii) follows from (i) and Lemma 13.1 (i). Corollary 13.4. Let be a finite connected valued graph without loops and multiple edges. Then is a Dynkin graph if and only if the Cartan matrix C. / associated to admits a subadditive but not additive function. Proof. In case is a Dynkin graph, C. / is strictly smaller than the Cartan matrix z associated to the Euclidean graph . z Then the required equivalence follows C. / from Lemma 13.2 and Theorem 13.3.

14 Canonical mesh algebras of Dynkin type Let K be a field and be a Dynkin graph An .n 1/, Bn .n 2/, Cn .n 3/, Dn .n 4/, E6 , E7 , E8 , F4 , or G2 . We associate to the double quiver Q as follows. QAn W

.n 1/

o 0

a0 aN 0

/ o 1

a1 aN 1

/ o 2

: : :

/ o n2

an2 aN n2

/ n1

14. Canonical mesh algebras of Dynkin type

QBn W

.n 2/

QCn W

.n 3/

QDn W

.n 4/

QE6 W

QE7 W

QE8 W

QF4 W

2n 5o a2n3 2n 3 0 00 G 00 aN 00 2n2 : : : 00 000 aN 2n3 00 : : : oa 2n 4 2n2 2n 2

1 o a3 3 o 0 > } G a1 }}}} 00 00 aN }}}} 00 4 ~}}}}} aN 2 0 0 ÀAA 000 AAAAaN 1 AAAA 00aN 3 0 a2 AAA o a o 4 2 4 0 ÀAA AAAAa0 AAAA aN 1 AAA a2 / o o > } } } a N aN 0 } } 2 2 3 }}}}}a } 1 ~}}} 1 0 ÀAA AAAAa0 AAAA aN 0 AAA } > o aN 1 }}}}2 }}}} ~}}}}} a1 1

a2 aN 2

o 1

aN 1

/ o 2

a2

/ o 2

a2

aN 2

aN 0

o 1

a1 aN 1

aN 2

aN 0

o 1

o 1

a1 aN 1

a1 aN 1

: : :

/ o 3

: : :

/ o n1

an1

/

aN n1 n

an2 aN n2

/ n

a0

/ o 3 O 0

a3 aN 3

/ o 4

a2

a4 aN 4

/ 5

a0

/ o 3 O 0

a3 aN 3

/ o 4

a4 aN 4

/ o 5

a5

/ o 5

a5

aN 5

/ 6

a0

/ o a3 / o aN 2 3 aN 3 4 2 o a4 4 0 > } G a2 }}}} 00 }} 00 aN }}}}}aN 3 00 5 } ~ } / A 00 0 ÀAAAAAaN 2 000 aN AAAA 00 4 a3 AAA o a 5 3 5 / o 2

/ o n1

O 0 aN 0

a1

: : :

a4 aN 4

aN 5

/ o 6

a6 aN 6

/ 7

453

454


> 2 }}}} } }} }}}}}a2 } a1 } / ~ o QG2 W A 1 aN 3 0 ÀAAAAAaN 2 AAAA a3 AAA 3 Our rule for the labelling of arrows comes from the automorphism of Q where: aN 1

(1) D idQ for D An ; Dn ; E6 ; E7 ; E8 ; (2) is the canonical automorphism of Q of order 2 for D Bn ; Cn ; F4 ; (3) is the canonical automorphism of Q of order 3 for D G2 , with .1/ D 2, .2/ D 3, .3/ D 1, .0/ D 0. Then for each arrow a of Q there is a unique arrow in Q of the form t .a/

aN

/ .s.a// :

Therefore, for a given vertex v of Q , we have in Q a mesh of the form t .a1 / MMMaN 1 t: t MMM t t t & tt :: .v/ V JJ : aN r qq8 JJaJr q JJ $ qqq t .ar / a1

given by all arrows a1 ; : : : ; ar of Q1 having v as the source, where r 2 f1; 2; 3g. We define the canonical mesh algebra (see [ES]) associated to to be the bound quiver algebra ƒ. / D KQ =I , where KQ is the path algebra of Q and I is the admissible ideal of KQ generated by the elements of the form X aaN a;s.a/Dv

for all vertices v of Q . When is a Dynkin graph of type An .n 1/, Dn .n 4/, E6 , E7 , E8 (with trivial valuations .1; 1/), then the mesh algebra ƒ. / coincides with the preprojective algebra P . /, introduced by I. M. Gelfand and V. A. Ponomarev [GePo]. We note that ƒ.A1 / D K, and hence is a simple algebra. On the other hand, for ¤ A1 , the set of arrows of Q is nonempty, and hence ƒ. / D KQ =I is a basic, indecomposable but not semisimple K-algebra, by Proposition I.3.15,

15. The Riedtmann–Todorov theorem

455

Lemma I.6.16, and Lemma II.6.17. In particular, it follows from Propositions 5.4 and 11.8 that, for ¤ A1 , ƒ. / is an indecomposable nonprojective module in mod ƒ. /e . Theorem 14.1. Let be a Dynkin graph different from A1 , K a field, and ƒ. / the canonical mesh algebra of over K. Then the following statements hold. (i) ƒ. / is a finite dimensional Frobenius K-algebra. (ii) 3ƒ./ .S / is simple for any simple module S in mod ƒ. /. (iii) ƒ. / is a periodic algebra. We have also the following theorem proved by A. Schofield [Sch], and K. Erdmann and N. Snashall [ESn]. Theorem 14.2. Let P . / be the preprojective algebra of a Dynkin type An .n 2/, Dn .n 4/, E6 , E7 , E8 . Then P6 ./e .P . // Š P . / in mod P . /e .

15 The Riedtmann–Todorov theorem The aim of this section is to describe a general shape of the Auslander–Reiten quiver A of a finite dimensional selfinjective K-algebra A of finite representation type over a field K. Let A be a finite dimensional selfinjective K-algebra over a field K. We know from Proposition III.8.6 that, for an indecomposable projective (equivalently, injective) right A-module P we have in mod A an almost split sequence of the form 0 ! rad P ! .rad P = soc P / ˚ P ! P = soc P ! 0: We denote by As the translation quiver, called the stable Auslander–Reiten quiver of A, obtained from A by removing all indecomposable projective-injective modules and the arrows attached to them. Observe that the Auslander–Reiten translations A D D Tr and A1 D Tr D are mutually inverse operations defined on all modules of As . We also note that we may recover the whole Auslander–Reiten quiver A of A from its stable Auslander–Reiten As if we know the positions of all indecomposable modules P = soc P (equivalently, rad P ) in As , for all indecomposable projective right A-modules P . Lemma 15.1. Let A be a finite dimensional selfinjective K-algebra of finite representation type over a field K. Then there is a common bound on the length of sectional paths in A .

456


Proof. Since A is of finite representation type, there is a common bound on the length of indecomposable modules in mod A. Take a positive integer m such that `.X/ m for any indecomposable module X in mod A. Let .a1 ;b1 /

.a2 ;b2 /

.an1 ;bn1 /

.an ;bn /

X0 ! X1 ! : : : ! Xn1 ! Xn be a sectional path in A . Then there is a sectional path of irreducible homomorphisms in mod A, f1

f2

fn1

fn

X0 ! X1 ! ! Xn1 ! Xn ; and we have fn : : : f2 f1 ¤ 0, by Theorem III.11.2. Applying now the Harada–Sai Lemma III.2.1, we obtain that n < 2m 1. Therefore, the length of any sectional path in A is bounded by 2m 1. Proposition 15.2. Let A be a finite dimensional selfinjective K-algebra over a field K. Assume there exist in A different sectional paths .a1 ;b1 /

.ar ;br /

X D M0 ! M1 ! Mr1 ! Mr D Y; .c1 ;d1 /

.cs ;ds /

X D N0 ! N1 ! Ns1 ! Ns D Z; with Y D Am Z for some integer m. Then A is of infinite representation type. Proof. We may assume that Mi ¤ At Nj in A for all i 2 f1; : : : ; r 1g, j 2 f1; : : : ; s 1g, and integers t. Moreover, in order to simplify notation, in the considerations below the valuations of arrows in A will be omitted. Since A is a selfinjective algebra, it follows from Proposition III.8.6 that every sectional path in A passing through an indecomposable projective module P has the target arrow rad P ! P or the source arrow P ! P = soc P , and rad P D A .P = soc P / in A . Hence, we conclude that all indecomposable modules Mi and Nj , for i 2 f0; 1; : : : ; rg and j 2 f0; 1; : : : ; sg, are nonprojective. Therefore, we obtain, using again the selfinjectivity of A, the families of indecomposable nonprojective right A-modules An Mi and An Nj , for all i 2 f0; 1; : : : ; rg, j 2 f0; 1; : : : ; sg, and integers n. Moreover, by the imposed assumption, we have also Ap Mi ¤ Aq Nj in A , for all integers p; q and i 2 f1; : : : ; r 1g, j 2 f1; : : : ; s 1g. Observe now that we have in A a sectional path As Z D As Ns ! As1 Ns1 ! ! A N1 ! N0 D X; induced by the sectional path X D N0 ! N1 ! ! Ns1 ! Ns D Z:


457

Then we obtain in A a sectional path As Z D As Ns GF @A

/X

/ s1 Ns1 A

/ :::

/ A N1

/ M1

/ :::

/ Mr1

BC ED / Mr D Y

of length r C s, because A M1 ¤ A N1 . Applying Ams to this path, we obtain in A a sectional path Y D Am Z GF @A

/ ms X A

/ m1 Ns1 A

/ :::

/ msC1 N1 A BC ED

/ ms M1 A

/ :::

/ ms Mr1 A

/ ms Mr D ms Y A A

of length r C s. Let a D m s and b D m 1. Then the above sectional path in A is of the form Y ! Ab Ns1 ! ! Aa Mr1 ! Aa Y; and is of length r C s. Hence, we obtain in A sectional paths Ata Y ! AtaCb Ns1 ! ! A.tC1/a Mr1 ! A.tC1/a Y of length r C s, for all nonnegative integers t . Since Ap Mi ¤ Aq Nj in A for all integers p and q, combining these sectional paths, we conclude that, for any positive integer t , we have in A a sectional path Y ! Aa Y ! ! A2a Y ! Ata Y of length t.r C s/. Applying Lemma 15.1, we conclude that A is of infinite representation type. x is one A valued quiver is said to be a Dynkin quiver if its underlying graph of the Dynkin graphs An .n 1/, Bn .n 2/, Cn .n 3/, Dn .n 4/, E6 , E7 , E8 , F4 , or G2 . Proposition 15.3. Let be a Dynkin quiver and G an admissible group of automorphisms of the valued translation quiver Z . Then G is an infinite cyclic group and contains a power m , with m 1, of the translation of Z . Proof. The proof is obtained by a careful analysis of the shapes of the valued translation quivers Z for all Dynkin quivers (see Exercises III.12.38 and III.12.40– III.12.46) and is left to the reader.

458


Proposition 15.4. Let A be a nonsimple indecomposable finite dimensional selfinjective K-algebra of finite representation type over a field K. The following conditions are equivalent. (i) As is not connected. (ii) As has exactly one A -orbit. (iii) A is a Nakayama algebra of Loewy length 2. (iv) As Š ZA1 =. n / for a positive integer n. Proof. It follows from the assumption and Proposition 6.8 that every simple module in mod A is nonprojective, and hence As admits at least one A -orbit. Then As is not connected if and only if the almost split sequences in mod A are of the form 0 ! rad P ! P ! P = soc P ! 0; where P are projective-injective indecomposable modules and rad P and soc P are simple modules. Moreover, by Theorem III.8.7, the almost split sequences of Nakayama selfinjective algebras of Loewy length 2 are exactly of the above form. Then the conditions (i), (ii), (iii), (iv) are easily equivalent. Lemma 15.5. Let A be a nonsimple indecomposable finite dimensional selfinjective K-algebra of finite representation type over a field K, and X , Y be indecomposable nonprojective right A-modules lying in different A -orbits of A . Then there is in As a sectional path from X to Am Y , for some integer m. Proof. It follows from the assumption that As has at least two different A -orbits, and hence As is connected, by Proposition 15.4. Then there is in As a walk X D Z0

.a1 ;b1 /

Z1

.a2 ;b2 /

Z2

.ai ;bi /

:::

Zn1

.an ;bn /

Zn D Y

.ai ;bi /

.ai ;bi /

with n 1, where Zi1 Zi means Zi1 ! Zi or Zi1 Zi for any i 2 f1; : : : ; ng. Observe that the claim is obvious for n D 1. Let n 2. We may assume by induction that there is in As a sectional path .c1 ;d1 /

.c2 ;d2 /

.cs ;ds /

Z1 D N0 ! N1 ! N2 ! ! Ns1 ! Ns D Ap Y with s 1 and p an integer. Observe that, if X D A N1 , then we have in As a sectional path .c2 ;d2 /

.cs ;ds /

X D A N1 ! A N2 ! ! A Ns1 ! A Ns D ApC1 Y:


459

Similarly, if X D A1 N1 , then we have in As a sectional path .c2 ;d2 /

.cs ;ds /

X D A1 N1 ! A1 N2 ! ! A1 Ns1 ! A1 Ns D Ap1 Y: Clearly, if X D N1 , then we have in As a sectional path .c2 ;d2 /

.cs ;ds /

X D N1 ! N2 ! ! Ns1 ! Ns D Ap Y: Therefore, assume that X ¤ A N1 , X ¤ A1 N1 and X ¤ N1 in As . Then, if Z0

.a1 ;b1 /

.a1 ;b1 /

Z1 is an arrow Z0 ! Z1 , we have in As a sectional path .a1 ;b1 /

.c1 ;d1 /

.cs ;ds /

X D Z0 ! N0 ! N1 ! ! Ns1 ! Ns D Ap Y: Finally, if Z0 path .b1 ;a1 /

.a1 ;b1 /

Z1 is an arrow Z0

.c1 ;d1 /

.a1 ;b1 /

Z1 , then we have in As a sectional .cs ;ds /

X ! A1 N0 ! A1 N1 ! ! A1 Ns1 ! A1 Ns D Ap1 Y: Summing up, in all above cases, we have a sectional path in As from X to Am Y for an integer m. The following theorem has been proved by C. Riedtmann in [Rie] for algebras over an algebraically closed field and extended by G. Todorov in [Tod] to algebras over an arbitrary field. Theorem 15.6. Let A be a nonsimple indecomposable finite dimensional selfinjective K-algebra of finite representation type over a field K. Then there exist a Dynkin quiver and an admissible infinite cyclic group G of automorphisms of the valued translation quiver Z such that the stable Auslander–Reiten quiver As of A is isomorphic to the orbit valued translation quiver Z =G. Proof. Since A is a selfinjective algebra of finite representation type, we conclude from Proposition 10.2 that the module category mod A is A -periodic. Hence, there exists a positive integer m such that Am X Š X for any indecomposable nonprojective module X in mod A. Further, it follows from Lemma III.9.1 and Proposition III.9.6 that, if X; Y are indecomposable nonprojective modules in mod A and there is an arrow from X to Y in A , then we have in As the valued arrows 1 X A X: X :: .d ;d 0 / A === .d ;d 0 / A@ ??? .dX Y ;d 0 / XY ?? == X Y X Y :: XY XY ?? == :: ?? = :: 0 == .d 0 ;dX Y / ? ;dXY / : .dXY XY A1 Y . A Y Y

460


Therefore, A is an automorphism of the stable valued translation quiver As such that Am is the identity automorphism of As . Further, since A is an indecomposable algebra of finite representation type, it follows from Theorem III.10.2 that A is connected. Moreover, by Proposition 15.4, we may assume that As is connected and has at least two A -orbits. Fix a vertex (module) M in As . We denote by D M the full valued subquiver of the valued quiver As formed by all modules N in As lying on sectional paths .a1 ;b1 /

.a2 ;b2 /

.as1 ;bs1 /

.as ;bs /

M D M0 ! M1 ! ! Ms1 ! Ms D N in As starting from M . It follows from Lemmas 15.1, 15.5 and Proposition 15.2 that is a finite valued tree and intersects every A -orbit in As exactly once. Clearly, M is a unique source of . We will show that is a Dynkin quiver. x the underlying graph (tree) of and by C D C. / x the We denote by x which is also the set 0 of associated Cartan matrix on the set I of vertices of , vertices of D M . We consider also the transposed Cartan matrix C T of C . Observe that for each arrow 0 .dXY ;dXY /

X ! Y 0 T 0 in , we have CXY D dXY and CYX D dXY , and hence CXY D dXY and T T CYX D dXY . Clearly, we have CXX D 2 D CXX for any module X on . We claim that C T admits a subadditive but not additive function. Let Y be an indecomposable nonprojective module in mod A, equivalently a vertex of As . Then it follows from Lemma III.9.1 and Proposition III.9.6 that As admits a valued mesh of the form ;dX1 Y /nnn7

X1 OO OOO.dX Y ;d 0 / n n OOO1 X1 Y n n n n 2 n X X e X e XXXXXXOOOOO n 2 e XXXXO' neneneeee0eeee A Y SSS .dX2 Y ;dX2 Y / :: .dX2 Y ;dX0 2 Y / lXl,5 Y SSS : lll SSS lll 0 l S l S l S 0 ll .dXr Y ;dXr Y / .dX ;d / S) r Y Xr Y Xr 0 .dX

1Y

such that there is in mod A an almost split sequence 0 ! A Y !

r M

0 dX

Xj

jY

˚ PY ! Y ! 0;

j D1

where PY D 0 or PY is an indecomposable projective module. In particular, we obtain the inequalities `.Y / C `.A Y / D `.PY / C

r X j D1

`.Xj /dX0 j Y

r X j D1

`.Xj /dX0 j Y ;

461


and the right inequality is strict if and only if PY ¤ 0. Consider now the function d W I ! NC D f1; 2; 3; : : : g defined for any module X in I by m1 X `.Ai X /: d.X / D iD0

Observe that d.X / D d.A X / for any module X in , because Am Z D Z for any module Z in As . Then, for any module Y in I , we have the inequalities X X T T d.X/CXY D d.Y /CYTY C d.X /CXY X2I nfY g

X2I

X

D 2d.Y /

0 d.X /dXY

X2I nfY g

D

D

m1 X

`.Ai Y /

C

`.AiC1 Y /

iD0

X2I nfY g

m1 X

m1 X

`.Ai Y / C `.AiC1 Y /

iD0

D

m1 X

X m1 `.Ai Y / C `.AiC1 Y /

X

X X2I nfY g

iD0

0 `.Ai X /dXY

`.Ai X /d0 i X i Y A

X2I nfY g

iD0

`.Ai Y / C `.AiC1 Y /

iD0

X

m1 X

0 `.Ai X / dXY

X2I nfY g

iD0

iD0

D

m1 X

X

`.Ai X /d0 i X i Y A

A

A

0; because there are in mod A almost split sequences 0 ! AiC1 Y !

r M d0 i i .Ai Xj / A XA Y ˚ P i Y ! Ai Y ! 0; A

j D1

with d 0 i

iY A XA

0 D dXY , P i Y D 0 or P i Y an indecomposable projective module, A

A

for all i 2 f0; 1; : : : ; m 1g. Moreover, X T d.X /CXY > 0; X2I

if P i Y ¤ 0, for some i 2 f0; 1; : : : ; m 1g. Since the valued quiver intersects A every A -orbit of As and A contains projective modules, we conclude that there are modules Y on such that P i Y ¤ 0 for some i 2 f0; 1; : : : ; m 1g. Therefore, we A

462


conclude that d W I ! NC is a subadditive but not additive function for the Cartan matrix C T . Then it follows from Corollary 13.4 that the valued graph .C T / of C T is a Dynkin graph. Observe now that .C T / is the graph obtained from the x of by changing the valuations of all edges underlying graph X

0 / .dXY ;dXY

on

Y

X

0 .dX Y ;dX Y /

Y :

x is also a Dynkin Hence, looking on the shapes of Dynkin graphs, we conclude that graph, and consequently is a Dynkin quiver. Consider now the valued translation quiver Z of D M . Recall that .Z /0 D Z 0 D f.i; X / j i 2 Z; X 2 0 g is the set of vertices of Z . Further, the set .Z /1 of arrows of Z consists of the valued arrows 0 / .dXY ;dXY

0 .dX Y ;dX Y /

.i; X / ! .i; Y / and .i C 1; Y / ! .i; X / for all arrows

0 .dXY ;dXY /

X ! Y in and all i 2 Z. Moreover, the translation W .Z /0 ! .Z /0 is defined by .i; X/ D .i C 1; X /. Then we have the canonical covering of valued translation quivers W Z ! As such that 0 W .Z 0 / ! .As /0 is given by 0 .i; X / D Ai X for any i 2 Z and any X 2 0 , and 1 W .Z /1 ! .As /1 assigns to a valued arrow 0 .dXY ;dXY /

.i; X / ! .i; Y / the valued arrow

0 .dXY ;dXY /

Ai X ! Ai Y 0 with dXY D d i X i Y and dXY D d 0i A

iY A XA

A

of Z of As ;

, and to a valued arrow

0 .dXY ;dXY /

.i C 1; Y / ! .i; X / the valued arrow

0 ;dXY / .dXY

AiC1 Y ! Ai X 0 with dXY D d i C1 Y i X and dXY D d 0 i C1 A

A

A

iX Y A

of Z of As ; . Observe also that evidently

W Z ! As is a covering of valued translation quivers.


463

Finally, let G be the group of all automorphisms g of the valued translation quiver Z such that, for vertices .i; X / and .j; Y / in .Z /0 , we have g.i; X / D .j; Y / if and only if Ai X D 0 .i; X / D 0 .j; Y / D Aj Y in As . We note that, for X and Y in 0 , the equality Ai X D Aj Y in As forces X D Y , because intersects every A -orbit in As exactly once. A simple checking shows that G is an admissible group of automorphisms of Z , and clearly contains m because Am Z D Z for any module Z in As . Moreover, by Proposition 15.3, G is an infinite cyclic group. It follows from definition of G, that the covering W Z ! As of valued translation quivers induces an isomorphism Z =G ! As of valued translation quivers.

Let A be a nonsimple indecomposable finite dimensional selfinjective K-algebra of finite representation type over a field K. Then it follows from Exercises III.12.40– III.12.46 and Theorem 15.6 that we may associate to A a Dynkin graph .A/, called the Dynkin type of A, such that As Š Z =G for a valued Dynkin quiver with x D .A/ and an admissible infinite cyclic group G of automorphisms of Z . We also note that in view of Proposition III.9.12, Theorem 15.6 gives a complete description of the shapes of the stable Auslander–Reiten quivers of finite dimensional selfinjective algebras of finite representation type over an arbitrary field. Example 15.7. Let A be a nonsimple indecomposable finite dimensional selfinjective Nakayama K-algebra over a field K. Then it follows from Theorems I.10.5 and I.10.7 that A is of finite representation type and every indecomposable module in mod A is isomorphic to a module of the form P = radr P for some indecomposable projective module P in mod A and an integer r with 1 r ``.P / D `.P /. Moreover, if r < ``.P /, then we have in mod A an almost split sequence of the form 0 ! rad P = radrC1 P ! .rad P = radr P / ˚ P =.radrC1 P / ! P = radr P ! 0; by Theorem III.8.7. We note that, for r D ``.P /, we have radr P D 0, and hence P = radr P is a projective, and hence also injective, module in mod A. Since A is indecomposable as K-algebra and of finite representation type, it follows from the Auslander’s Theorem III.10.2 that the Auslander–Reiten quiver A of A is connected. Moreover, since A is nonsimple, all simple modules in mod A are nonprojective (see Proposition 6.8). Let ``.A/ D m C 1 and n be the number of pairwise nonisomorphic simple modules in mod A. Observe that, by the description of almost split sequences in mod A, we have `.A X / D `.X / for any indecomposable nonprojective module X in mod A. Therefore, we conclude that the stable Auslander–Reiten quiver As has exactly m A -orbits, and every A -orbit

464


in As consists of n indecomposable modules having a fixed length r 2 f1; : : : ; mg. Moreover, the maximal sectional paths in A are of the forms soc.P / ! ! rad P ! P; P ! P = soc.P / ! ! top.P /; and hence, the maximal sectional paths in As are of the forms soc.P / ! ! rad P; P = soc.P / ! ! top.P /; for indecomposable projective modules P in mod A. In particular, we obtain that As is isomorphic to the orbit translation quiver Z =. n /, where is the linear quiver / / ::: / / ; m 1 2 m1 and hence A is of Dynkin type .A/ D Am , with m D ``.A/ 1. Example 15.8. Let Q be the quiver 3 4 2 @@ @@ ~~ ~ @ˇ ~~ ˛ @@ ~~ 1 and A D KQ the path algebra of Q over a field K. It follows from Example III.9.11 (b) thatA is of finite representation type and the Auslander–Reiten quiver A of A is of the form P .2/ S.2/ D I.2/ > X2 CC DD ; < CC xx }} yy DD x y } CC x y } DD CC xx yy DD }} yy xx ! ! }} / X3 / S.3/ D I.3/ / P .3/ /M / I.1/ S.1/ D P .1/ EE = FF z= AAA { z FF E { AA EE zz FF {{ AA EE zz FF {{ z A { E" # z { P .4/ S.4/ D I.4/ , X4 where S.i /, P .i /, I.i / denote the simple, projective, injective module in mod A D repK .Q/ at the vertex i 2 f1; 2; 3; 4g, respectively. Consider the trivial extension algebra T.A/ D A Ë D.A/. Then T.A/ is iso-


465

morphic to the bound quiver algebra K =J , where is the quiver 3 2 @_@@ @@@@˛0 ˇ ~~~~~? @@@@ ~~~~ ˛ @@@ ~~~~ ˇ 0 ~ 1 O 0

4

and J is the admissible ideal of the path algebra K of generated by the elements ˛ 0 ˛ ˇ 0 ˇ, ˇ 0 ˇ 0 , ˛ˇ 0 , ˛ 0 , ˇ˛ 0 , ˇ 0 , ˛ 0 , ˇ 0 . Observe that the path algebra A0 D KQ0 of the quiver O 3 4 2 @_ @ ~? @@ 0 ~ @@ˇ ~~ @ ~~~ 0 ˛0 1 is also a quotient algebra of T.A/, and we have isomorphism T.A/ Š T.A0 / of K-algebras. Since Q0 D Qop , we have A0 Š Aop as K-algebras. Hence, A0 is an algebra of finite representation type and the Auslander–Reiten quiver A0 of A0 is of the form S.2/0 D P .2/0

Y2 = EE == {= { EE == {{ EE { == EE { { = " { / P .1/0 /N / Y3 S.3/0 D P .3/0 CC @ y< C y CC yy CC yy y C y ! Y4 S.4/0 D P .4/0

I.2/0 > DD } DD }} DD } } DD } " }} 0 / I.1/0 D S.1/0 , / I.3/ AA < AA zz AA zz z AA z zz 0 I.4/

where S.i /0 , P .i /0 , I.i /0 denote the simple, projective, injective module in mod A0 D repK .Q0 / at the vertex i 2 f1; 2; 3; 4g, respectively. Clearly, in mod T.A/ D repK . ; J / D mod T.A0 /, we have S.i / D S.i /0 for any i 2 f1; 2; 3; 4g. Further, for i 2 f1; 2; 3; 4g, let Pz .i / be the indecomposable projective-injective module in mod T.A/ with top.Pz .i // D S.i / and soc.Pz .i // D S.i /, since T.A/ is a symmetric, hence weakly symmetric K-algebra (Corollary 6.4). Applying Proposition III.8.6, we conclude that we have in mod T.A/ almost split sequences of the forms 0 ! rad Pz .1/ ! .rad Pz .1/=S.1// ˚ Pz .1/ ! Pz .1/=S.1/ ! 0; 0 ! rad Pz .2/ ! .rad Pz .2/=S.2// ˚ Pz .2/ ! Pz .2/=S.2/ ! 0; 0 ! rad Pz .3/ ! .rad Pz .3/=S.3// ˚ Pz .3/ ! Pz .3/=S.3/ ! 0; 0 ! rad Pz .4/ ! .rad Pz .4/=S.4// ˚ Pz .4/ ! Pz .4/=S.4/ ! 0:

466


Invoking now Proposition I.8.27, we deduce that the indecomposable modules occurring in the above four almost split sequences are of the form as shown below: K AÀA 1 0 01 } > K AAAA }}}} AAA ~}}}}}1 0 0 Pz .1/ W 1 K 2O 10

K

K K K ?_?? K ?_?? ????1 0 ? ????0 1 ? ? ? ? ? 0 ? 1 ? 0 1 z z K KO rad P .1/ W P .1/=S.1/ W O

0 1

0 K 2AÀA 01 @ AAAA AAA 10 KO Pz .2/ W

0

1

0 K ?_?? @ ????1 ??? 0 KO rad Pz .2/ W

1

0 ; :: >> .. 88 >> ;; >> : > > >> 88 ;; : > . > 0 / S.1/ / P .3/ 0 / / / / / / N / I.3/0 / I.1/ S.3/ X Y I.3/ P .1/ M 2 3 A AA .2;1/ ~? } ~ AA ~ }} AA ~~ }} }} ~~ .1;2/ P1 I1 . .1;2/

Moreover, S1 D P1 is a simple projective but noninjective module and S2 D I2 is a simple injective but nonprojective module in mod A. Further, rad P2 D S1 ˚ S1 is semisimple and projective. Therefore, A is a hereditary R-algebra (see Theorem I.9.1).

468


We will describe the Auslander–Reiten quiver T.A/ of the trivial extension algebra T.A/ of A. We have in T.A/ the standard primitive idempotents eQ1 D .e1 ; 0/ and eQ2 D .e2 ; 0/ with 1T.A/ D .1A ; 0/ D eQ1 C eQ2 . Hence, we have in mod T.A/ the indecomposable projective modules Pz1 D eQ1 T.A/ D e1 A ˚ e1 D.A/ D e1 A ˚ D.Ae1 /; Pz2 D eQ2 T.A/ D e2 A ˚ e2 D.A/ D e2 A ˚ D.Ae2 /; Since T.A/ is a symmetric algebra, Pz1 and Pz2 are also indecomposable injective modules in mod T.A/ with soc.Pz1 / Š top.Pz1 / and soc.Pz2 / Š top.Pz2 /, because every symmetric algebra is weakly symmetric (see Corollary 6.4). Moreover, A is the quotient algebra T.A/ by the two-sided ideal D.A/ D 0 ˚ D.A/ with D.A/2 D 0. Further, rad A ˚ D.A/ is a two-sided ideal of T.A/ such that .rad A ˚ D.A//.rad A ˚ D.A// rad2 A ˚ D.rad A/: Hence, rad A ˚ D.A/ is a nilpotent ideal of T.A/, because rad A is a nilpotent ideal of A (Corollary I.3.4), and T.A/=.rad A˚D.A// ! A= rad A D RC. Applying Lemma I.3.5, we conclude that rad T.A/ D rad A ˚ D.A/. As a consequence, we obtain that rad Pz1 D eQ1 .rad A ˚ D.A// D e1 .rad A/ ˚ e1 D.A/ D rad e1 A ˚ D.Ae1 /; rad Pz2 D eQ2 .rad A ˚ D.A// D e2 .rad A/ ˚ e2 D.A/ D rad e2 A ˚ D.Ae2 /: Similarly, we have Pz1 = soc.Pz1 / D Pz1 =S1 D .e1 A ˚ D.Ae1 // =S1 D e1 A ˚ .D.Ae1 /=S1 /; Pz2 = soc.Pz2 / D Pz2 =S2 D .e2 A ˚ D.Ae2 // =S2 D e2 A ˚ .D.Ae2 /=S2 /: Since P1 D e1 A is simple, we have rad e1 A D 0, and so rad Pz1 D D.Ae1 / D I1 . Similarly, since I2 D D.Ae2 / is simple, we have D.Ae2 /=S2 D 0, and hence Pz2 = soc.Pz2 / D e2 A D P2 . Moreover, since Pz1 and Pz2 are indecomposable projective-injective right T.A/-modules, applying Proposition III.8.6, we conclude that there are in mod T.A/ almost split sequences of the form 0 ! rad Pz1 ! .rad Pz1 =S1 / ˚ Pz1 ! Pz1 =S1 ! 0; 0 ! rad Pz2 ! .rad Pz2 =S2 / ˚ Pz2 ! Pz2 =S2 ! 0: Further, it follows from Proposition 10.12 that T.A/ I1 D A I1 D P1 and T.A/ I2 D A I2 D P2 , and consequently the almost split sequences 0 ! P1 ! P2 ! I1 ! 0; 0 ! P2 ! I1 ˚ I1 ! I2 ! 0


469

1 I1 D in mod A remain almost split sequences in mod T.A/. Observe that T.A/ .1;2/

1 T.A/ .rad Pz1 / D Pz1 =S1 and T.A/ has the valued arrow I1 ! I2 . Hence, .2;1/

applying Proposition III.9.6, we infer that T.A/ admits the valued arrow I2 ! Pz1 =S1 , and it is the unique arrow of T.A/ with source I2 . This leads to an almost split sequence in mod T.A/ of the form 1 I2 ! 0: 0 ! I2 ! .Pz1 =S1 / ˚ .Pz1 =S1 / ! T.A/

Hence, we conclude that ` 1 I2 D 2`.Pz1 =S1 / `.I2 / D 2.`.P1 / C `.I2 // `.I2 / D 3; T.A/

because P1 D S1 and I2 D S2 are simple modules in mod T.A/. Further, applying .1;2/ Proposition III.9.6 again, we infer that T.A/ admits the valued arrow Pz1 =S1 ! 1 I2 , and hence we have in mod T.A/ an almost split sequence T.A/ 1 1 0 ! Pz1 =S1 ! T.A/ I2 ! T.A/ .Pz1 =S1 / ! 0:

Hence, we obtain that 1 1 .Pz1 =S1 / D ` T.A/ I2 `.Pz1 =S1 / D 3 2 D 1; ` T.A/ 1 and so T.A/ .P1 =S1 / is simple right T.A/-module. Observe that S2 D I2 and 1 T.A/ S2 D A S2 D P2 . Moreover, T.A/ T.A/ .P1 =S1 / D P1 =S1 . Since 1 top.P2 / D S2 and top.P1 =S1 / D S1 , we conclude that T.A/ .P1 =S1 / D S1 D P1 . Summing up, we have proved that T.A/ admits a connected component C of the form

z ? P2 ?? ~ ?? ~~ ?? ~~ ?? ~ ~ z P2 rad ? I2 ?? >} P2@@ ? >>> ?? .2;1/ @@ .1;2/ .1;2/ }} .1;2/ >> ?? @@ } >> } ?? @ .2;1/ .2;1/ > @ }} z z P I P1 =S1 P? 1 =S1 : 1 1> >> >> >> > Pz1 Then using the indecomposability of T.A/ and Auslander’s Theorem III.10.2, we conclude that C D A and A is of finite representation type. Moreover, the stable Auslander–Reiten quiver is isomorphic to the orbit valued translation quiver

470


Z =. 3 /, where is the Dynkin quiver

.1;2/

/ .

In particular, T.A/ is of Dynkin type .T.A// D B2 . In the second volume of the book we will develop techniques allowing us to show that, for any Dynkin quiver and admissible group G of automorphisms of the valued translation quiver Z , there exists a finite dimensional selfinjective K-algebra A of finite representation type over a field K such that As Š Z =G as valued translation quivers.

16 Exercises 1. Let K be a field, Q the quiver o 1

˛

/ o 2

ˇ

/ ; 3

I the ideal in KQ generated by ˇ˛, , ˛ ˇ, and A D KQ=I the associated bound quiver algebra. Prove that A is a symmetric K-algebra and describe a nondegenerate associative symmetric K-bilinear form .; / W A A ! K. 2. Let K be a field, Q the quiver o 1

˛

/ o 2

ˇ

/ ; 3

I the ideal in KQ generated by ˛, ˇ, ˛ ˇ, and A D KQ=I the associated bound quiver algebra. Prove that A is a nonsymmetric Frobenius K-algebra and describe a nondegenerate associative K-bilinear form .; / W A A ! K. 3. Let K be a field and Q the quiver 2 1 @_@@ @@@@ˇ1 ˛2 ~~~~~? @@@@ ~ ~ ~~ ˛1 @@@ ~~~~ ˇ2 ~ 0 O ˇ3

˛3

3:

Consider in KQ the ideals • I1 generated by ˛1 ˇ2 , ˛1 ˇ3 , ˛2 ˇ1 , ˛2 ˇ3 , ˛3 ˇ1 , ˛3 ˇ2 , ˇ1 ˛1 ˇ2 ˛2 , ˇ2 ˛2 ˇ3 ˛3 ,

16. Exercises

471

• I2 generated by ˛1 ˇ1 , ˛1 ˇ2 , ˛2 ˇ1 , ˛2 ˇ3 , ˛3 ˇ2 , ˛3 ˇ3 , ˇ1 ˛1 ˇ2 ˛2 , ˇ2 ˛2 ˇ3 ˛3 , • I3 generated by ˛1 ˇ1 , ˛1 ˇ2 , ˛2 ˇ2 , ˛2 ˇ3 , ˛3 ˇ1 , ˛3 ˇ3 , ˇ1 ˛1 ˇ2 ˛2 , ˇ2 ˛2 ˇ3 ˛3 , and the associated bound quiver algebras A1 D KQ=I1 , A2 D KQ=I2 , A3 D KQ=I3 . Prove that A1 , A2 , A3 are pairwise nonisomorphic Frobenius algebras and describe the associated Nakayama permutations of f0; 1; 2; 3g. 4. Let K be a field, Q the quiver o 1

˛1

o 2

˛2

o 3

˛3

o 4

˛4

; 5

I the ideal in KQ generated by ˛2 ˛1 , ˛3 ˛2 , ˛4 ˛3 , and A D KQ=I the associated bound quiver algebra. Prove that the trivial extension algebra T.A/ D A Ë D.A/ is isomorphic to the bound quiver algebra K =J , where is the quiver o 1

˛1 ˇ1

/ o 2

˛2 ˇ2

/ o 3

˛3 ˇ3

/ o 4

˛4 ˇ4

/ 5

and J the ideal in K generated by the elements ˛2 ˛1 , ˛3 ˛2 , ˛4 ˛3 , ˇ1 ˇ2 , ˇ2 ˇ3 , ˇ3 ˇ4 , ˛1 ˇ1 ˇ2 ˛2 , ˛2 ˇ2 ˇ3 ˛3 , ˛3 ˇ3 ˇ4 ˛4 . 5. Let K be a field, Q the quiver o 1

˛1

o 2

˛2

o 3

˛3

o 4

˛4

o 5

˛5

; 6

I the ideal in KQ generated by the elements ˛3 ˛2 ˛1 and ˛5 ˛4 ˛3 ˛2 , and A D KQ=I the associated bound quiver algebra. Prove that the trivial extension algebra T.A/ D A Ë D.A/ is isomorphic to the bound quiver algebra K =J , where is the quiver 3o ˇ 1 O ? // ~ ˛3 ~~ // ~~ / //˛2 ~~ 4 _@ 6 /// ˛1 @@ @@ ˛5 // // ˛4 @@ o 5 2 and J is the ideal in K generated by ˛3 ˛2 ˛1 , ˛5 ˛4 ˛3 ˛2 , ˛1 ˇ ˛4 ˛3 , ˛2 ˛5 , ˇ, ˛4 ˛3 ˛2 ˛4 .

472


6. Let K be a field, Q the quiver 4 5 @ @ ~ ~ @@ ~ ˛1 ~~ ~ ~~ ˛2 ˛ @@@ ~~~ ~ ~ 3 ~ 1 2 3 and A D KQ. Prove the following statements. (a) The trivial extension algebra T.A/ D A Ë D.A/ is isomorphic to the bound quiver algebra K =J , where is the quiver 1 @_@@ @@@@˛1 @@@@ ˇ1 @@@ ? o ~ ~ ~ ˇ2 ~ ~ 4 ~~~~ ~~~~~ ˛2 2

˛3 ˇ3

/ o 3

/ 5

and J is the ideal in K generated by ˇ1 ˛2 , ˇ1 ˛3 , ˇ2 ˛1 , ˇ2 ˛3 , ˇ3 ˛1 , ˇ3 ˛2 , ˛3 , ˇ3 , ˛1 ˇ1 ˛2 ˇ2 , ˛2 ˇ2 ˛3 ˇ3 , ˇ3 ˛3 . (b) T.A/ is isomorphic to the trivial extension algebra T.A0 / D A0 Ë D.A0 / of the bound quiver algebra A0 D KQ0 =I 0 , where Q0 is the quiver

1

ˇ1

3? ~ ˛3 ~~ ~ ~ ~~ / @ 4 @@@ @ ˛2 @@ 2

/ 5

and I 0 is the ideal in KQ0 generated by ˇ1 ˛2 , ˇ1 ˛3 , ˛3 . 7. Let K be a field, Q the quiver 1 ~ _@@@ ˇ ˛1 ~~ @@ 1 @@ ~~ ~ ~ ˛2 ˇ2 o o 4 ; 0 _@ @@ ~ @@ 2 ~~~ ~ ˛3 @@ ~ ~ ˇ3 3

16. Exercises

473

I the ideal in KQ generated by ˇ1 ˛1 Cˇ2 ˛2 Cˇ3 ˛3 , and A D KQ=I the associated bound quiver algebra. Prove that the trivial extension algebra T.A/ D A Ë D.A/ is isomorphic to the bound quiver algebra K =J , where is the quiver 1 ~ _@@@ ˇ ˛1 ~~ @@ 1 @ ~~ ~ ~ ˛2 ˇ2 @% o o 0 _@ 9 4 @@ ~~ @@ 2 ~ ~~ ˛3 @@ ~~ ˇ3 3 and J is the ideal in K generated by ˇ1 ˛1 C ˇ2 ˛2 C ˇ3 ˛3 , ˛1 , ˇ1 , ˛3 , ˇ3 , ˛2 ˛2 , ˇ2 ˇ2 . 8. Let K be a field, Q the quiver 1 O

˛1

˛4

4 o

/ 2 ˛2

˛3

3;

I the ideal in KQ generated by ˛2 ˛3 and ˛4 ˛1 , and A D KQ=I the associated bound quiver algebra. Prove that the trivial extension algebra T.A/ D A Ë D.A/ is isomorphic to the bound quiver algebra K =J , where is the quiver ˛1

1 O @_@@ / 2 @@@@ @@@@ ˛2 ˛4 @@@ 4 o ˛3 3 and J is the ideal in K generated by ˛2 ˛3 , ˛4 ˛1 , , , ˛3 ˛4 ˛1 ˛2 , ˛1 ˛2 ˛3 ˛4 . 9. Let K be a field, Q the quiver 1 O

˛1

˛4

4 o

/ 2 ˛2

˛3

3;

I the ideal in KQ generated by ˛1 ˛2 ˛3 , ˛2 ˛3 ˛4 , ˛3 ˛4 ˛1 , ˛4 ˛1 ˛2 , and A D KQ=I the associated bound quiver algebra. Prove that the trivial extension algebra

474


T.A/ D A Ë D.A/ is isomorphic to the bound quiver algebra K =J , where is the quiver ˇ / 1 Y2Do D 3 22 DD ˛1 ˛2 zzz < z 22 DD z 22 DD" 2 zzz 2 ˛4 2 22 O ˛3 2ı2 2 4

and J is the ideal in K generated by ˛1 ˛2 ˛3 , ˛2 ˛3 ˛4 , ˛3 ˛4 ˛1 , ˛4 ˛1 ˛2 , ˇ˛3 ˛1 ı, ˛1 ˛3 , ˛2 ˛4 ˇ, ı˛4 ˛2 . 10. Let K be a field, Q the quiver 4

1

˛

/ 2

ˇ

/ o 3

ı

5

/ o 6

7

/ 8

and A D KQ. Find a finite quiver and an admissible ideal J in K such that the trivial extension algebra T.A/ D A Ë D.A/ is isomorphic to the bound quiver algebra K =J . 11. Let K be a field, Q the quiver 4 5 6 @ @ ~ ~ @@ ~ ˛ ~~ ~ @@ ~~ ˇ ~~ ~ ~ @ ~~ ı 1 2 3 and A D KQ. Find a finite quiver and an admissible ideal J in K such that the trivial extension algebra T.A/ D A Ë D.A/ is isomorphic to the bound quiver algebra K =J . 12. Let K be a field, Q the quiver O 7

6 o 1

˛

o 2

ˇ

3

/ o 4

5

16. Exercises

475

and A D KQ. Find a finite quiver and an admissible ideal J in K such that the trivial extension algebra T.A/ D A Ë D.A/ is isomorphic to the bound quiver algebra K =J . 13. Let K be a field, Q the quiver

oo 1

˛ ˇ

3 @ @@ ~ ~~ @@ ~ @@ ~ ~ ~ _@ 5 ? ~ 2 @@@ ~ @ ~~ @@ ~~~ 4

and A D KQ. Find a finite quiver and an admissible ideal J in K such that the trivial extension algebra T.A/ D A Ë D.A/ is isomorphic to the bound quiver algebra K =J . 14. Let K be a field, Q the quiver 1 2 3 4 5 5 D D 55 5

55 5

55

˛ 55

ˇ

GG ; w GG 5 ww 6 GG ww G w GG ww # w ; GG w GG w ı ww 7 GG GG ww w G# 9 w 8 w D 44

D 444 44! %

'

44 44

44

; 13 10 11 12 I the ideal in KQ generated by ˛ˇ, , , ı, %!, ' , and A D KQ=I the associated bound quiver algebra. Find a finite quiver and an admissible ideal J in K such that the trivial extension algebra T.A/ D A Ë D.A/ is isomorphic to the bound quiver algebra K =J .

476


15. Let K be a field, Q the quiver 2 5 _@ @ ~ ~ @@ˇ ~~ ˛ ~~ @@ ~ ~~ @ ~~~ ~ ~ _? 1 _? ?? 4 ??? ?? ? ?? ??? ; 6 3 I the ideal in KQ generated by ˇ˛ , and A D KQ=I the associated bound quiver algebra. Find a finite quiver and an admissible ideal J in K such that the trivial extension algebra T.A/ D A Ë D.A/ is isomorphic to the bound quiver algebra K =J . 16. Let K be a field and T D TS10 be the Brauer tree

1

2

3

4

'&%$ !"# S

5

with the exceptional vertex S of multiplicity 10. Describe (a) The Brauer tree algebra A.TS10 / D KQT 10 =IT 10 . S

S

(b) The bound quiver algebra A.TS10 / D KQT 10 =IT10 isomorphic to A.TS10 /. S

S

17. Let K be a field and T D TS5 the Brauer tree

3

2

5

'&%$ !"# S

4

8

1

6 7

9

10

with the exceptional vertex S of multiplicity 5. Describe (a) The Brauer tree algebra A.TS5 / D KQT 5 =IT 5 . S

S

(b) The bound quiver algebra A.TS5 / D KQT 5 =IT5 isomorphic to A.TS5 /. S

S

16. Exercises

477

18. Let K be a field and T D TS3 be the Brauer tree D DD z DD zz z 1 z D 13 z 12 DD zz 2 GG w 11 3 GG ww GG w w G ww 10 GGG ww 4 w '&%$ !"# S 5

9

5 5557 55 8

6

with the exceptional vertex S of multiplicity 3. Describe (a) the Brauer tree algebra A.TS3 / D KQT 3 =IT 3 ; S

S

(b) the bound quiver algebra A.TS3 / D KQT 3 =IT3 isomorphic to A.TS3 /. S

S

19. Let K be a field and T D a Brauer tree with the exceptional vertex S of multiplicity m. Prove that the following conditions are equivalent. TSm

(a) m D 1. (b) A.TSm / D KQTSm =ITSm is isomorphic to the trivial extension algebra T.B/ D B Ë D.B/ of a finite dimensional K-algebra B. 20. Let K be a field of characteristic 2 and G D Z2 Z2 the Klein 4-group. Prove that the group algebra KG is isomorphic to the trivial extension algebra T.A/ D A Ë D.A/ of the quotient polynomial algebra A D KŒx=.x 2 /. 21. Let K be a field of characteristic 2 and Q8 D f˙1; ˙i; ˙j; ˙kg the quaternion group of order 8 in the quaternion division R-algebra H D R ˚ Ri ˚ Rj ˚ Rk. Find a finite quiver and an admissible ideal J in K such that the group algebra KQ8 is isomorphic to the bound quiver algebra K =J . 22. Let K be a field of characteristic 2. For a positive integer m, consider the generalized quaternion group ˝ ˛ m Q2mC2 D x; y j x 2 D y 2 ; xyx D x of order 2mC2 . Find a finite quiver and an admissible ideal J in K such that the group algebra KQ2mC2 is isomorphic to the bound quiver algebra K =J .

478


23. Let K be a field of characteristic 2. For a positive integer m 2, consider the dihedral 2-group ˝ ˛ m m D2mC1 D x; y j x 2 D y 2 D 1; yx D xy 2 1 of order 2mC1 D 2 2m . Find a finite quiver and an admissible ideal J in K such that the group algebra KD2mC1 is isomorphic to the bound quiver algebra K =J . 24. Let K be a field of characteristic p > 0, m1 ; : : : ; mr a sequence of positive integers and G D Zpm1 Zpmr the product of the associated cyclic groups Zpm1 ; : : : ; Zpmr of orders p m1 ; : : : ; p mr , respectively. Find a finite quiver and an admissible ideal J in K such that the group algebra KG is isomorphic to the bound quiver algebra K =J . 25. Let K be a field, Q the quiver o 1

˛1 ˇ1

/ o 2

˛2 ˇ2

/ o 3

˛3 ˇ3

/ ; 4

I the ideal in KQ generated by ˛2 ˛1 , ˛3 ˛2 , ˇ1 ˇ2 , ˇ2 ˇ3 , ˛1 ˇ1 ˇ2 ˛2 , ˛2 ˇ2 ˇ3 ˛3 , and A D KQ=I the associated bound quiver algebra. Prove that A is a selfinjective algebra of finite representation type and describe the Auslander–Reiten quiver of A . 26. Let K be a field, Q the quiver 1 ?~ @@ @@% ˛ ~~ @@ ~ ~ @ ~o ~ / o / 4 ; 2 _@ @@ˇ 3 ~~ @@ ~ ~ ı @@ ~~~ 5 I the ideal in KQ generated by ˛, %, ıˇ, , ˇ, , ˇ , ˛% ˇ , ı , and A D KQ=I the associated bound quiver algebra. Prove that A is a selfinjective algebra of finite representation type and describe the Auslander–Reiten quiver of A . 27. Let K be a field, Q the quiver 1 @_@@ @@@@˛1 @@@@ ˇ1 @@@ ? o ~ ~ ˇ2 ~~~ 4 ~~~~ ~~~~~ ˛2 2

˛3 ˇ3

/ o 3

/ ; 5

16. Exercises

479

I the ideal in KQ generated by ˇ1 ˛1 , ˇ2 ˛2 , ˇ1 ˛3 , ˇ2 ˛3 , ˇ3 ˛1 , ˇ3 ˛2 , ˇ3 , ˛3 , ˛1 ˇ1 ˛2 ˇ2 , ˛2 ˇ2 ˛3 ˇ3 , ˇ3 ˛3 , and A D KQ=I the associated bound quiver algebra. Prove that A is a selfinjective algebra of finite representation type and describe the Auslander–Reiten quiver of A . 28. Let K be a field, Q the quiver 1 O 2 eKK ˛ ˇ t9 4 KK KK tttt t 9 eK tt 0 KKKK t t tt ı % K 5, 3 I the ideal in KQ generated by ˛ˇ ı, ˛ˇ %, ˇ , ı˛, ˇ , %˛, ı , %, ı , % , and A D KQ=I the associated bound quiver algebra. Prove that A is a selfinjective algebra of finite representation type and describe the Auslander–Reiten quiver of A . 29. Let K be a field, Q the quiver

˛

2 D ˇ1

$

Z5 1 555 ˇ6 55

o 6

ˇ2

ˇ5

3 / 55 55ˇ3 55 4 ;

ˇ

4 5

I the ideal in KQ generated by ˛ 2 ˇ1 ˇ2 ˇ3 ˇ4 ˇ5 ˇ6 , ˇj ˇj C1 : : : ˇ6 ˛ˇ1 : : : ˇj 1 ˇj , for j 2 f2; 3; 4; 5g, ˇ6 ˇ1 , and A D KQ=I the associated bound quiver algebra. Prove that A is a symmetric algebra of finite representation type and describe the Auslander–Reiten quiver of A . 30. Let K be a field, Q the quiver

˛

2 D ˇ1

$

Z5 1 555 ˇ6 55

o 6

ˇ2

ˇ5

3 / 55 55ˇ3 55 4 ;

ˇ

4 5

480


I the ideal in KQ generated by the elements ˛ 2 .ˇ1 ˇ2 ˇ3 ˇ4 ˇ5 ˇ6 /2 , ˛ˇ1 , ˇ6 ˛, ˇj ˇj C1 : : : ˇ6 ˇ1 : : : ˇj 1 ˇj , for j 2 f2; 3; 4; 5g, and A D KQ=I the associated bound quiver algebra. Prove that A is a symmetric algebra of infinite representation type. 31. Let K be a field, Q the quiver

˛

2 D ˇ1

$

Z5 1 555 ˇ6 55

o 6

ˇ2

ˇ5

3 / 55 55ˇ3 55 4 ;

ˇ

4 5

I the ideal in KQ generated by ˛ 2 ˇ1 ˇ2 ˇ3 ˇ4 ˇ5 ˇ6 , ˇ6 ˇ1 , ˛ˇ1 ˇ2 ˇ3 ˇ4 ˇ5 ˇ6 C ˇ1 ˇ2 ˇ3 ˇ4 ˇ5 ˇ6 ˛, ˇj ˇj C1 : : : ˇ6 ˛ˇ1 ˇ2 : : : ˇj , for j 2 f2; 3; 4; 5; 6g, and A D KQ=I the associated bound quiver algebra. Prove that A is a nonsymmetric selfinjective algebra of infinite representation type. p p 32. Let Q. 3 2/ be the field extension of Q by 3 2 and consider the Q-algebra ³ ² p p ˇ Q 0p a 0 3 3 ˇ p AD .Q. 2// a 2 Q; b; c 2 Q. 2/ : D 2 M 2 c b Q. 3 2/ Q. 3 2/ Prove that the trivial extension algebra T.A/ D A Ë D.A/ is a symmetric algebra of finite representation type and the Dynkin type G2 . 33. Let A be the R-algebra 2 3 82 R 0 0 0 ˆ ˆ a 0 6 R R 0 0 7 > = :

> > ;

Prove that the trivial extension algebra T.A/ D A Ë D.A/ is a symmetric algebra of finite representation type and the Dynkin type F4 . 34. Let A be the R-algebra 2 3 82 R 0 0 < a A D 4C C 0 5 D 4b : C C C x

0 c y

9 3 0 ˇ = ˇ a2R 05 ˇ : b; c; x; y; z 2 C ; z

Prove that the trivial extension algebra T.A/ D A Ë D.A/ is a symmetric algebra of finite representation type and the Dynkin type B3 .

16. Exercises

35. Let A be the R-algebra 2 3 82 R 0 0 < a 4 5 A D R R 0 D 4b : C C C x

481

9 3 0 ˇ = ˇ a; b; c 2 R 05 ˇ : x; y; z 2 C ; z

0 c y

Prove that the trivial extension algebra T.A/ D A Ë D.A/ is a symmetric algebra of finite representation type and the Dynkin type C3 . 36. Let K be a field, Q the quiver

$

h 1

˛

(

; 2

ˇ

I1 the ideal in KQ generated by ˛ 2 ˇ, ˇ , I2 the ideal in KQ generated by ˛ 2 ˇ, ˇ ˇ˛, ˇ˛ˇ, and A1 D KQ=I1 , A2 D KQ=I2 the associated bound quiver algebras. Prove the following assertions. (a) A1 and A2 are finite dimensional symmetric K-algebras. (b) A1 and A2 are socle equivalent. (c) A1 and A2 are isomorphic if and only if K is of characteristic different from 2. (d) A1 and A2 are of finite representation type and the Dynkin type D6 . (e) The Auslander–Reiten quivers A1 and A2 are isomorphic. 37. Let K be a field, Q the quiver ˛

$

h 1

ˇ

(

; 2

I1 the ideal in KQ generated by ˛ 2 , ˛ˇ C ˇ˛, ˇ , ˇ˛ˇ, I2 the ideal in KQ generated by ˛ 2 ˛ˇ, ˛ˇ C ˇ˛, ˇ, ˇ˛ˇ, and A1 D KQ=I1 , A2 D KQ=I2 the associated bound quiver algebras. Prove that (a) A1 and A2 are finite dimensional selfinjective K-algebras. (b) A1 and A2 are not isomorphic. (c) A1 and A2 are socle equivalent. (d) For each i 2 f1; 2g, the simple Ai -module S1 at the vertex 1 is not Ai -periodic. (e) For each i 2 f1; 2g, the simple Ai -module S2 at the vertex 2 is Ai -periodic.

482


38. Let K be a field, Q the quiver $

˛

h 1

ˇ

(

; 2

I1 the ideal in KQ generated by ˛ 3 ˇ, ˇ , ˇ˛ 2 , ˛ 2 , I2 the ideal in KQ generated by ˛ 3 ˇ, ˇ ˇ˛, ˇ˛ 2 , ˛ 2 , and A1 D KQ=I1 , A2 D KQ=I2 the associated bound quiver algebras. Prove the following statements. (a) A1 and A2 are finite dimensional symmetric K-algebras. (b) A1 and A2 are socle equivalent. (c) A1 and A2 are isomorphic if and only if K is of characteristic different from 3. (d) The simple A1 -modules S1 and S2 at the vertices 1 and 2 are A1 -periodic. (e) The simple A2 -modules S1 and S2 at the vertices 1 and 2 are A2 -periodic. 39. Let K be a field, Q the quiver $

˛

h 1

(

ˇ

h 2

(

%

; 3

I the ideal in KQ generated by ˛ 2 C ˇ, ˇ C %, % , and A D KQ=I the associated bound quiver algebra. Prove the following statements. (a) A is a finite dimensional symmetric K-algebra of infinite representation type. (b) Every simple right A-module is A -periodic of period 3. 40. Let K be a field, 2 K n f0; 1g, Q the quiver ˛

$

o 1

/ d 2

ˇ

;

I the ideal in KQ generated by ˛ 2 , ˇ 2 , ˛ ˇ , ˇ ˛ , and A D KQ=I the associated bound quiver algebra. Prove the following statements. (a) A is a finite dimensional symmetric K-algebra of infinite representation type. (b) A4 .S/ Š S for any simple right A-module S.

16. Exercises

483

41. Let K be a field, Q the quiver 2 1 @_@@ @@@@ˇ1 ˛2 ~~~~~? @@@@ ~~~~ ˛1 @@@ ~~~~ ˇ2 ~ 0 O ˛3

ˇ3

; 3 I the ideal in KQ generated by ˇ1 ˛1 C ˇ2 ˛2 C ˇ3 ˛3 , ˛1 ˇ1 , ˛2 ˇ3 , ˛3 ˇ2 , and A D KQ=I the associated bound quiver algebra. Prove the following statements. (a) A is a finite dimensional symmetric K-algebra of infinite representation type. (b) A6 .S/ Š S for any simple right A-module S. (c) A3 .T / © T for a simple right A-module T . 42. Let K be a field, Q the quiver 2 1 @_@@ @@@@ˇ1 ˛2 ~~~~~? @@@@ ~~~~ ˛1 @@@ ~~~~ ˇ2 ~ 0 O ˇ3

˛3

; 3 I the ideal in KQ generated by ˇ1 ˛1 C ˇ2 ˛2 C ˇ3 ˛3 , ˛1 ˇ1 , ˛2 ˇ2 , ˛3 ˇ3 , and A D KQ=I the associated bound quiver algebra. Prove the following statements. (a) A is a finite dimensional weakly symmetric K-algebra of infinite representation type. (b) A is a symmetric K-algebra if and only if K is of characteristic 2. (c) A3 .S/ Š S for any simple right A-module S. 43. Let K be a field, Q the quiver 2 C 77 77 77 ˛ 77 77 ı / 3; 1 o ˇ

484


I the ideal in KQ generated by ıˇı ˛ , ˇ˛, .ˇı/3 ˇ, and A D KQ=I the associated bound quiver algebra. Prove the following statements. (a) A is a finite dimensional symmetric K-algebra. (b) A8 .S/ Š S for any simple right A-module S. (c) A4 .T / © T for a simple right A-module T . 44. Let K be a field, Q the quiver 5 ? @@ ~ @@ ~~ @@ ~~ ~ ı @ ~o o / / 3 ; 2 _@ @@ 1 ~~ @@ ~ ~~ ˛ @@ ~~ ˇ 4 I the ideal in KQ generated by ˇ, ˛ , ı , ˇ˛ı , , and A D KQ=I the associated bound quiver algebra. Prove the following statements. (a) A is a finite dimensional selfinjective K-algebra. (b) A is not a weakly symmetric K-algebra. (c) A12 .S/ Š S for any simple right A-module S. 45. Let K be a field, Q the quiver 1 _@@@ @@@@ˇ @@@@ ˛ @@@ ~? o ~~~~ 3 ~~~~ ~~~~~ 2

~? 5 ~~~~~ ~ ~~~~% / ~~~ @ 4 _@@@@@@@@' @@@@ @ 6; !

I the ideal in KQ generated by ˇ, ˇ, ˛, , ˛, , % , %', , !, ˛ˇ , , !%, !% ' , and A D KQ=I the associated bound quiver algebra. Prove the following statements. (a) A is isomorphic to the trivial extension algebra T.H / D H Ë D.H / of the path algebra H D K of the quiver of the form 3 5 6 @ ~ @@ ~~ ˛ ~~ @ ~ @@% ~~ ~~ ~~ @ ~~ 1 2 4.

16. Exercises

485

(b) The stable Auslander–Reiten quiver As contains two different connected components C and D isomorphic to the stable translation quiver Z such that C contains the simple modules S1 , S2 , S4 , and D contains the simple modules S3 , S5 , S6 . (c) The simple modules S1 , S2 , S3 , S4 , S5 , S6 are not A -periodic. (d) The indecomposable right A-modules

M1 W

M2 W

K ÀAA AAAA0 AAAA 1 AAA o ~~> K 0 ~~~ ~ ~ ~~~~ ~~~~~~ 1 K

@ 0 / 0 ^=== ==== ==== ==== = 0;

0 _??? ???? ???? ???? ? ?Ko

? 0 / K _>>> >>>> >>>> >>>> > 0;

0

1 0

0 ^=== }> K ==== 0 }}}} ==== } } }} === }~}}}} 1 / K ÀAA @ 0o M3 W AAAA0 AAAA 1 AAA 0 K form the mouth of a stable tube T of A of rank 3 and A M1 D M3 , A M2 D M1 , A M3 D M2 . (e) The modules M1 , M2 , M3 are A -periodic of period 6. (f) The indecomposable right A-modules

N1 W

K ÀAA AAAA0 AAAA 1 AAA >Ko ~~~~ ~ ~~ ~~~~ ~~~~~~ 0

1 0

~? 0 ~~~~~ ~ ~~~~ ~~~~~ / K _??? ???? 0 ???? ?? 1 ??? K;

486


N2 W

0 ÀAA AAAA AAAA AAAA A o ~~> K 0 ~~~ ~ ~ ~~~~ ~~~~~~ 1 K

>K }}}} } }}}} }~}}}} 1 / K `@@@ @@@@ @@@@ @@@@ @ 0; 0

1 0

form the mouth of a stable tube T of A of rank 2 and A N1 D N2 , A N2 D N1 . (g) The modules N1 , N2 are A -periodic of period 4. 46. Let n 2 be a natural number, A D KŒX =.X n / the quotient polynomial K-algebra over a field K, and Ae D Aop ˝K A. Consider the homomorphism of right Ae -modules ! W Ae ! Ae such that !.1 ˝ 1/ D x ˝ 1 1 ˝ x, where and x is the coset of X in A. Prove that Ker ! is the right Ae -module 1 D 1A D 1Aop P e n1i yA with y D n1 ˝ xi . iD1 x 47. Let K be a field, Q D .Q0 ; Q1 ; s; t / a finite quiver, Qop D .Q00 ; Q10 ; s 0 ; t 0 / the opposite quiver of Q, %1 ; : : : ; %r a set of relations in KQ generating an admissible op op ideal I in KQ, %1 ; : : : ; %r the associated set of relations in KQop generating the opposite ideal I op in KQop , and A D KQ=I and Aop D KQop =I op the associated bound quiver algebras. Consider the quiver Qe D .Q0e ; Q1e ; s e ; t e / such that Q0e D Q00 Q0 , Q1e D .Q00 Q1 / [ .Q10 Q0 / with s e ; t e W Q1e ! Q0e defined for .a0 ; ˛/ 2 Q00 Q1 and .ˇ 0 ; b/ 2 Q10 Q0 by s e .a0 ; ˛/ D .a0 ; s.˛//; t e .a0 ; ˛/ D .a0 ; t .˛//; s e .ˇ 0 ; b/ D .s 0 .ˇ 0 /; b/; t e .ˇ 0 ; b/ D .t 0 .ˇ 0 /; b/: Further, denote by I e the ideal in the path algebra KQe of Qe generated by the relations .a0 ; %1 /; : : : ; .a0 ; %r / for all vertices a0 2 Q00 ; op .%1 ; b/; : : : ; .%op r ; b/ for all vertices b 2 Q0 ; and the relations in KQe of the forms .ˇ 0 ; s.˛//.t 0 .ˇ 0 /; ˛/ .s 0 .ˇ 0 /; ˛/.ˇ 0 ; t .˛// for all arrows ˇ 0 2 Q10 and ˛ 2 Q1 . Prove that (a) I e is an admissible ideal of the path algebra KQe ; (b) the bound quiver algebra KQe =I e is isomorphic to the enveloping algebra Ae D Aop ˝K A of A.

16. Exercises

487

48. Let K be a field, Q D .Q0 ; Q1 ; s; t / a finite acyclic quiver, Qop D .Q00 ; Q10 ; s 0 ; t 0 / the opposite quiver of Q, and A D KQ, Aop D KQop the associated path algebras of Q and Qop , respectively. Prove that the enveloping algebra Ae D Aop ˝K A of A is isomorphic to the bound quiver algebra KQe =I e , where Qe D .Q0e ; Q1e ; s e ; t e / is the quiver defined in Exercise 16.47 and I e is the ideal in KQe generated by the relations in KQe of the forms .ˇ 0 ; s.˛//.t 0 .ˇ 0 /; ˛/ .s 0 .ˇ 0 /; ˛/.ˇ 0 ; t .˛// for all arrows ˇ 0 2 Q10 and ˛ 2 Q1 . 49. Let K be a field, Q the quiver ˇ

/ ; o 3 1 2 and A D KQ the associated path algebra. Prove that the enveloping algebra Ae D Aop ˝K A of A is of infinite representation type. ˛

50. Let K be a field, Q the quiver ˇ ˛ ; o o 3 1 2 I the ideal in KQ generated by ˇ˛, and A D KQ=I the associated bound quiver Nakayama algebra. Prove that the enveloping algebra Ae D Aop ˝K A is of finite representation type.

51. Let K be a field, Q the quiver o 1

˛ ˇ

/ ; 2

I the ideal in KQ generated by ˇ˛, ˛ˇ, and A D KQ=I the associated bound quiver selfinjective Nakayama algebra. Prove that the enveloping algebra Ae D Aop ˝K A is of infinite representation type. 52. Let A be a finite dimensional K-algebra over a field K. Then A is a symmetric algebra if and only if Ae is a symmetric algebra. 53. Let K be a field, Q the quiver 1 C 88 88 ˛ 88 8 2; 3 o ˇ

I the ideal in KQ generated by ˛ˇ, ˇ, ˛, and A D KQ=I the associated bound quiver algebra. Prove that A is a periodic algebra.

488


54. Let K be a field, Q the quiver ˛

$

h 1

ˇ

(

; 2

I the ideal in KQ generated by ˛ 2 ˇ, ˇ, and A D KQ=I the associated bound quiver algebra. Prove that A is a periodic algebra. 55. Let A be a finite dimensional K-algebra over a field K and W Ae ! A the canonical epimorphism of right Ae -modules (A-bimodules) such that .a0 ˝ b/ D ab for a; b 2 A. Prove the following assertions. (a) A is a separable K-algebra if and only if there exists an element f 2 Ae such that .f / D 1A and af D f a for all a 2 A. (b) If f is an element f 2 Ae with .f / D 1A and af D f a for all a 2 A, then f is an idempotent of Ae and Ae D .1 f /ae ˚ eAe as right Ae -modules, with .1 f /Ae D Ker and fAe Š A in mod Ae . 56. Prove that H ˝R C Š M2 .C/ as R-algebras. 57. Prove that He D Hop ˝R H is a semisimple R-algebra and H is a projective right He -module.

Chapter V

Hecke algebras

In this chapter we associate to a finite group G generated by a finite number of reflections of a real Euclidean space V and a nonzero element q of a field K a finite dimensional K-algebra HK;q ŒG, called a Hecke algebra of G, and prove that it is a symmetric algebra. Moreover, a classification of finite reflection groups, established in 1934 by H. S. M. Coxeter [Cox], will be presented. We will follow the algebraic account of finite reflection groups presented in the book [GrBe] by L. C. Grove and C. T. Benson. Further, in our treatment of Hecke algebras, we will show an essential application of the article of N. Iwahori [Iwa].

1 Finite reflection groups Let V be a real Euclidean space with an inner product .; / W V V ! R, which is a symmetric p R-bilinear form with .x; x/ > 0 for all nonzero vectors x in V . The length .x; x/ of a vector x 2 V will be denoted by kxk. A vector x 2 V with kxk D 1 is said to be a unit vector of V . We denote by O.V / the group of orthogonal transformations of V , that is, the group of R-linear automorphisms T of V such that .T .x/; T .y// D .x; y/ for all vectors x; y 2 V . A distinguished class of orthogonal transformations of V is given by reflections of V . An R-linear transformation S W V ! V of V is said to be a reflection of V if S carries each vector of V to its mirror image with respect to a fixed hyperplane P of V . Then V admits an orthogonal decomposition V D P ˚ P? , where dimR P D dimR V 1 and P? D fx 2 V j .x; y/ D 0 for all y 2 Pg is a line orthogonal to P. Then, for each nonzero vector r 2 P? , we have S D Sr , where Sr is defined as Sr .x/ D x

2.x; r/ r .r; r/

for all x 2 V . Then Sr is called the reflection of V along the vector r or through the hyperplane P D .Rr/? orthogonal to the line Rr. We note that Sr 2 O.V / and Sr2 D idV (see Exercise 6.3). The following lemma will be useful. Lemma 1.1. Let r be a nonzero vector of a real Euclidean space V and T an element of O.V /. Then T Sr T 1 D ST .r/ . Proof. Let P D .Rr/? and P0 D T .P/. Then P0 is a hyperplane of V and P0 D .RT .r//? since T 2 O.V /. Moreover, for y D T .x/ with x 2 P, we have .T Sr T 1 /.y/ D T .Sr .T 1 .T .x//// D T .Sr .x// D T .x/ D y. Finally,

490

Chapter V. Hecke algebras

.T Sr T 1 /.T .r// D T .Sr .r// D T .r/ D T .r/. Therefore, T Sr T 1 D ST .r/ . Let G be a subgroup of O.V /. Then the set VG D fx 2 V j T .x/ D x for all T 2 Gg is the subspace of V such that the restriction T jVG of every T 2 G is the identity transformation of VG , and VG is the largest subspace of V with this property. In particular, V has an orthogonal decomposition V D VG ˚ VG? and every T 2 G can be represented as idVG 0 T D D idVG ˚T 0 ; 0 T0 where T 0 D T jV ? . The group G 0 D fT 0 j T 2 Gg is a subgroup of the orthogonal G group O.VG? / of the Euclidean space VG? with VG? G 0 D 0, and clearly isomorphic to G. A subgroup G of O.V / with VG D 0 is called effective. It follows from the above discussion that studying subgroups of O.V / reduces to studying effective subgroups of O.W / for real Euclidean subspaces W of V . We also note that a subgroup G of O.V / generated by a finite number of reflections Sr1 ; : : : ; Srm along vectors r1 ; : : : ; rm 2 V is effective if and only if fr1 ; : : : ; rm g contains a basis of the R-vector space V (see Exercise 6.4). For a real Euclidean space V , a finite effective subgroup G of O.V / generated by a finite number of reflections of V is said to be a Coxeter group in V . Hence, if G is generated by reflections Sr1 ; : : : ; Srm of V along the vectors r1 ; : : : ; rm of V then fr1 ; : : : ; rm g contains a basis of V over R and every element T 2 G is of the form T D Sri1 : : : Srik for some i1 ; : : : ; ik 2 f1; : : : ; mg. Let G be a Coxeter group in a real Euclidean space V , say n D dimR V . For a reflection S 2 G there are exactly two unit vectors r and r in V such that Sr D S D Sr , called (unit) roots of G. The set of all unit vectors r 2 V with Sr 2 G will be called the (unit) root system of G. Clearly, is a finite set of unit vectors of V . Moreover, for any T 2 G, we have T . / D . Indeed, for r 2 and T 2 G, T .r/ is a unit vector of V and ST .r/ D T Sr T 1 2 G, by Lemma 1.1. Since is a finite set of vectors in V , its orthogonal set ? is a proper subspace of V (see Exercise 6.1), and hence we may choose a vector t 2 V such that .t; r/ ¤ 0 for all vectors r in (see Exercise 6.2). This allows us to divide into two disjoint subsets C t D fr 2 j .t; r/ > 0g; t D fr 2 j .t; r/ < 0g; which are subsets of lying on the two sides of the hyperplane .Rt /? of V . Observe C C C that t D t , and hence j t j D j t j. The elements of t (respectively, t )

1. Finite reflection groups

491

are called t-positive roots (respectively, t-negative roots) of G. Further, we may choose a subset … t of C t which is minimal with respect to the property that every r 2 C is a linear combination of vectors from … t with all coefficients nonnegative. t C D , every r 2 Then, since t t t is a linear combination of vectors from … t with all coefficients nonpositive. The set … t is called a t -basis of . Example 1.2. Let m 3 be a natural number and V D R2 be the Euclidean space with the canonical inner product .x; y/ D x1 y1 C x2 y2 for x D .x1 ; x2 /; y D .y1 ; y2 / 2 V . Consider the reflection S D Sr1 of R2 along the vector r1 D .0; 1/ and the counterclockwise rotation R of R2 with the center .0; 0/ through the angle 2 . Thus S and R are given in the canonical basis e1 D .1; 0/, e2 D .0; 1/ of R2 m by the matrices " # cos 2 sin 2 1 0 m m AD and B D : 0 1 sin 2 cos 2 m

m

Then the group H2m D hR; Si D f1; R; : : : ; Rm1 ; S; SR; : : : ; SRm1 g;

generated by R and S, is called the dihedral group of order 2m in R2 . Moreover, T D RS, given by the matrix " # cos 2 sin 2 m m BA D ; sin 2 cos 2 m m is the reflection Sr2 along the vector r2 D sin m ; cos m . Indeed, we have the equalities #" # " sin m cos 2 sin 2 m m T .r2 / D cos m sin 2 cos 2 m m

2 2 2 2 D cos sin sin cos ; sin sin C cos cos m m m m m m m m

2 2 D sin ; cos D sin ; cos m m m m m m D r2 ; and so T D Sr2 . Hence, H2m is generated by two reflections Sr1 D S and Sr2 D T , and is easily seen to be an effective subgroup of O.R2 /. Therefore, H2m is a Coxeter group in R2 . Moreover, the set of all roots of H2m is of the form ²

³ k k ˇˇ D sin ; cos k D 0; 1; : : : ; 2m 1 : m m

492


Further, choosing t D cos 4m ; sin 4m , we have ²

k k D D sin ; cos m m ²

k k D t D sin ; cos m m C

C t

³ ˇ ˇ0k m1 ; ³ ˇ ˇ m k 2m 1 ;

and … t D fr1 ; r2 g is a t-basis for (see Exercise 6.10). The following theorem describes distinguished properties of bases of the root systems of Coxeter groups. Theorem 1.3. Let G be a Coxeter group in a real Euclidean space V and … t D fr1 ; : : : ; rm g a t-basis of the root system of G, for some vector t 2 V . Then the following statements hold. (i) .ri ; rj / 0 for all i ¤ j in f1; : : : ; mg. (ii) r1 ; : : : ; rm is a basis of the vector space V over R. C (iii) Sri C t n fri g D t n fri g, for any i 2 f1; : : : ; mg. Proof. (i) Take i ¤ j in f1; : : : ; mg. Then we have ri 2.ri ; rj /rj D Srj .ri / 2 . Hence, in order to prove that .ri ; rj / 0, it is enough to show that, for any positive real numbers and , the vector ri rj does not belong to . Suppose ri rj 2 for some positive real numbers and . Since D C t [ t , we C have two cases to consider. Assume first that ri rj 2 t . Then ri rj D ˛1 r1 C C ˛m rm for some nonnegative real numbers ˛1 ; : : : ; ˛m . For ˛i , we have X ˛k rk ; 0 D .˛i /ri C .˛j C /rj C k¤i;j

and hence the inequalities X 0 D t; .˛i /ri C .˛j C /rj C ˛k rk k¤i;j

X

D .˛i /.t; ri / C .˛j C /.t; rj / C

˛k .t; rk /

k¤i;j

.t; rj / > 0; a contradiction. For ˛i < , we have . ˛i /ri D . C ˛j /rj C

X k¤i;j

˛k rk ;


or equivalently, ri D

493

X ˛k C ˛j rj C rk ; ˛i ˛i k¤i;j

C˛

˛k with ˛ji and ˛ , k ¤ i; j , nonnegative real numbers, which contradicts the i minimality of the t -basis … t D fr1 ; : : : ; rm g of . Hence, ri rj … C t . C . Since D , we obtain that r r Assume now that ri rj 2 j i 2 t t t C , which leads to a contradiction (as above). Therefore, .r ; r / 0. i j t (ii) We prove first that r1 ; : : : ; rm are linearly independent vectors of the real space V . Assume r1 ; : : : ; rm are linearly dependent vectors of V . Then, after a renumbering of the vectors r1 ; : : : ; rm , we have an equality m X

i ri D 0

iD1

with 1 ; : : : ; k positive and kC1 ; : : : ; m nonpositive real numbers, for some P k 2 f1; : : : ; mg. We claim that kiD1 i ri D 0. Indeed, for k < m, we have the inequalities k k m

X

2 X X

0 i ri D i ri ; j rj iD1

D

k X

iD1 m X

j DkC1

i .j / ri ; rj 0;

iD1 j DkC1

because i > 0, .j / 0 and .ri ; rj / 0 for i 2 f1; : : : ; kg and j 2 fk C P 1; : : : ; mg. This shows that kiD1 i ri D 0. But then we obtain that k k X X 0 D t; i r i D i .t; ri / > 0; iD1

iD1

a contradiction. Therefore, r1 ; : : : ; rm are linearly independent vectors of V over R. We know also that G is generated by a finite number of reflections Sv1 ; : : : ; Svp along some unit roots v1 ; : : : ; vp of V , which clearly belong to . Since G is effective, the vectors v1 ; : : : ; vp generate the vector space V over R. On the other hand, every root r of is a combination of vectors r1 ; : : : ; rm with real coefficients. Hence, the vectors r1 ; : : : ; rm also generate the vector space V over R. Summing up, r1 ; : : : ; rm form a basis of the vector space V over R. In particular, we have m D dimR V . (iii) Let r 2 C t n fri g. Assume first that r 2 … t , so r D rj for some j ¤ i in f1; : : : ; mg. Then Sri .rj / D rj 2.rj ; ri /ri with at least one coefficient positive.

494


Since Sri .rj / 2 D C t [ t and r1 ; : : : ; rm form a basis of V over R, we C conclude that Sri .rj / 2 t , and clearly Sri .rj / ¤ ri , because Sri .Sri .rj // D rj ¤ ri . Assume now that r … … t . Then

rD

m X

j rj

j D1

with 1 0; : : : ; m 0 and at least two coefficients j positive. Without loss of generality, we may assume that ri ¤ r1 and 1 > 0. Then we obtain the equalities Sri .r/ D Sri

m X

j rj

j D1

D

m X

m X

j rj 2

j D1

j D1

D 1 r1 C

j rj ; ri ri

m X

j rj 2

j D2

m X

j rj ; ri ri :

j D1

C Since Sri .r/ 2 D C t [ t and 1 > 0, we conclude that Sri .r/ 2 t . Obviously, Sri .r/ ¤ ri , because Sri .Sri .r// D r ¤ ri .

The following theorem is fundamental for understanding the structure of Coxeter groups. Theorem 1.4. Let G be a Coxeter group in a real Euclidean space V and … t D fr1 ; : : : ; rn g a t-basis of the root system of G, for some vector t 2 V . Then the following statements hold. (i) The group G is generated by the reflections Sr1 ; : : : ; Srn along the vectors r1 ; : : : ; rn . (ii) For every reflection S 2 G, there exist T 2 G and i 2 f1; : : : ; ng such that S D ST .ri / . Proof. Denote by H the subgroup of G generated by the reflections Sr1 ; : : : ; Srn along the unit roots r1 ; : : : ; rn of … t . Clearly, H is a finite group. Moreover, by Theorem 1.3 (ii), r1 ; : : : ; rn is a basis of V over R. We prove first that for any vector v 2 V there exists an element T 2 H such that .T .v/; ri / 0 for all i 2 f1; : : : ; ng. Take v 2 V . Consider the vector v0 D

1 X r; 2 C r2 t


495

and take an element T 2 H such that .T .v/; v0 / is maximal among all inner products .R.v/; v0 /, for R 2 H . Applying Theorem 1.3 (iii), we obtain, for any i 2 f1; : : : ; ng, the equalities

1 1 X Sri .v0 / D Sri Sri .r/ ri C 2 2 C r2 t nfri g

1 1 D ri C 2 2

X

r

r2C t nfri g

D v0 ri : Then, invoking the maximality of .T .v/; v0 /, we obtain that .T .v/; v0 / .Sri T /.v/; v0 D .Sr2i T /.v/; Sri .v0 / D .T .v/; v0 ri / D .T .v/; v0 / .T .v/; ri / ; and hence .T .v/; ri / 0, for all i 2 f1; : : : ; ng. We prove now that every r 2 C t is of the form r D T .ri / for some T 2 H and i 2 f1; : : : ; ng. For r 2 … t , we may take T D idV . Hence, assume that r … … t . Since r1 ; : : : ; rn is a basis of V over R, r1 ; : : : ; rn ; r form a linearly dependent set of vectors of V . Further, r1 ; : : : ; rn ; r belong to C t . Hence, it follows from the proof of (ii) in Theorem 1.3 that .r; ri1 / > 0 for some i1 2 f1; : : : ; ng. Consider the vector v1 D Sri1 .r/ D r 2.r; ri1 /ri1 : C Applying Theorem 1.3 (iii), we conclude that v1 2 C t , because r 2 t n fri1 g. Moreover, we have

.t; v1 / D .t; r/ 2.r; ri1 /.t; ri1 / < .t; r/: If v1 D ri for some i 2 f1; : : : ; ng then, for T D Sri1 2 H , we have r D Sr2i .r/ D Sri1 .v1 / D Sri1 .ri / D T .ri /. Assume v1 … … t . Then, as above, there 1 exists i2 2 f1; : : : ; ng such that .v1 ; ri2 / > 0. Taking the vector v2 D Sri2 .v1 / D Sri2 Sri1 .r/ we conclude that v2 2 C t and .t; v2 / D .t; v1 / 2.v1 ; ri2 /.t; ri2 / < .t; v1 /: If v2 D ri for some i 2 f1; : : : ; ng, then, taking T D Sri1 Sri2 2 H , we obtain that r D Sri1 Sri2 Sri2 Sri1 .r/ D T .v2 / D T .ri /. For v2 … … t , we may continue

496


the above process. Since C t is a finite set, the process terminates with some vk D ri 2 … t of the form vk D Srik : : : Sri2 Sri1 .r/; for some i1 ; i2 ; : : : ; ik 2 f1; : : : ; ng. Then, for T D Sri1 Sri2 : : : Srik 2 H , we obtain that r D Sri1 Sri2 : : : Srik Srik : : : Sri2 Sri1 .r/ D T .vk / D T .ri /: We note that this shows the statement (ii), because every reflection S 2 G is of the form S D Sr D Sr for some r 2 C t . For the statement (i), observe that the group G is generated by the reflections Sr for all r 2 C t . Since, by the above C considerations, every r 2 t is of the form r D T .ri / for some T 2 H and i 2 f1; : : : ; ng, we obtain that Sr D ST .ri / D T Sri T 1 2 H , by Lemma 1.1. This shows that G D H . Let G be a Coxeter group in a real Euclidean space V . It follows from the above theorem that we may fix a vector t … ? and abbreviate C D C t , D , … D … . In fact, it is known that there exists a unique t -basis of t t (see Exercise 6.6). The elements of C (respectively, ) are called the positive roots (respectively, negative roots) of the Coxeter group G, and … the basis of . Moreover, … D fr1 ; : : : ; rn g, where r1 ; : : : ; rn are unit vectors forming a basis of the R-vector space V . Further, the roots r1 ; : : : ; rn are called the fundamental roots (or simple roots) of G and the associated reflections Sr1 ; : : : ; Srn are said to be the fundamental reflections of G. Hence, G is generated by its fundamental reflections. The following proposition is fundamental for combinatorial description of Coxeter groups. Proposition 1.5. Let G be a Coxeter group in a real Euclidean space V , and … t D fr1 ; : : : ; rn g a t -basis of the root system of G, for some vector t 2 V . Then for any i; j 2 f1; : : : ; ng there is a positive integer pij such that .ri ; rj / D cos

; pij

and pij is the order of the rotation Sri Srj in G. Proof. For i D j , we may take pij D 1. Moreover, if .ri ; rj / D 0, then Sri Srj D Srj Sri and we may take pij D 2. Therefore, assume i ¤ j and .ri ; rj / ¤ 0. Consider the 2-dimensional subspace W of V generated by the vectors ri and rj . Denote by H the subgroup of G generated by the reflections Sri and Srj . Observe that for the orthogonal decomposition V D W ˚ W ? of V given by W we have Sri jW ? D idW ? and Srj jW ? D idW ? . Further, the restrictions Si D Sri jW and Sj D Srj jW of Sri and Srj to W are reflections of W and generate the Coxeter


497

group H D fT jW j T 2 H g in W , which is isomorphic (see Exercise 6.11) to the dihedral group H2m with m 3 the order of Si Sj (equal to the order of Sri Srj ), described in Example 1.2. Let t D t1 C t2 with t1 2 W and t2 2 W ? . We claim that fr1 ; r2 g is the t1 -basis of the root system of the Coxeter group H . Suppose it is not the case. Observe that .t1 ; ri / D .t; ri / > 0 and .t1 ; rj / D .t; rj / > 0. Then we may choose a root r 2 and an element t10 2 W such that .t10 ; r/ > 0, .t10 ; ri / > 0, .t10 ; rj / > 0, and fr; rj g is a t10 -basis of , as illustrated below: t10 K t1 X2 22 ri 22 ? rj fM MMM 22 MMM 2 MM2 /r Observe that, by Theorem 1.3 (i) and our assumption, we have .ri ; rj / < 0, and hence the angle between ri and rj belongs to the interval 2 ; . Clearly, the root r of is also a root of , when it is considered as a vector of V , because the reflection Sr of V along r belongs to the subgroup H of G. On the other hand, r D i ri j rj for some i > 0 and j > 0, and this contradicts the fact that every root in is either in C t or in t . Summing up, it follows from Example 1.2 that indeed we have .ri ; rj / D cos pij where pij D m is the order of Sri Srj .

The above proposition allows us to assign to a Coxeter group G in a real Euclidean space V a marked graph .G/ as follows. The basis … D fr1 ; : : : ; rn g of the root system of G is the set of vertices of .G/, two vertices ri and rj of .G/ are joined by an edge if pij 3, and then we have in .G/ the marked edge ri

pij

rj :

3 rj of .G/ is usually replaced by the edge For pij D 3, the edge ri rj . ri Two Coxeter groups G and G 0 in a real Euclidean space V are said to be geometrically isomorphic if there exists T 2 O.V / such that G 0 D T GT 1 . The following lemma shows that a Coxeter group G is uniquely determined by its marked graph .G/ up to geometric isomorphism.

Lemma 1.6. Let G and G 0 be Coxeter groups in a real Euclidean space V . Then G and G 0 are geometrically isomorphic if and only if the marked graphs .G/ and .G 0 / are the same (up to permutation of vertices).

498


Proof. Let and 0 be the root systems of G and G 0 , and … and …0 the bases of and 0 , respectively. Assume .G/ D .G 0 /. This means that … and …0 are of the form … D fr1 ; : : : ; rn g and …0 D fr10 ; : : : ; rn0 g with n D dimR V and .ri ; rj / D .ri0 ; rj0 / for all i; j 2 f1; : : : ; ng. Consider the R-linear map T W V ! V such that T .ri / D ri0 for any i 2 f1; : : : ; ng. Obviously then T 2 O.V /. Moreover, for any i 2 f1; : : : ; ng, we have T Sri T 1 D ST .ri / D Sri0 , by Lemma 1.1. Further, by Theorem 1.4, G is generated by Sr1 ; : : : ; Srn and G 0 is generated by Sr10 ; : : : ; Srn0 . Since T Sri1 Sri2 : : : Srim T 1 D .T Sri1 T 1 /.T Sri2 T 1 / : : : .T Srim T 1 / D Sri0 Sri0 : : : Sri0 ; 1

2

m

for any i1 ; i2 ; : : : ; im 2 f1; : : : ; ng, we conclude that G 0 D T GT 1 , and hence G and G 0 are geometrically isomorphic. Conversely, if G 0 D T GT 1 for some T 2 O.V /, then 0 D T . / and we may take …0 D T .…/ as the basis of 0 . Hence, for … D fr1 ; : : : ; rn g, we have …0 D fr10 ; : : : ; rn0 g with ri0 D T .ri / for i 2 f1; : : : ; ng. Then we obtain .ri0 ; rj0 / D .ri ; rj / for all i; j 2 f1; : : : ; ng, which shows that .G/ D .G 0 /. Let G be a Coxeter group in a real Euclidean space V , the root system of G and … a basis of . Then G is said to be irreducible if … is not a union of two nonempty orthogonal subsets. We note that G is irreducible if and only if the marked graph .G/ is connected (see Exercise 6.7). We have the following general fact whose proof we leave to the reader (Exercises 6.8 and 6.9). Proposition 1.7. Let G be a Coxeter group in a real Euclidean space V . Then there is an orthogonal direct sum decomposition V D V1 ˚ V2 ˚ ˚ Vm of V such that the following statements hold. (i) T .Vi / Vi for any i 2 f1; : : : ; mg and T 2 G. (ii) For each i 2 f1; : : : ; mg, Gi D fT jVi j T 2 Gg is an irreducible Coxeter group in the real Euclidean space Vi . (iii) The canonical group homomorphism G ! G1 G2 Gm which assigns to T 2 G the m-tuple .T jV1 ; T jV2 ; : : : ; T jVm / is an isomorphism. It follows from the above proposition that the classification of Coxeter groups in a real Euclidean space V reduces to the classification of irreducible Coxeter groups in Euclidean spaces of dimensions smaller than or equal to dimR V .

2. Coxeter graphs

499

2 Coxeter graphs The aim of this section is to introduce a distinguished class of marked graphs called the Coxeter graphs, and provide their combinatorial characterization. In general, by a marked graph we mean a finite graph without multiple edges and loops, say with the set of vertices f1; : : : ; mg, such that every edge joining two vertices i and j is marked by a real number pij > 2, which we denote by i

pij

j :

3

j will be denoted simply by i j . As before, an edge i Let be a marked graph with the vertices 1; : : : ; m. We associate to the symmetric m m real matrix A D Œaij defined as follows: ai i D 1 for all i 2 pij

j joining i and f1; : : : ; mg, aij D cos pij if there is in a marked edge i j , and aij D 0 otherwise. Then we may consider the quadratic form q W Rm ! R defined by A , that is, q .1 ; : : : ; m / D

m X

aij i j

i;j D1

for .1 ; : : : ; m / 2 Rm . Recall also that, by the Silvester criterion, the quadratic form q is positive definite if and only if the principal minors det A.k/ of all principal submatrices 3 2 a11 a12 : : : a1k 6a21 a22 : : : a2k 7 7 6 A.k/ :: :: 7 D 6 :: 4 : : : 5 ak1 ak2 : : : akk k 2 f1; : : : ; mg, of A are positive. The following simple lemma will be useful. Lemma 2.1. Let be a marked graph and i1 a vertex of joined only with one other vertex i2 . Denote by 1 the marked graph obtained from by removing the vertex i1 and the unique edge joined to i1 , and by 2 the marked graph obtained from by removing the vertices i1 ; i2 and the edges attached to them. Moreover, let p D pi1 i2 . Then we have det A D det A1 .cos2 =p/ det A2 : Proof. We may assume (without loss of generality) that 1; 2; : : : ; m is the set of vertices of , i1 D 1 and i2 D 2. Then the matrix A has the form 3 2 1 cos =p 0 1 C 5 A D 4 cos =p 0 B A2

500


with A 1

1 D B

C : A 2

Hence we obtain the equalities cos =p 1 C det A D det C .cos =p/ det 0 B A2

C A2

D det A1 .cos2 =p/ det A2 :

Let be a marked graph. By a marked subgraph of we mean a marked graph † obtained from by the following operations: removing some vertices (and the edges attached to them), removing some edges, or decreasing the marks of some edges. For example, the marked graph 5

555

554

5 5

2 1 2 3 4 is a subgraph of the marked graph 5 6 6 5

55

4

554

5 5 10 1 2 3 4. Proposition 2.2. Let † be a nonempty marked subgraph of a marked graph , and assume that the quadratic form q is positive definite. Then the quadratic form q† is also positive definite. Proof. We may assume (without loss of generality) that 1; 2; : : : ; m are the vertices of and 1; 2; : : : ; k are the vertices of †, for some k m. Moreover, let A D .aij / and A† D .bij / be the real matrices associated to and †, respectively. Since † is a marked subgraph of , we have aij bij for all i; j 2 f1; : : : ; kg. Suppose that the quadratic form q† W Rk ! R is not positive definite. Then there exists a nonzero vector D .1 ; : : : ; k / 2 Rk such that q† .1 ; : : : ; k / 0. Consider the nonzero vector D .1 ; : : : ; m / 2 Rm with i D ji j for i 2 f1; : : : ; kg and i D 0 for i 2 fk C 1; : : : ; mg. Hence we obtain the inequalities 0 q† .1 ; : : : ; k / D

k X i;j D1

k X i;j D1

bij i j

k X i;j D1

aij ji jjj j D q .1 ; : : : ; m / > 0;

bij ji jjj j

2. Coxeter graphs

501

a contradiction, because ¤ 0 and q is positive definite. Therefore, q† is positive definite. By an integral marked graph we mean a marked graph for which every mark pij is integral. Clearly, the marked graph .G/ of a Coxeter group G is an integral marked graph. Consider the following family of integral marked graphs: ::: (n vertices), n 1 An W 4 (n vertices), n 2 Bn W ::: MMM MM ::: Dn W (n vertices), n 4 qq qqq E6 W E7 W E8 W 4 F4 W G2 W Hn2 W I3 W I4 W

6

n

n 5, n ¤ 6

5

5

, 3

means . The above graphs are called the where, as above, Coxeter graphs. The following theorem gives a combinatorial characterization of the Coxeter graphs. Theorem 2.3. Let be a connected integral marked graph. The following statements are equivalent. (i) is a Coxeter graph. (ii) q is a positive definite quadratic form. Proof. We prove first that (i) implies (ii), showing that the principal minors of the matrices associated to the Coxeter graphs are positive. We consider the possible cases.

502


Case An .n 1/. Observe that the matrix of the form 2 0 1 12 6 1 1 12 6 2 6 0 1 1 6 2 6 :: :: :: 6 : : : 6 4 0 0 0 0 0 0

matrix AAn of the graph An is the n n ::: ::: :::

0 0 0 :: :

::: :::

1 12

0 0 0 :: :

3

7 7 7 7 7; 7 7 15 2 1

since cos 3 D 12 . Hence, for k 2 f1; : : : ; ng, the k-th principal minor det A.k/ An of AAn is the determinant det AAk of the matrix AAk of Ak . We claim that det AAn D

nC1 2n

for any n 1. Clearly, det AA1 D 1 D 22 and det AA2 D 34 D applying Lemma 2.1, we obtain inductively that det AAn2 det AAn D det AAn1 cos2 3 n 1n1 nC1 D n1 D : n2 2 42 2n Case Bn .n 2/. For n D 2, we have " 1p 1 cos 4 D AB2 D cos 4 1 2 and hence det AB2 D 1 edge

4

p 2 2 2

2

D

1 2

3 . 22

For n 3,

p # 2 2

1

> 0. For n 3, applying Lemma 2.1 to the

, we obtain that

det ABn D det AAn1 cos2 det AAn2 4 n 1n1 1 D n1 D n1 > 0: n2 2 22 2

Observe also that det A.k/ Bn D det ABk , for k 2 f2; : : : ; ng, and the suitable numbering of the vertices of Bn . Case Dn .n 4/. Applying Lemma 2.1, we obtain that 1 det AAn3 4 n2 1 n1 D n2 > 0: 2 2

det ADn D det AAn1 D

n 2n1

2. Coxeter graphs

503

.1/ Moreover, we have also det A.k/ Dn D det ADk , for k 2 f4; : : : ; ng, det ADn D 1, .3/ 1 det A.2/ Dn D 1, and det ADn D det AA3 D 2 , for the corresponding numbering of the vertices of Dn . Case En .n D 6; 7; 8/. Fix n 2 f6; 7; 8g. Applying Lemma 2.1 to the left edge of En , we obtain that

1 det AAn2 4 1 1n1 9n D n3 D > 0: 2 4 2n2 2n Moreover, for the suitable numbering of the vertices of En , the remaining principal minors of AEn are principal minors of ADn1 , and hence are positive. Case F4 . Applying Lemma 2.1 to the left edge of F4 , we conclude that det AEn D det ADn1

1 1 3 1 det AA2 D D > 0: 4 4 16 16 Further, for the suitable numbering of the vertices of F4 , the remaining three principal minors of AF4 are principal minors of AB3 , and hence are positive. Case G2 . We have det AG2 D 1 cos2 6 D 1 34 D 14 > 0. Case Hn2 .n 5; n ¤ 6/. We have det AHn2 D 1 cos2 n > 0. In order to show the required claim for the graphs I3 and I4 , we calculate first cos2 5 . We set x D sin 5 and y D cos 5 . Then we have the equalities det AF4 D det AB3

2 5 2 cos 5 4 sin 5 4 cos 5 and consequently sin

D 2xy; D y2 x2; D 4xy.y 2 x 2 / D 4xy 3 4x 3 y; D .y 2 x 2 /2 .2xy/2 D y 4 6x 2 y 2 C x 4 ;

4 1 D cos D cos C 5 5 4 2 2 4 D y.y 6x y C x / x.4xy 3 4x 3 y/ D y 5 10x 2 y 3 C 5x 4 y: Since x 2 C y 2 D 1, this leads to the equality 1 D y 5 10.1 y 2 /y 3 C 5.1 y 2 /2 y D 16y 5 20y 3 C 5y:

504


Observe that 16y 5 20y 3 C 5y C 1 is divisible by y C 1, and y D cos 5 ¤ 1. Hence, we obtain the equality 16y 4 16y 3 4y 2 C 4y C 1 D 0: Substituting y D z C 14 , we then obtain

1 4 1 3 1 2 16 z C 4 zC C4 0 D 16 z C 4 4 4

3 1 1 D 16 z 4 C z 3 C z 2 C z C 16 z 3 C 8 16 256

1 1 1 2 4 z C zC C4 zC C1 2 16 4

2 25 5 D 16z 4 10z 2 C : D 16 z 2 16 16 Hence we get z D that z D

p 5 4

p 5 4

and y D

or z D

p 1C 5 . 4

p 5 . 4

Since z C

1 4

1 zC C1 4 3 2 3 1 z C zC 4 16 64

D y D cos 5 > 0, we conclude

This leads to p 2 p p 1C 5 6C2 5 3C 5 2 2 cos Dy D D D : 5 16 16 8

Cases I3 and I4 . Applying Lemma 2.1 to the edge

5

, we obtain that

det AI3 D det AA2 cos2 det AA1 5 p p 3 3C 5 3 5 D D > 0; 4 8 8! det AI4 D det AA3 cos2 det AA2 5 p ! p 1 3C 5 3 73 5 D D > 0: 2 8 4 32 Moreover, for the suitable numbering of the vertices of I3 (respectively, I4 ), the remaining minors of AI3 (respectively, AI4 ) are minors of AA2 (respectively, AA3 ), and hence are positive. This finishes the proof that (i) implies (ii). We prove now that (ii) implies (i). Consider the following family of marked graphs ::: MMM qq MM q q q z (n C 1 vertices), n 1 An W MMM q MM qqq q :::

2. Coxeter graphs

zn W 4 B MMM MM z Dn W qq qqq MMM MM f BDn W q q qqq

:::

:::

z6 W E

:::


4

qq qqq MMM MM

4



z F4 W z2W G Iz3 W

z7 W E z8 W E

Iz4 W

6

5

4

505

q

with cos q D 5 2

3 4

z n .n 1/, B z n .n 2/, We show that, if is one of the marked graphs A f z z z z z z z z Dn .n 4/, BDn .n 3/, E6 , E7 , E8 , F4 , G2 , I3 , or I4 , then det A D 0, and consequently the quadratic form q is not positive definite. z n .n 1/, the sum of all rows (respectively, columns) of the matrix For D A AAz n is zero, and hence det AAz n D 0. Applying Lemma 2.1 and calculations from the first part of the proof, we obtain the equalities 1 1 1 1 det ABn1 D n1 D 0 for n 3; 2 2 2 2n2 1 1 1 D det AB2 det AA1 D D 0; 2 2 2 1 1 1 1 D det ADn det ADn2 D n2 D 0 for n 6; 4 2 4 2n4 3 1 1 1 D det AD4 det AA1 D D 0; 4 4 4

det ABzn D det ABn det ABz2 det ADz n det ADz 4

506


1 1 11 det AA3 D D 0; 4 8 42 1 1 1 1 det ABD e n D det ABn 4 det ABn2 D 2n1 4 2n3 D 0 for n 4; 1 1 1 det ABD e 3 D det AB3 4 det AA1 D 4 4 D 0; 1 3 1 6 det AEz 6 D det AE6 det AA5 D 6 D 0; 4 2 4 25 1 2 1 1 det AEz 7 D det AE7 det AD6 D 7 D 0; 4 2 4 24 1 1 1 2 det AEz 8 D det AE8 det AE7 D 8 D 0; 4 2 4 27 1 1 1 1 det AFz4 D det AF4 det AB3 D D 0; 4 16 4 22 1 1 1 det AG det AA1 D D 0; z 2 D det AG2 4 4 4 p p p

73 5 3 53 5 2 2 det AI3 D D 0; det AIz4 D det AI4 cos 5 32 8 8 det ADz 5 D det AD5

D 2 cos2 because cos 2 5 p 3 5 . 8

5

p 5

1 D 2 3C8

1 D

p 1C 5 , 4

and hence cos2

2 5

D

Moreover, we have also 2

det AIz3

1 6 1 2 D det 6 4 0 0

12 1 34 0

0 34 1 12

3 0 0 7 7 D 0: 12 5 1

Assume now that q is positive definite. Then it follows from Proposition 2.2 z n , and hence is a marked that does not contain a marked subgraph of the form A tree. Further, if contains a marked subgraph of the form G2 (respectively, Hn2 , with n 7), then D G2 (respectively, D Hn2 ), because is connected and z 2 . Since is an integral marked does not contain a marked subgraph of the form G graph, we may assume that the edges of are marked by 3; 4, or 5. Suppose now that has an edge marked by 4 or 5, or equivalently B2 is a marked subgraph of . Then has only one edge with the mark greater than 3, because does not contain z n . Moreover, since does not contain a marked a marked subgraph of the form B f n , we conclude that every vertex of has at most two subgraph of the form BD neighbours. Further, if H52 is a marked subgraph of , then we infer that is one of the forms H52 , I3 , or I4 , because does not contain a marked subgraph of the form Iz3 or Iz4 . Finally, if H52 is not a marked subgraph of , then is one of the forms Bn or F4 , because does not contain Fz4 as a marked subgraph. Therefore, it remains to consider the case when all edges of are marked by 3. Clearly, if every vertex

3. The Coxeter theorems

507

of has at most two neighbours, then D An for some n 1. Assume ¤ An . z n , we conclude that Since does not contain a marked subgraph of the form D admits exactly one vertex having three neighbours, and the remaining vertices of have at most two neighbours. Since does not contain a marked subgraph of one z 6, E z 7 , or E z 8 , we then infer that is one of the forms Dn .n 4/, E6 , of the forms E E7 , or E8 .

3 The Coxeter theorems The following classification theorem has been proved by H. C. M. Coxeter in 1934 in [Cox] (see Theorems 5.1.3, 5.1.7, 5.3.1 in [GrBe]). Theorem 3.1. (i) Let G be an irreducible Coxeter group in a real Euclidean space V . Then the marked graph .G/ of G is one of the Coxeter graphs An .n 1/, Bn .n 2/, Dn .n 4/, E6 , E7 , E8 , F4 , G2 , Hn2 .n 5; n ¤ 6/, I3 , or I4 . (ii) For every Coxeter graph G, there exists an irreducible Coxeter group G in a real Euclidean space V such that .G/ D G. Proof. The statement (i) follows from Theorem 2.3 and the following simple observations. Let … D fr1 ; r2 ; : : : ; rn g be the basis of the root system of the Coxeter group G. It follows from the definition of the marked graph .G/ of G that its associated matrix A.G/ D Œaij is given by aij D .ri ; rj / for all i; j 2 f1; : : : ; ng. Moreover, we know that r1 ; r2 ; : : : ; rn is a basis of the vector P space V over R. Then, for any nonzero vector D .1 ; : : : ; n / 2 Rn , v D niD1 i ri is a nonzero vector of V , and we have q.G/ .1 ; : : : ; n / D

n X

aij i j D

i;j D1

D

n X iD1

n X

.ri ; rj /i j

i;j D1

i ri ;

n X

j rj D .v ; v / > 0:

j D1

Therefore, the quadratic form q.G/ W Rn ! R associated to the marked graph .G/ of G is positive definite. The proof of the statement (ii) is more involving, and will be provided by the examples below and Exercises 6.14–6.25. Observe that for m 3 the dihedral group H2m in R2 is an irreducible Coxeter group such that .H23 / D A2 , .H24 / D B2 , .H26 / D G2 , and .H2m / D Hm 2 for m 5, m ¤ 6. The following example shows that the permutation groups may be considered as Coxeter groups.

508


Example 3.2. Let n be a positive integer and SnC1 the permutation group of f1; 2; : : : ; n C 1g. Moreover, let RnC1 be the Euclidean space with the canonical inner product and basis e1 ; e2 ; : : : ; enC1 . Then we may view SnC1 as the subgroup of O.RnC1 / given by the R-linear automorphisms T of RnC1 with 2 SnC1 such that T .ei / D e.i/ for any i 2 f1; : : : ; n C 1g. Consider the hyperplane V of RnC1 consisting of all vectors x D .x1 ; x2 ; : : : ; xnC1 / 2 RnC1 with x1 Cx2 C CxnC1 D 0. Since V ? D R.1; 1; : : : ; 1/ ˚ D x 2 RnC1 j T .x/ D x for all 2 SnC1 ; we conclude that SnC1 may be viewed as the effective subgroup of O.V / consisting of the restrictions T D T jV of all T , 2 SnC1 , to V . For each i 2 f1; : : : ; ng, let ri D p1 .eiC1 ei / and Si D Sri be the reflection of V along the vector ri . Observe 2 that, for each transposition .i; i C1/ 2 SnC1 , i 2 f1; : : : ; ng, we have Si D T.i;iC1/ . Therefore, SnC1 is a Coxeter group in V generated by the reflections S1 ; : : : ; Sn . Further, ² ³ ˇ 1 ˇ D p .ei ej / i ¤ j; 1 i; j n C 1 2 is the set of all roots of SnC1 , and ² ³ ˇ 1 ˇ C D C D e / 1 j < i n C 1 .e p i j t 2 and

D

t

²

³

ˇ 1 D p .ei ej / ˇ 1 i < j n C 1 2

for an element t 2 V with .t; ri / > 0, for all i 2 f1; : : : ; ng (see Exercise 6.13). Moreover, … D … t D fr1 ; : : : ; rn g is the t-basis of . Since .ri ; riC1 / D

1 D cos 2 3

for i 2 f1; : : : ; ng, and .ri ; rj / D 0 for i; j 2 f1; : : : ; ng with ji j j 2, we conclude that .SnC1 / D An . We refer to [GrBe], (5.3), (see also Exercises 6.14–6.25) for constructions of irreducible Coxeter groups corresponding to the remaining Coxeter graphs. For an irreducible Coxeter group G its order jGj, depending on the Coxeter


509

graph .G/ of G, is described by the table below (see Table 5.4 in [GrBe]). .G/

jGj

An .n 1/ .n C 1/Š Bn .n 2/ 2n nŠ Dn .n 4/ 2n1 nŠ E6 27 34 5 E7 210 34 5 7 : E8 214 35 52 7 F4 27 32 G2 12 n H2 .n 5; n ¤ 6/ 2n 3 I3 2 35 I4 26 32 52 In general, a Coxeter group G in a real Euclidean space V is uniquely determined (up to a geometric isomorphism) by its marked graph .G/ which is a disjoint union of a finite number of Coxeter graphs. We will present also (Theorem 3.11) an alternative description of the Coxeter groups as abstract groups established by H. S. M. Coxeter [Cox]. For further considerations in this section, we need some technical results on Coxeter groups and their roots. Let G be a Coxeter group in a real Euclidean space V , the root system of G, t a fixed element of V with .t; r/ ¤ 0 for all roots r 2 , C D C t , and D the sets of positive roots and negative roots (with respect to t) t in , and … D … t D fr1 ; : : : ; rn g the unique t -basis of . We abbreviate the fundamental reflections of G as follows Si D Sri for any i 2 f1; : : : ; ng. It follows from Theorem 1.4 that S1 ; : : : ; Sn generate the group G, and hence every element T 2 G can be written as T D Si1 : : : Sik ; for some i1 ; : : : ; ik 2 f1; : : : ; ng. We note that in general T may have several presentations as product of fundamental reflections of G. By the length `.T / of an element T 2 G nf1g we mean the minimal positive integer k such that T is a product of k fundamental reflections of G. Moreover, the length `.1/ of the identity 1 of G is defined to be 0. For an element T 2 G n f1g, a presentation T D Si1 : : : Sik of T as product of fundamental reflections of G with k D `.T / is said to be a reduced presentation of T (in G). Our next aim is to provide an alternative description of the length `.T / of an element T 2 G given by N. Iwahori in [Iwa]. We need some preliminary results. We keep the notation introduced above.

510


For an element T 2 G, we introduce the subset (see [Iwa]) C 1 . / C T D \T

ˇ C ˇ 1 ˇ . / ˇ D of C and denote n.T / D j C T j. Observe that n.T / D T \ T jT . C / \ j. Lemma 3.3. Let T 2 G and T . C / D C . Then T D 1. Proof. Suppose T ¤ 1. Since T . C / D C , we have also T .…/ D …, by the uniqueness of the basis … of D C [ . Let T D Si1 : : : Sik be a reduced presentation of T . Observe that then k 2, because Si .ri / D ri for any i 2 f1; : : : ; ng. Then T .rik / D .Si1 : : : Sik1 /.Sik .rik // D .Si1 : : : Sik1 /.rik / belongs to …, and so .Si1 : : : Sik1 /.rik / 2 . Set a0 D .Si1 : : : Sik1 /.rik /; a1 D Si1 .a0 / D .Si2 : : : Sik1 /.rik /; a2 D Si2 .a1 / D .Si3 : : : Sik1 /.rik /; :: : ak1 D Sik1 .ak2 / D rik : Observe that a0 2 and ak1 2 C . Hence there exists a positive integer j 2 f1; : : : ; k 1g such that a0 ; a1 ; : : : ; aj 1 2 but aj 2 C . Then we have aj D Sij .aj 1 / 2 C and Sij .aj / D aj 1 2 . Since Sij . C n frij g/ D C n frij g, by Theorem 1.3 (iii), we conclude that aj D rij . Thus rij D .Sij C1 : : : Sik1 /.rik /, and hence, by Lemma 1.1, we have Sij D Srij D S.Si

j C1

:::Sik1 /.rik /

D .Sij C1 : : : Sik1 /Srik .Sij C1 : : : Sik1 /1

D .Sij C1 : : : Sik1 /Sik .Sij C1 : : : Sik1 /1 : This gives the equality Sij .Sij C1 : : : Sik1 / D .Sij C1 : : : Sik1 /Sik : But then we obtain that T D Si1 : : : Sik D .Si1 : : : Sij 1 /.Sij : : : Sik1 /Sik D .Si1 : : : Sij 1 /.Sij C1 : : : Sik /Sik D Si1 : : : Sij 1 Sij C1 : : : Sik1 is a product of k 2 fundamental reflections, which is in contradiction to the fact that k D `.T /.


511

Lemma 3.4. Let T 2 G and i 2 f1; : : : ; ng. Then C (i) Si . C T n fri g/ D T Si n fri g. C C C (ii) ri 2 C T [ T Si but ri … T \ T Si . C Proof. (i) Let r 2 C T n fri g. Then r 2 n fri g, and, applying Theorem 1.3 C (iii), we conclude that Si .r/ 2 n fri g. Since r 2 C T we have .T Si /.Si .r// D C C T .r/ 2 , and hence Si .r/ 2 T Si . This shows that Si . C T nfri g/ T Si nfri g. C C Replacing T by T Si , we obtain the inclusion Si . T Si n fri g/ T n fri g, and C so the inclusion C T Si n fri g Si . T n fri g/. Therefore, we get the equality C Si . C T n fri g/ D T Si n fri g.

(ii) Observe that .T Si /.ri / D T .ri / D T .ri /. Then ri … C T forces T .ri / 2 , and then .T Si /.ri / 2 , which contradicts (i). This shows that ri 2 C T [ C C C T Si . Similarly, if ri 2 T \ T Si , then T .ri / 2 and T .ri / D .T Si /.ri / 2 , a contradiction. C

Corollary 3.5. Let T 2 G and i 2 f1; : : : ; ng. Then we have ´ n.T / 1 if ri 2 C T, n.T Si / D n.T / C 1 if ri … C T. C C Proof. Assume ri 2 C T . Then ri … T Si , by Lemma 3.4 (ii), and hence T Si D C T Si n fri g. Applying Lemma 3.4 (i), we obtain the equalities

ˇ ˇ ˇ ˇ ˇ ˇ C C ˇ ˇ ˇ ˇ ˇ n.T Si / D ˇ C T Si D T Si n fri g D Si . T n fri g/ ˇ C ˇ ˇ Cˇ D ˇ n fri gˇ D ˇ ˇ 1 D n.T / 1: T

T

C C Assume now that ri … C T . Then T D T n fri g and, applying Lemma 3.4 (ii), we conclude that ri 2 C T Si . Hence, applying Lemma 3.4 (i) again, we obtain the equalities ˇ ˇ ˇ C ˇ ˇ ˇ C ˇ ˇ ˇ ˇ ˇ n.T Si / D ˇ C T Si D T Si n fri g C 1 D Si . T n fri g/ C 1 ˇ C ˇ ˇ Cˇ D ˇ n fri gˇ C 1 D ˇ ˇ C 1 D n.T / C 1: T

T

Proposition 3.6. For any T 2 G, we have `.T / D n.T /. Proof. We first show that for every T 2 G the inequality n.T / `.T / holds. Indeed, if `.T / D k then T D Si1 : : : Sik for some i1 ; : : : ; ik 2 f1; : : : ; ng, and,

512


applying Corollary 3.5, we obtain the inequalities n.Si1 / D 1; n.Si1 Si2 / D n.Si1 / ˙ 1 2; :: : n.Si1 : : : Sik / D n.Si1 : : : Sik1 / ˙ 1 .k 1/ ˙ 1 k; and so n.T / k D `.T /. We will show now by induction on m D n.T / that n.T / D `.T /. If m D 0, then it follows from Lemma 3.3 that T D 1, and hence `.T / D 0. Assume m 1 and n.R/ D `.R/ for all R 2 G with n.R/ < m. Since n.T / 1, the C 1 intersection C . / is and so we may choose r 2 C T D \T Pnonempty, n such thatP T .r/ 2 . Hence, r D iD1 i ri with all coefficients i 0, and T .r/ D niD1 i T .ri / 2 . This forces T .rj / 2 for some j 2 f1; : : : ; ng, and consequently rj 2 C T . Then it follows from Corollary 3.5 that n.T Sj / D n.T / 1 D m 1. Hence, by the induction hypothesis, we obtain that n.T Sj / D `.T Sj /. But then we obtain the inequalities `.T / D `..T Sj /Sj / `.T Sj / C 1 D n.T Sj / C 1 D .m 1/ C 1 D m D n.T /: Summing up, we have n.T / D `.T /.

As an immediate consequence of Proposition 3.6 and Corollary 3.5 we obtain the following fact. Corollary 3.7. Let T 2 G and i 2 f1; : : : ; ng. Then we have ´ `.T / C 1 if T .ri / 2 C , `.T Si / D `.T / 1 if T .ri / 2 . We note that for any T 2 G we have T . / D , and D C [ . Lemma 3.8. Let T 2 G and i 2 f1; : : : ; ng. Then there exists a reduced presentation T D Si1 : : : Sik of T with ik D i if and only if ri 2 C T. Proof. Assume that T D Si1 : : : Sik is a reduced presentation of T with ik D i . Then T Si D Si1 : : : Sik1 is a reduced presentation of T Si . Indeed, if it is not the case, then n.T Si / D `.T Si / < k 1 and hence n.T / < k, by Corollary 3.5. This contradicts the equality n.T / D `.T / D k, established in Proposition 3.6. Hence we have n.T Si / D `.T Si / D k 1 D `.T / 1. Applying Corollary 3.5 again, we conclude that ri 2 C T. Conversely, assume that ri 2 C T . Then it follows from Corollary 3.5 that n.T Si / D n.T / 1. Let T Si D Si1 : : : Sik1 be a reduced presentation of T Si . Clearly, then n.T Si / D `.T Si / D k 1, and hence `.T / D n.T / D n.T Si /C1 D k. Then T D Si1 : : : Sik1 Si is a reduced presentation of T .


513

For two fundamental reflections Si and Sj of G, we denote, as before, by pij the order of the rotation Si Sj . Then, for m 2 f1; : : : ; pij g, we denote by .: : : Si Sj /m the product : : : Si Sj Si Sj of m alternating factors Si and Sj , ending on the right with Sj . Similarly, .Si Sj : : : /m is the product Si Sj Si Sj : : : of m alternating factors Si and Sj , beginning on the left with Si . For m D 0, we set .: : : Si Sj /m D 1 and .Si Sj : : : /m D 1. Moreover, we denote by .: : : Si Sj : : : /m a product of m alternating factors Si and Sj . Lemma 3.9. Let Si and Sj be fundamental reflections of G. Then we have .: : : Si Sj /m1 .ri / 2 C for any m 2 f1; : : : ; pij g. Proof. If i D j then pij D 1 and .: : : Si Sj /0 D 1, so the claim is trivial. Assume i ¤ j , or equivalently, pij 2. Suppose the claim in the lemma is false. Take the smallest m in f1; : : : ; pij g such that .: : : Si Sj /m1 .ri / 2 . Observe that then m 2, by Theorem 1.3 (iii). We have two cases to consider. Assume that m is even. Then we have .: : : Si Sj /m1 .ri / D .Sj : : : Si Sj /m1 .ri / D Sj .: : : Si Sj /m2 .ri /: Hence Sj .: : : Si Sj /m2 .ri / 2 and .: : : Si Sj /m2 .ri / 2 C by the minimality of m. Applying Theorem 1.3 (iii), we conclude that .: : : Si Sj /m2 .ri / D rj . Then it follows from Lemma 1.1 that Sj D .: : : Si Sj /m2 Si .: : : Si Sj /1 m2 ; or equivalently, Sj .: : : Si Sj /m2 D .: : : Si Sj /m2 Si : This gives the equality .Sj : : : Si Sj /m1 D .Si : : : Sj Si /m1 : But then we obtain that .Si Sj /m1 D .Si Sj : : : /2m2 D .Si : : : Sj Si /m1 .Sj : : : Si Sj /m1 D .Si : : : Sj Si /m1 .Si : : : Sj Si /m1 D 1; which contradicts the fact that Si Sj has order pij , since m 1 < pij . For m odd the proof is similar.

Lemma 3.10. Let i; j 2 f1; : : : ; ng and T 2 G be such that `.T Si / D `.T / 1 D `.T Sj /. Then `.T .: : : Si Sj : : : /m / D `.T / m for any m 2 f0; 1; : : : ; pij g.

514


Proof. We use induction on m 2 f0; 1; : : : ; pij g. Observe that the claim is obvious for m D 0 and m D 1. Assume m 2 and that the claim holds for m 1. Since `.T Si / D `.T / 1 D `.T Sj /, applying Corollary 3.7, we conclude that T .ri / and T .rj / belong to . Moreover, from Lemma 3.9, we have .: : : Si Sj /m1 .ri / 2 C . On the other hand, it follows from the formulae on the reflections Si D Sri and Sj D Srj that .: : : Si Sj /m1 .ri / D ri C rj for some ; 2 R. Then .: : : Si Sj /m1 .ri / 2 C forces 0, 0, and so ¤ 0 or ¤ 0. Further, we have T .: : : Si Sj /m1 .ri / D T .ri C rj / D T .ri / C T .rj / 2 : Then, applying Corollary 3.7 and the induction hypothesis, we obtain that ` T .: : : Sj Si /m D ` T .: : : Si Sj /m1 Si D ` T .: : : Si Sj /m1 1 D `.T / .m 1/ 1 D `.T / m: Similarly, one proves that ` T .: : : Si Sj /m D `.T / m.

We denote by ˝

˛

S1 ; : : : ; Sn j .Si Sj /pij D 1; 1 i; j n

the group given by the generators S1 ; : : : ; Sn and the relations .Si Sj /pij D 1, for some positive integers pij , 1 i; j n, that is, the factor group of the free group generated by S1 ; : : : ; Sn by the minimal normal subgroup containing the elements of the form .Si Sj /pij , 1 i; j n. The following theorem proved by H. S. M. Coxeter in [Cox] gives a purely algebraic description of the Coxeter groups. The proof below is taken from [GrBe], Theorem 6.1.4. We will provide in Section 5 an alternative proof of this theorem, invoking the Hecke algebras. Theorem 3.11. Let G be a Coxeter group in a real Euclidean space V of dimension n. Then G is isomorphic to the group ˝ ˛ S1 ; : : : ; Sn j .Si Sj /pij D 1; 1 i; j n ; where S1 ; : : : ; Sn are fundamental reflections of G and pij the associated orders of the rotations Si Sj , 1 i; j n. Proof. We have to show that every relation W D Si1 : : : Sik D 1 in G is a consequence of the relations .Si Sj /pij D 1, 1 i; j n. Let W D Si1 : : : Sik D 1 for some i1 ; : : : ; ik 2 f1; : : : ; ng. For j 2 f1; : : : ; kg, Si1 : : : Sij is said to be a partial product of W . Let u be the maximal length of

4. The Iwahori theorem

515

partial products of W . Then we may write W D W1 Si Sj W2 , where `.W1 Si / D u and every partial product of W1 has length less than u. Let p D pij . Consider W 0 D W1 .Sj Si : : : /2p2 W2 . Observe that we have in G the equality W 0 D W1 .Sj Si /p1 W2 D W1 .Si Sj /W2 D W; since .Si Sj /p D 1 forces .Sj Si /p1 D Si Sj . Then the partial products of W , with exception of W1 Si , coincide, as the group elements, with the corresponding partial products of W 0 . Moreover, W 0 has the partial products W1 Sj ; W1 Sj Si ; : : : ; W1 .Sj Si : : : /2p3 in place of the partial product W1 Si of W . Taking T D W1 Si and using the relation Si2 D 1, we see that the above partial products of W 0 coincide, as the group elements, with the products T .Si Sj : : : /q ;

2 q 2p 2:

Since `.T / D u is the maximal among the length of partial products of W , we have `.T Si / D `.T / 1 D `.T Sj /. For m 2 f2; : : : ; pg, we get `.T .Si Sj : : : /m / < u, by Lemma 3.10. For m 2 fp C 1; : : : ; 2p 2g, we have 2 2p m < p and hence `.T .Si Sj : : : /m / D `.T .Sj Si : : : /2pm / < u, by the equality .Si Sj /p D 1 and Lemma 3.10. Therefore, by applying the relation .Si Sj /pij D 1 in G, we have replaced the product W D W1 Si Sj W2 by another product W 0 D W1 .Sj Si /p1 W2 all of whose partial products have length less than or equal to u, and there are fewer partial products of length u. Repeating the above procedure we may replace the relation W D Si1 : : : Sik D 1 by the relation 1 D 1, and the theorem is proved. We note that, for i ¤ j in f1; : : : ; ng, the relation .Si Sj /pij D 1, describing the order of the rotation Si Sj in G, can be written equivalently as .Si Sj /pij =2 D .Sj Si /pij =2 .Si Sj /

.pij 1/=2

Si D .Sj Si /

if pij is even;

.pij 1/=2

Sj

if pij is odd:

Moreover, the positive integers pij , i; j 2 f1; : : : ; ng, are fully visible in the marked graph .G/ of G, which is a disjoint union of Coxeter graphs, describing the Coxeter group G up to a geometric isomorphism.

4 The Iwahori theorem We are now in position to introduce the Hecke algebras of Coxeter groups. Let K be a field and q 2 K n f0g. Moreover, let G be a Coxeter group in a real Euclidean space V , S1 ; : : : ; Sn a set of fundamental reflections generating G, and

516


pij the order of Si Sj , for any i; j 2 f1; : : : ; ng. The Hecke algebra HK;q ŒG is the factor algebra KhX1 ; : : : ; Xn i=IK;q ŒG of the polynomial algebra KhX1 ; : : : ; Xn i in n noncommuting variables X1 ; : : : ; Xn over the field K by the ideal IK;q ŒG generated by the following elements: (1) .Xi q/.Xi C 1/ for i 2 f1; : : : ; ng, (2) .Xi Xj /pij =2 .Xj Xi /pij =2 if pij is even, .Xi Xj /.pij 1/=2 Xi .Xj Xi /.pij 1/=2 Xj if pij is odd, for all i ¤ j in f1; : : : ; ng. We will prove that HK;q ŒG is a finite dimensional symmetric K-algebra. In fact, we are going to show that HK;q ŒG is as K-vector space isomorphic to the K-vector space of the group algebra KG of G. For each i 2 f1; : : : ; ng, we denote by xi the coset Xi C IK;q ŒG of Xi in HK;q ŒG. Denote by R.G/ the set of all finite sequences .i1 ; : : : ; ir / with i1 ; : : : ; ir 2 f1; : : : ; ng such that `.Si1 : : : Sir / D r. We say that two elements .i1 ; : : : ; ir / and .j1 ; : : : ; j t / in R.G/ are equivalent, and write .i1 ; : : : ; ir / .j1 ; : : : ; j t /, if Si1 : : : Sir D Sj1 : : : Sj t . Observe that if .i1 ; : : : ; ir / .j1 ; : : : ; j t / then r D `.Si1 : : : Sir / D `.Sj1 : : : Sj t / D t . An element .i1 ; : : : ; ir / of R.G/ is said to be an admissible sequence and r is called its length. Obviously is an equivalence relation in R.G/. We note that, if .i1 ; : : : ; ir / 2 R.G/, then .i1 ; : : : ; ik / 2 R.G/ for any k 2 f1; : : : ; rg. We denote by x W R.G/ ! HK;q ŒG the map which assigns to an admissible sequence D .i1 ; : : : ; ir / in R.G/ the monomial x./ D xi1 : : : xir in HK;q ŒG. The following combinatorial theorem, proved by N. Iwahori in [Iwa], will be fundamental for describing the dimension of HK;q ŒG. Theorem 4.1. For all sequences and in R.G/ with , we have x./ D x./ in HK;q ŒG. Proof. Let D .i1 ; : : : ; ir / and D .j1 ; : : : ; jk / be sequences in R.G/ such as . We shall prove the theorem by induction on the length k of and . For k D 1, we have D .i1 / D .j1 / D , because Si1 D Sj1 forces i1 D j1 , and hence x./ D xi1 D xj1 D x./. Assume k 2 and the theorem is valid for all sequences and in R.G/ which are equivalent and of length < k. Observe that, if ik D jk , then Si1 : : : Sik1 Sik D Sj1 : : : Sjk1 Sjk gives the equality Si1 : : : Sik1 D Sj1 : : : Sjk1 , and `.Si1 : : : Sik1 / D k 1 D `.Sj1 : : : Sjk1 /, because `.Si1 : : : Sik / D k D `.Sj1 : : : Sjk /. Then 0 D .i1 ; : : : ; ik1 / and 0 D .j1 ; : : : ; jk1 / are equivalent sequences in R.G/, and, by the induction assumption, we have xi1 : : : xik1 D x.0 / D x.0 / D xj1 : : : xjk1 . Hence we obtain x./ D xi1 : : : xik1 xik D xj1 : : : xjk1 xjk D x./. Therefore, we may assume that ik ¤ jk . We divide the proof into several steps.


517

Since Si1 : : : Sik D Sj1 : : : Sjk , applying Lemma 3.8, we conclude that Sj1 : : : Sjk .rik / D Si1 : : : Sik .rik / 2 : Hence there is an integer l 2 f2; : : : ; kg such that Sjl : : : Sjk .rik / 2 C but Sjl1 Sjl : : : Sjk .rik / 2 . We know from Theorem 1.3 (iii) that the equality Sjl1 . C n frjl1 g/ D C n frjl1 g holds. Thus we get Sjl : : : Sjk .rik / D rjl1 . Then, applying Lemma 1.1, we obtain the equality .Sjl : : : Sjk /Sik .Sjl : : : Sjk /1 D Sjl1 ; and hence the equality Sjl : : : Sjk Sik D Sjl1 Sjl : : : Sjk :

(1)

We consider two cases: l > 2 and l D 2. Assume first that l > 2. Then the right side of the equality (1) is a reduced presentation, because Sjl : : : Sjk is a reduced presentation, and has length k l C 2 < k. Hence the left side of (1) is also a reduced presentation and has the same length k l C 2. This shows that .jl ; : : : ; jk ; ik / and .jl1 ; jl ; : : : ; jk / are equivalent sequences in R.G/ of length < k, and then by the induction assumption we get xjl : : : xjk xik D xjl1 xjl : : : xjk :

(2)

Now (1) and Si1 : : : Sik D Sj1 : : : Sjk imply that Sj1 : : : Sjl2 Sjl : : : Sjk Sik D Sj1 : : : Sjl2 Sjl1 Sjl : : : Sjk D Sj1 : : : Sjk D Si1 : : : Sik ; and hence we obtain the equality Sj1 : : : Sjl2 Sjl : : : Sjk D Si1 : : : Sik1 : Both sides of the above equality are reduced presentations of the same length k 1, and so .j1 ; : : : ; jl2 ; jl ; : : : ; jk / and .i1 ; : : : ; ik1 / are equivalent sequences in R.G/. Hence we get by our induction assumption that xj1 : : : xjl2 xjl : : : xjk D xi1 : : : xik1 :

(3)

Combining now (2) and (3), we obtain x./ D xi1 : : : xik D .xi1 : : : xik1 /xik D xj1 : : : xjl2 .xjl : : : xjk xik / D xj1 : : : xjl2 xjl1 xjl : : : xjk D x./: Therefore, we may assume that in (1) we have l D 2. Then we have Si1 : : : Sik D Sj1 : : : Sjk D Sj2 : : : Sjk Sik :

(4)

518


This gives the equality Si1 : : : Sik1 D Sj2 : : : Sjk , where both sides are reduced presentations of length k 1. Then, by our induction assumption, we obtain that xi1 : : : xik1 D xj2 : : : xjk : Therefore, in order to prove that x./ D xi1 : : : xik D xj1 : : : xjk D x./, it is sufficient to show the equality xj2 : : : xjk xik D xj1 : : : xjk :

()

Let p D pik jk . Since ik ¤ jk , we have p 2. Assume p D 2. Then Sik Sjk D Sjk Sik . Using (4) we then obtain that Sj1 : : : Sjk D Sj2 : : : Sjk Sik D Sj2 : : : Sjk1 Sik Sjk ; and hence the equality Sj1 : : : Sjk1 D Sj2 : : : Sjk1 Sik . Since the left side of this equality is a reduced presentation of length k 1, the right side is also a reduced presentation of length k 1. Hence, by our induction assumption, we get xj1 : : : xjk1 D xj2 : : : xjk1 xik :

(5)

On the other hand, we have in HK;q ŒG the equality xik xjk D xjk xik (as pik jk D 2). Combining this with (5), we obtain that xj2 : : : xjk1 xjk xik D xj2 : : : xjk1 xik xjk D xj1 : : : xjk ; and so the required equality () holds. Assume p 3. From Theorem 1.3 (iii) and Lemma 3.9 we have Sik .rjk / 2 C and Sjk Sik .rjk / 2 C . On the other hand, applying Lemma 3.8, we infer that Sj2 : : : Sjk Sik .rjk / D .Sj1 : : : Sjk1 /Sjk .rjk / 2 . Hence there exists m 2 f3; : : : ; kg such that Sjm : : : Sjk Sik .rjk / 2 C

and

Sjm1 Sjm : : : Sjk Sik .rjk / 2 :

Since Sjm1 . C n frjm1 g/ D C n frjm1 g, by Theorem 1.3 (iii), we obtain that Sjm : : : Sjk Sik .rjk / D rjm1 . Applying now Lemma 1.1, we conclude that Sjm : : : Sjk Sik Sjk D Sjm1 Sjm : : : Sjk Sik :

(6)

Assume now that m > 3. Then both sides of (6) are of length k m C 3 < k and are reduced presentations, since Sj1 : : : Sjk D Sj2 : : : Sjk Sik are reduced presentations. Hence, by our induction assumption, we obtain the equality xjm : : : xjk xik xjk D xjm1 xjm : : : xjk xik :

(7)


519

Observe also that we have from (4) and (6) the equalities Sj1 : : : Sjk D Sj2 : : : Sjk Sik D .Sj2 : : : Sjm2 /.Sjm1 Sjm : : : Sjk Sik / D Sj2 : : : Sjm2 Sjm : : : Sjk Sik Sjk ; and hence the equality Sj1 : : : Sjk1 D Sj2 : : : Sjm2 Sjm : : : Sjk Sik : Since the left side of this equality is a reduced presentation we infer that both sides are reduced presentations of length k 1. Hence, by the induction assumption, we obtain the equality xj1 : : : xjk1 D xj2 : : : xjm2 xjm : : : xjk xik :

(8)

Combining now (7) and (8), we obtain that xj2 : : : xjk xik D .xj2 : : : xjm2 /.xjm1 xjm : : : xjk xik / D xj2 : : : xjm2 xjm : : : xjk xik xjk D xj1 : : : xjk1 xjk ; and so the required equality () holds. Assume now that in (6) we have m D 3. Then we have Sj3 : : : Sjk Sik Sjk D Sj2 Sj3 : : : Sjk Sik D Sj1 : : : Sjk1 Sjk ;

(9)

and consequently the equality Sj1 : : : Sjk1 D Sj3 : : : Sjk Sik : Since the left side of this equality is a reduced presentation, then both sides are reduced presentations of length k 1. Then, by the induction assumption, we obtain that xj1 : : : xjk1 D xj3 : : : xjk xik :

(10)

Assume p D 3. Then we have Sjk Sik Sjk D Sik Sjk Sik . Invoking now the equalities (9), we obtain the equalities Sj2 : : : Sjk1 D .Sj2 : : : Sjk1 /.Sjk Sik Sik Sjk / D .Sj2 : : : Sjk1 Sjk Sik /Sik Sjk D .Sj3 : : : Sjk Sik Sjk /Sik Sjk D .Sj3 : : : Sjk /.Sik Sjk Sik /Sjk D .Sj3 : : : Sjk1 /Sjk .Sjk Sik Sjk /Sjk D Sj3 : : : Sjk1 Sik :

520


Since Sj2 : : : Sjk1 is a reduced presentation of length k 2, Sj3 : : : Sjk1 Sik is then also a reduced presentation of length k 2, and so we conclude, by our induction assumption, that xj2 : : : xjk1 D xj3 : : : xjk1 xik :

(11)

Moreover, we have in HK;q ŒG the equality xjk xik xjk D xik xjk xik , since p D 3. Applying now (10) and (11), we obtain the equalities xj2 : : : xjk1 xjk xik D xj3 : : : xjk1 xik xjk xik D xj3 : : : xjk1 xjk xik xjk D xj1 : : : xjk1 xjk ; and so the required equality () holds. Assume now p 4. Then it follows from Lemma 3.9 that Sjk .rik / 2 C , .Sik Sjk /.rik / 2 C , .Sjk Sik Sjk /.rik / 2 C . On the other hand, applying (9) and Lemma 3.8, we obtain that .Sj3 : : : Sjk Sik Sjk /.rik / D .Sj2 : : : Sjk Sik /.rik / 2 : Hence there exists q 2 f4; : : : ; kg such that .Sjq : : : Sjk Sik Sjk /.rik / 2 C

and

.Sjq1 Sjq : : : Sjk Sik Sjk /.rik / 2 ;

and consequently .Sjq : : : Sjk Sik Sjk /.rik / D rjq1 , by Theorem 1.3 (iii). Applying now Lemma 1.1, we conclude that Sjq : : : Sjk Sik Sjk Sik D Sjq1 Sjq : : : Sjk Sik Sjk :

(12)

Assume now that q > 4. Observe that then the right side of the above equality is a part of the reduced presentation Sj3 : : : Sjk Sik Sjk , and hence is a reduced presentation. Hence the left side of this equality is also a reduced presentation, and both sides have length k q C 4 < k. Hence, by our induction assumption, we obtain the equality xjq1 xjq : : : xjk xik xjk D xjq : : : xjk xik xjk xik : Further, it follows from (9) and (12) that Sj2 Sj3 : : : Sjk Sik D Sj3 : : : Sjk Sik Sjk D .Sj3 : : : Sjq2 /.Sjq1 Sjq : : : Sjk Sik Sjk / D Sj3 : : : Sjq2 Sjq : : : Sjk Sik Sjk Sik ; and hence we get Sj2 Sj3 : : : Sjk1 D Sj3 : : : Sjq2 Sjq : : : Sjk Sik :

(13)


521

Since the left side of this equality is a reduced presentation, we conclude that both sides are reduced presentations of length k 2 < k. Then, by the induction assumption, we conclude that xj2 xj3 : : : xjk1 D xj3 : : : xjq2 xjq : : : xjk xik :

(14)

Therefore, applying (10), (13) and (14), we obtain the equalities xj1 : : : xjk1 xjk D xj3 : : : xjk xik xjk D .xj3 : : : xjq2 /.xjq1 xjq : : : xjk xik xjk / D .xj3 : : : xjq2 /.xjq : : : xjk xik /xjk xik D xj2 xj3 : : : xjk1 xjk xik ; and hence the required equality () holds. Assume that in (12) we have q D 4. Then we have from (9) and (12) the equalities Sj2 Sj3 : : : Sjk1 Sjk Sik D Sj3 : : : Sjk Sik Sjk D Sj4 : : : Sjk Sik Sjk Sik ;

(15)

and hence we obtain that Sj2 Sj3 : : : Sjk1 D Sj4 : : : Sjk Sik : Since the left side of this equality is a reduced presentation, the right side is also a reduced presentation of the same length k 2 < k, we conclude, by the induction assumption, that xj2 xj3 : : : xjk1 D xj4 : : : xjk xik :

(16)

Assume p D 4. Then .Sik Sjk /4 D 1 gives the equality .Sik Sjk /2 D .Sjk Sik /2 . Applying (15), we obtain then Sj3 : : : Sjk1 Sjk Sik Sjk D Sj4 : : : Sjk1 .Sjk Sik /2 D Sj4 : : : Sjk1 .Sik Sjk /2 D Sj4 : : : Sjk1 Sik Sjk Sik Sjk ; and hence the equality Sj3 : : : Sjk1 D Sj4 : : : Sjk1 Sik : Since the left side is a reduced presentation, the right side is also a reduced presentation of the same length k 3 < k, and then we conclude, by our induction assumption, that xj3 : : : xjk1 D xj4 : : : xjk1 xik :

522


Moreover, we have in HK;q ŒG the equality .xik xjk /2 D .xjk xik /2 . Then, using (10) and (16), we obtain xj2 : : : xjk xik D .xj2 : : : xjk1 /xjk xik D .xj4 : : : xjk xik /xjk xik D xj4 : : : xjk1 .xjk xik /2 D xj4 : : : xjk1 .xik xjk /2 D .xj4 : : : xjk1 xik /xjk xik xjk D .xj3 : : : xjk1 /xjk xik xjk D .xj3 : : : xjk1 xjk xik /xjk D xj1 : : : xjk1 xjk ; and so the required equality () holds. Therefore, we may assume now that p 5. Assume p D 5. Then it follows from Lemma 3.9 that Sik .rjk / 2 C , .Sjk Sik /.rjk / 2 C , .Sik Sjk Sik /.rjk / 2 C , .Sjk Sik /2 .rjk / 2 C . On the other hand, by (15) and Lemma 3.8, we conclude that .Sj4 : : : Sjk Sik Sjk Sik /.rjk / D .Sj3 : : : Sjk Sik Sjk /.rjk / 2 : Hence there exists t 2 f5; : : : ; kg such that .Sj t : : : Sjk Sik Sjk Sik /.rjk / 2 C and .Sj t 1 Sj t : : : Sjk Sik Sjk Sik /.rjk / 2 : Then it follows from Theorem 1.3 (iii) that .Sj t : : : Sjk Sik Sjk Sik /.rjk / D rj t 1 : Hence, applying Lemma 1.1, we get the equality Sj t : : : Sjk Sik Sjk Sik Sjk D Sj t1 Sj t : : : Sjk Sik Sjk Sik : Further, since pik jk D p D 5, we have also the equality .Sik Sjk /2 Sik D .Sjk Sik /2 Sjk : This leads to the equality Sj t : : : Sjk1 Sik D Sj t1 Sj t : : : Sjk1 :

(17)

Since the right side of this equality is a reduced presentation, the left side of this equality is also a reduced presentation, and both of them have length k t C 1 < k. Hence, by our induction assumption, we obtain the equality xj t : : : xjk1 xik D xj t1 xj t : : : xjk1 :

(18)


523

Assume now that t > 5. Then we obtain from (15) and (17) the equalities Sj2 Sj3 : : : Sjk1 D Sj4 : : : Sjk Sik D Sj4 : : : Sj t2 .Sj t 1 Sj t : : : Sjk1 /Sjk Sik D Sj4 : : : Sj t2 Sj t : : : Sjk1 Sik Sjk Sik : Since the first term of these equalities is a reduced presentation, the last term is also a reduced presentation, and both of them have length k 2 < k. Hence, by the induction assumption, we get the equality xj2 xj3 : : : xjk1 D xj4 : : : xj t2 xj t : : : xjk1 xik xjk xik : This leads to the equality xj2 xj3 : : : xjk1 xjk xik D xj4 : : : xj t2 xj t : : : xjk1 .xik xjk /2 xik :

(19)

Further, it follows from (9), (15) and (17) that Sj1 : : : Sjk1 Sjk D Sj2 Sj3 : : : Sjk1 Sjk Sik D Sj4 : : : Sjk1 Sjk Sik Sjk Sik D Sj4 : : : Sj t2 .Sj t1 Sj t : : : Sjk1 /Sjk Sik Sjk Sik D Sj4 : : : Sj t2 Sj t : : : Sjk1 Sik Sjk Sik Sjk Sik : Since .Sik Sjk /2 Sik D .Sjk Sik /2 Sjk , we obtain the equality Sj1 : : : Sjk1 D Sj4 : : : Sj t2 Sj t : : : Sjk1 Sjk Sik Sjk Sik : Clearly, both sides of this equality are reduced presentations of length k 1, so the induction assumption implies that xj1 : : : xjk1 D xj4 : : : xj t2 xj t : : : xjk1 xjk xik xjk xik : Then, multiplying this equality by xjk , we obtain that xj1 : : : xjk1 xjk D xj4 : : : xj t2 xj t : : : xjk1 .xjk xik /2 xjk :

(20)

Observe that we have in HK;q ŒG the equality .xik xjk /2 xik D .xjk xik /2 xjk , because pik jk D p D 5. Therefore, combining this with (19) and (20), we obtain the required equality xj2 : : : xjk xik D xj1 : : : xjk . Assume now t D 5. Then, applying (15) and (17) again, we obtain that Sj2 Sj3 : : : Sjk1 D Sj4 : : : Sjk Sik D .Sj4 Sj5 : : : Sjk1 /Sjk Sik D Sj5 : : : Sjk1 Sik Sjk Sik :

524


Hence, by the induction assumption, we have the equality xj2 xj3 : : : xjk1 D xj5 : : : xjk1 xik xjk xik ; which leads to the equality xj2 xj3 : : : xjk1 xjk xik D xj5 : : : xjk1 .xik xjk /2 xik :

(21)

Similarly, using (9), (15) and (17), we obtain the equalities Sj1 : : : Sjk1 Sjk D Sj2 Sj3 : : : Sjk1 Sjk Sik D Sj4 : : : Sjk1 Sjk Sik Sjk Sik D Sj5 : : : Sjk1 Sik Sjk Sik Sjk Sik : Invoking again the equality .Sik Sjk /2 Sik D .Sjk Sik /2 Sjk , we get the equality Sj1 : : : Sjk1 D Sj5 : : : Sjk1 Sjk Sik Sjk Sik : Since both sides of this equality are reduced presentations of length k 1, the induction assumption implies that xj1 : : : xjk1 D xj5 : : : xjk1 xjk xik xjk xik : Hence, multiplying this equality by xjk , we get that xj1 : : : xjk1 xjk D xj5 : : : xjk1 .xjk xik /2 xjk :

(22)

Then, as above, the equality .xik xjk /2 xik D .xjk xik /2 xjk combined with (21) and (22) leads to the required equality xj2 : : : xjk xik D xj1 : : : xjk . Finally, assume that p 6. We know that the marked graph .G/ of the Coxeter group G is a disjoint union of Coxeter graphs, which are the connected components of .G/. Observe now that the Coxeter graphs G2 W

6

and Hn2 W

n

;

n 7;

are the unique Coxeter graphs having an edge marked by an integer greater than or equal to 6. This means that the edge rik

pik jk

rjk ;

with pik jk D p 6, forms a connected component of .G/. In particular, for any i 2 f1; : : : ; ng n fik ; jk g, we have pi ik D 2 D pijk , or equivalently, Si Sik D Sik Si and Si Sjk D Sjk Si . Observe also that .ri ; rik / D cos pii D cos 2 D 0 and k

D cos 2 D 0 for any i 2 f1; : : : ; ng n fik ; jk g. Since the .ri ; rjk / D cos pij k


525

basis … D fr1 ; : : : ; rn g of the root system of G is a basis of the Euclidean space V , we have the orthogonal direct sum decomposition V D V 0 ˚ V 00 , where V 0 is the R-vector subspace of V generated by the vectors ri , i 2 f1; : : : ; ng n fik ; jk g, and V 00 is the R-vector subspace of V generated by rik and rjk . Moreover, it follows from the formula on reflection that Si jV 00 D idV 00 for any i 2 f1; : : : ; ng n fik ; jk g, Sik jV 0 D idV 0 and Sjk jV 0 D idV 0 . Consider now T D Si1 : : : Sik D Sj1 : : : Sjk . Assume that l1 ; : : : ; la (respectively, m1 ; : : : ; mb ) are the consecutive indices from the sequence i1 ; : : : ; ik (respectively, j1 ; : : : ; jk ), different from ik and jk and such that Sl1 : : : Sla and Sm1 : : : Smb are reduced presentations. Then we have Si1 : : : Sik D Sl1 : : : Sla .: : : Sjk Sik /u D .: : : Sjk Sik /u Sl1 : : : Sla ; Sj1 : : : Sjk D Sm1 : : : Smb .: : : Sik Sjk /v D .: : : Sik Sjk /v Sm1 : : : Smb ; where u, v are from f1; : : : ; pik jk g. Then we have ˇ : : : .Sjk jV 00 /.Sik jV 00 / u D .: : : Sjk Sik /u ˇV 00 ˇ D T ˇV 00

ˇ D .: : : Sik Sjk /v ˇV 00 D : : : .Sik jV 00 /.Sjk jV 00 / v ;

where .Sik jV 00 / and .Sjk jV 00 / are reflections in V 00 generating the Coxeter group p H2 (with p D pik jk ) in V 00 . Since in the equality .: : : .Sjk jV 00 /.Sik jV 00 //u D .: : : .Sik jV 00 /.Sjk jV 00 //v both sides are reduced presentations, invoking the structure of the Coxeter group H2p , we conclude that u D v D p2 , if p is even, and u D v D pC1 , if p is odd. Therefore, we obtain the equalities 2 .: : : Sjk Sik /u D .Sjk Sik /p=2 D .Sik Sjk /p=2 D .: : : Sik Sjk /v ; if p is even, and .: : : Sjk Sik /u D .Sik Sjk /.p1/=2 Sik D .Sjk Sik /.p1/=2 Sjk D .: : : Sik Sjk /v ; if p is odd. Moreover, we obtain that Sl1 : : : Sla D Sm1 : : : Smb ; with both sides reduced presentations, and hence a D b. Since a < k, using the induction assumption, we conclude that xl1 : : : xla D xm1 : : : xma : Moreover, we have in HK;q ŒG the equalities .xik xjk /p=2 D .xjk xik /p=2

if p is even;

.xik xjk /.p1/=2 xik D .xjk xik /.p1/=2 xjk

if p is odd:

526


Further, for i 2 f1; : : : ; ng n fik ; jk g, we have pi ik D 2 D pijk , and hence xi xik D xik xi and xi xjk D xjk xi . Combining this together, we obtain the equalities xi1 : : : xik D xl1 : : : xla .: : : xjk xik /u D xm1 : : : xmb .: : : xik xjk /v D xj1 : : : xjk : Summing up, we have proved that x./ D xi1 : : : xik D xj1 : : : xjk D x./:

We note that in the proof the element q 2 K n f0g as well as the relations .xi q/.xi C 1/ D 0, i 2 f1; : : : ; ng, in HK;q ŒG did not play any role. Consider the K-vector space M HK;q ŒG D KuT ; T 2G

isomorphic to the K-vector space of the group algebra KG of G. The following corollary is an immediate consequence of Theorem 4.1. Corollary 4.2. There is an isomorphism of K-vector spaces 'G W HK;q ŒG ! HK;q ŒG

such that (i) 'G .u1 / D 1; (ii) 'G .uT / D xi1 : : : xik for any T 2 G n f1g and an arbitrary reduced presentation T D Si1 : : : Sik of T as product of fundamental reflections in G. In particular, we have dimK HK;q ŒG D jGj. The following theorem due to Iwahori [Iwa] provides a useful interpretation of Hecke algebras of Coxeter groups up to isomorphism. Theorem 4.3. The K-vector space HK;q ŒG admits a unique K-algebra structure such that the following statements hold. (i) u1 is the identity 1 D 1HK;q ŒG of HK;q ŒG.

(ii) If T D Si1 : : : Sik is a reduced presentation of an element T 2 G n f1g, then uT D uSi1 : : : uSik . (iii) u2Si D q C .q 1/uSi for any i 2 f1; : : : ; ng, where q D q1.


527

(iv) If T and R are elements of G with `.TR/ D `.T /C`.R/, then uTR D uT uR . (v) If T 2 G and i 2 f1; : : : ; ng, then ´ uT Si uT uSi D quT Si C .q 1/uT and

´ uSi uT D

uSi T quSi T C .q 1/uT

if `.T Si / D `.T / C 1; otherwise if `.Si T / D `.T / C 1; otherwise.

(vi) For any T 2 G n f1g, uT is an invertible element of HK;q ŒG with inverse 1 1 1 uT D uSi : : : uSi , where T D Si1 : : : Sik is a reduced presentation of T , 1 k and u1 Si D .1 q/=q C .1=q/uSi ;

for any i 2 f1; : : : ; ng. Proof. It follows from definition of the K-algebra HK;q ŒG and Corollary 4.2 that there is a unique K-algebra structure on HK;q ŒG satisfying the conditions (i)–(iii), and it is the K-algebra structure for which the K-linear isomorphism 'G W HK;q ŒG ! HK;q ŒG is an isomorphism of K-algebras. We will show that ŒG. then the statements (iv)–(vi) hold in such K-algebra HK;q (iv) Let T; R 2 G and `.TR/ D `.T / C `.R/. Then, for reduced presentations T D Si1 : : : Sik and R D Sj1 : : : Sjl , the presentation TR D Si1 : : : Sik Sj1 : : : Sjl is reduced, because k C l D `.T / C `.R/ D `.TR/. Then we obtain 'G .uTR / D xi1 : : : xik xj1 : : : xjl D .xi1 : : : xik /.xj1 : : : xjl / D 'G .uT /'G .uR / D 'G .uT uR /; and hence uTR D uT uR . (v) Let T 2 G and i 2 f1; : : : ; ng. Clearly, if `.T Si / D `.T / C 1, then we have uT uSi D uT Si , by (iv), because `.Si / D 1. Assume `.T Si / ¤ `.T / C 1. For the root ri corresponding to Si D Sri , we have T .ri / 2 D C [ and T Si T 1 D ST .ri / (see Lemma 1.1). On the other hand, by Corollary 3.7, we have also ´ `.T / C 1 if T .ri / 2 C , `.T Si / D `.T / 1 if T .ri / 2 . Hence our assumption `.T Si / ¤ `.T / C 1 forces `.T Si / D `.T / 1 and T .ri / 2 , or equivalently, ri 2 C T . Applying now Lemma 3.8, we conclude that T admits a reduced presentation T D Si1 : : : Sik1 Sik with ik D i . Then

528


we get T Si D .Si1 : : : Sik1 Sik /Si D Si1 : : : Sik1 , and the last term is a reduced presentation of T Si . Invoking now (iii) and (iv), we obtain the equalities uT uSi D .uSi1 : : : uSik1 uSi /uSi D .uSi1 : : : uSik1 /.u2Si / D uT Si .q C .q 1/uSi / D quT Si C .q 1/uT Si uSi D quT Si C .q 1/uT ; because uT Si uSi D u.T Si /Si D uT due to `.T / D `.T Si / C 1. The proof of the second part of (v) is similar by using the corresponding versions of Corollary 3.7 and Lemma 3.8, observing that Si T D .T 1 Si /1 . (vi) For each i 2 f1; : : : ; ng, we have, by (iii), the equalities .1 q/=q C .1=q/uSi uSi D .1 q/=q uSi C .1=q/ u2Si D .1 q/=q uSi C .1=q/ q C .q 1/uSi D 1 C .1 q/=q uSi .1 q/=q uSi D 1; ŒG. Then for T 2 G n f1g so .1 q/=q C .1=q/uSi is the inverse of uSi in HK;q and a reduced presentation T D Si1 : : : Sik of T , we have the equalities 1 1 1 uT u1 Si : : : uSi D .uSi1 : : : uSik /.uSi : : : uSi / D 1; k

1

k

1

1 1 so indeed u1 T D uSi : : : uSi . k

1

5 Hecke algebras Consider now the K-linear automorphism ŒG ! HK;q ŒG W HK;q

such that .uT / D uT 1 for any T 2 G. Lemma 5.1. is a K-algebra antiautomorphism of HK;q ŒG of order 2.

Proof. Clearly, we have 2 .uT / D .uT 1 / D uT for any T 2 G. We have to show that .uT uR / D .uR / .uT / for all T; R 2 G. We may assume that T ¤ 1 ¤ R, since .u1 / D u1 . We first show that .uT uSi / D .uSi / .uT / for any T 2 G and i 2 f1; : : : ; ng. Fix i 2 f1; : : : ; ng. Applying Theorem 4.3 (v), we obtain that ´ if `.T Si / D `.T / C 1; uT Si uT uSi D quT Si C .q 1/uT otherwise. We note also that .uSi / D uSi , since Si2 D 1 in G.

5. Hecke algebras

529

Assume `.T Si / D `.T / C 1. Then we obtain that .uT uSi / D .uT Si / D u.T Si /1 D uSi T 1 D uS 1 uT 1 D .uSi / .uT /; i

because `.Si T 1 / D `..T Si /1 / D `.T Si / D `.T / C 1 D `.Si / C `.T 1 /, and the property (iv) in Theorem 4.3 can be applied. Assume `.T Si / ¤ `.T / C 1. Then we obtain that .uT uSi / D .quT Si C .q 1/uT / D q .uT Si / C .q 1/ .uT / D quSi T 1 C .q 1/uT 1 D uSi uT 1 D .uSi / .uT /; because `.Si T 1 / D `..T Si /1 / D `.T Si / ¤ `.T / C 1 D `.T 1 / C 1. Take now T; R 2 G n f1g, and let R D Sj1 : : : Sjl be a reduced presentation of R. Then we get the equalities .uT uR / D .uT uSj1 : : : uSjl / D .uSjl / : : : .uSj1 / .uT / D uSjl : : : uSj1 .uT / D uSjl :::Sj1 .uT / D uR1 .uT / D .uR / .uT /; because R1 D Sjl : : : Sj1 is a reduced presentation of R1 .

Consider the symmetric K-bilinear form .; /0 W HK;q ŒG HK;q ŒG ! K

´

such that .uT ; uR /0 D

q `.T / 0

if T D R; otherwise.

Observe that the form .; /0 is nondegenerate, because for a nonzero element X hD T uT T 2G

with T 2 K, for T 2 G, and R ¤ 0, we have .h; uR /0 D R q `.R/ ¤ 0. ŒG, we have .h1 h2 ; h3 /0 D Lemma 5.2. For all elements h1 ; h2 ; h3 2 HK;q 0 .h1 ; h3 .h2 // .

Proof. It is enough to show that .uT1 uT2 ; uT3 /0 D .uT1 ; uT3 .uT2 //0 for all elements T1 ; T2 ; T3 2 G. We may assume that T2 ¤ 1, since u1 is the identity of ŒG. HK;q

530


Assume first that T2 D Si , for some i 2 f1; : : : ; ng. Observe that .uSi / D uSi , since Si 2 D 1 in G. We have two cases to consider. Let `.T1 Si / D `.T1 / C 1. Then uT1 uSi D uT1 Si , by Theorem 4.3 (iv), and we get ´ q `.T1 /C1 if T3 D T1 Si ; .uT1 uSi ; uT3 /0 D .uT1 Si ; uT3 /0 D 0 otherwise. On the other hand, if `.T1 Si / ¤ `.T1 / C 1, then `.T1 Si / D `.T1 / 1, by Corollary 3.7. Applying Theorem 4.3 (v), we obtain then uT1 uSi D quT1 Si C.q 1/uT1 , and hence 8 `.T / ˆ if T3 D T1 Si ; 0. A Lie algebra over K is a pair .L; Œ; /, where L is a K-vector space and Œ; W L L ! L is a K-bilinear map, called the Lie bracket, satisfying the following conditions: (1) Œx; x D 0 for any x 2 L; (2) ŒŒx; y; z C ŒŒy; z; x C ŒŒz; x; y D 0 for all x; y; z 2 L.

580

Chapter VI. Hopf algebras

The condition (2) is known as the Jacobi identity. Observe that the condition (1) forces the anticommutativity condition: Œx; y D Œy; x for any x; y 2 L: Indeed, for x; y 2 L, we have the equalities 0 D Œx C y; x C y D Œx; x C Œx; y C Œy; x C Œy; y D Œx; y C Œy; x: A homomorphism .L; Œ; L / ! .M; Œ; M / of two Lie algebras over K is a K-linear map f W L ! M such that f .Œx; yL / D Œf .x/; f .y/M for all elements x; y 2 L. Given a Lie algebra .L; Œ; / over K and an element x 2 L, we have the K-linear map ad x W L ! L such that .ad x/.y/ D Œx; y for all y 2 L. Moreover, for a Lie algebra .L; Œ; / over K and a commutative K-algebra R we may consider the Lie algebra .L ˝K R; Œ; / with the Lie bracket Œ; defined by Œx ˝ r; y ˝ s D Œx; y ˝ rs for x; y 2 L and r; s 2 R. Then a p-map of a Lie algebra .L; Œ; / over K is a map Œp W L ! L, x 7! x Œp for x 2 L, satisfying the following conditions: (3) ad x Œp D .ad x/p for any x 2 L; (4) .x/Œp D p x Œp for any 2 K and x 2 L; P the si .x; y/ are given by the (5) .x C y/Œp D x Œp C y Œp C p1 iD1 si .x; y/, where P i1 identity .ad .x ˝ X C y ˝ 1//p1 .x ˝ 1/ D p1 in the iD1 i si .x; y/ ˝ X tensor Lie algebra L ˝K KŒX of L and the polynomial algebra KŒX in one variable X over K, with the Lie bracket described above. Following N. Jacobson [Jac] a restricted Lie algebra over K is a triple L; Œ; ; Œp Œp where .L; Œ; / is a Lie algebra over K and W L ! L is with p > 0 a p-map, the characteristic of K. A homomorphism L; Œ; L ; Œp ! M; Œ; M ; Œp of restricted Lie algebras isa K-linear map f W L ! M such that f .Œx; yL / D Œf .x/; f .y/M and f x Œp D f .x/Œp for all x; y 2 L. Examples 2.21. Let K be a field of characteristic p > 0. (a) Let L D K 2 and e1 D .1; 0/, e2 D .0; 1/ be the canonical basis of L over K. Consider the K-bilinear map Œ; W L L ! L given by Œe1 ; e1 D 0;

Œe2 ; e2 D 0;

Œe1 ; e2 D e2 ;

Œe2 ; e1 D e2 ;

p and the map Œp W L ! L given by x Œp D 1 e1 for x D 1 e1 C 2 e2 2 L. Then a direct checking shows that L; Œ; ; Œp is a restricted Lie algebra over K (see Exercise 7.19).

2. Hopf algebras

581

(b) Let n be a positive integer, Hn the .2n C 1/-dimensional K-vector space with the basis x1 ; : : : ; xn ; y1 ; : : : ; yn ; z. Consider the K-bilinear map Œ; W Hn Hn ! Hn given by Œxi ; yj D ıij z; Œyj ; xi D ıij z; Œxi ; xj D 0; Œyi ; yj D 0; Œxi ; z D 0; Œz; xi D 0; Œyi ; z D 0; Œz; yi D 0; for all i; j 2 f1; : : : ; ng. Then .Hn ; Œ; / is a Lie algebra over K, called the Heisenberg algebra over K. The Lie algebra .Hn ;Œ; / may be endowed with two different structures of restricted Lie algebras Hn ; Œ; ; Œp given by two different p-maps Œp W Hn ! Hn defined on the basis elements as follows: (i) xiŒp D 0, yiŒp D 0, z Œp D 0 for i 2 f1; : : : ; ng; (ii) xiŒp D 0, yiŒp D 0, z Œp D z for i 2 f1; : : : ; ng. The standard checking is left to the reader (see Exercise 7.20). (c) Let A be an arbitrary finite dimensional K-algebra. Consider the maps Œ; W A A ! A

and

Œp

W A ! A

given by Œa; b D ab ba; and aŒp D ap for all a; b 2 A. Then A; Œ; ; Œp is a restricted Lie algebra over K. Observe first that .A; Œ; / is a Lie algebra over K. Indeed, for a; b; c 2 A, we have ŒŒa; b; c C ŒŒb; c; a C ŒŒc; a; b D Œab ba; c C Œbc cb; a C Œca ac; b D .ab ba/c c.ab ba/ C .bc cb/a a.bc cb/ C .ca ac/b b.ca ac/ D abc bac cab C cba C bca cba abc C acb C cab acb bca C bac D 0: Moreover, Œa; a D a2 a2 D 0 for any a 2 A. We show now that Œp W A ! A is a p-map of the Lie algebra .A; Œ; /. For any x 2 A, consider the linear maps Lx ; Rx W A ! A given by Lx .a/ D xa and Rx .a/ D ax for a 2 A. Since the maps Lx and Rx commute, we obtain, for any positive integer m and a 2 A, the equalities .ad x/m .a/ D .Lx Rx /m .a/ D D

m X iD0

.1/mi

m X

.1/mi

iD0

m i mi x ax : i

m .Lix Rxmi /.a/ i

582


Then for m D p, we have .ad x/p .a/ D x p a ax p D .ad x p / .a/ D ad x Œp .a/; since K is of characteristic p and pi is divisible by p for i 2 f1; : : : ; p 1g. This proves that ad x Œp D .ad x/p for any x 2 A. Further, for 2 K and x 2 A, we have .x/Œp D .x/p D p x p D p x Œp . Observe also that A ˝K KŒX is the K-algebra AŒX of polynomials in one variable X and coefficients in A. Denote by D W AŒX ! AŒX the A-linear map (differentiation) such that D aX i D i aX i1 for any positive integer i and a 2 A. Take a; b 2 A and consider the polynomial in AŒX of the form .aX C b/ D a X C b C p

p

p

p

p1 X

si .a; b/X i ;

iD1

with si .a; b/ 2 A. Applying the differentiation map D to this equality, we obtain p1 X

p1 X

iD0

iD1

.aX C b/i a.aX C b/p1i D

Since p is the characteristic of K, we have equality may be written as p1 X

.1/i

iD0

i si .a; b/X i1 :

p1

1K D .1/i 1K . Hence the above

i

p1 X p1 .aX C b/i a.aX C b/p1i D i si .a; b/X i1 : i iD1

Applying the first formula for m D p 1 and x D aX C b, we obtain the equalities .ad.aX C b//p1 .a/ D

p1 X

.1/i

iD0

D

p1 X

p1 .aX C b/i a.aX C b/p1i i

i si .a; b/X i1 :

iD1

Taking X D 1 in the third formula, we get the required equality .a C b/Œp D aŒp C b Œp C

p1 X

si .a; b/:

iD1

The restricted Lie algebra A; Œ; ; Œp is frequently called the restricted commutator algebra of the K-algebra A, and denoted by A .

2. Hopf algebras

583

Let L D L; Œ; ; Œp be a finite dimensional restricted Lie algebra over a field K of positive characteristic p. Consider the quotient K-algebra u.L/ D TK .L/=I.L/ of the tensor K-algebra TK .L/ of L over K by the two-sided ideal I.L/ generated by the elements • Œx; y x ˝ y C y ˝ x for all x; y 2 L; • x Œp x p for all x 2 L. The K-algebra u.L/ is called the restricted enveloping algebra of the restricted Lie algebra L. Proposition 2.22. Let L D L; Œ; ; Œp be a finite dimensional restricted Lie algebra over a field K of positive characteristic p. Then u.L/ is a Hopf algebra over K. Proof. We know from Example 2.18 that TK .L/ is a Hopf algebra over K with the comultiplication, counit and antipode W TK .L/ ! TK .L/ ˝ TK .L/;

" W TK .L/ ! K;

S W TK .L/ ! TK .L/

such that .x/ D x ˝ 1 C 1 ˝ x;

".x/ D 0;

S.x/ D x;

for any element x 2 L D TK1 .L/. In particular, TK .L/ is a cocommutative Hopf algebra generated as K-algebra by any choice of primitive elements forming a basis of L over K. We claim that I.L/ is a Hopf ideal of TK .L/, and consequently u.L/ is a Hopf algebra over K. x introduced in Example 2.18, we have Let x; y 2 L. Then, in the notation for ˝ the equalities .Œx; y x ˝ y C y ˝ x/ D .Œx; y/ .x ˝ y/ C .y ˝ x/ x C 1˝Œx; x y .x/ .y/ C .y/ .x/ D Œx; y˝1 x C 1˝Œx; x y .x ˝1 x C 1˝x/.y x x C 1˝y/ x D Œx; y˝1 ˝1 x C 1˝y/.x x x C 1˝x/ x C .y ˝1 ˝1 x x .Œx; y x ˝ y C y ˝ x/ ; D .Œx; y x ˝ y C y ˝ x/ ˝1 C 1˝ ".Œx; y x ˝ y C y ˝ x/ D ".Œx; y/ ".x/".y/ C ".y/".x/ D 0; S.Œx; y x ˝ y C y ˝ x/ D S.Œx; y/ S.x ˝ y/ C S.y ˝ x/ D Œx; y S.xy/ C S.yx/ D Œx; y S.y/S.x/ C S.x/S.y/ D Œx; y .y/ ˝ .x/ C .x/ ˝ .y/ D .Œx; y x ˝ y C y ˝ x/ ;

584


x Œp x p D x Œp .x p / D x Œp .x/p D x Œp ˝ 1 C 1 ˝ x Œp .x ˝ 1 C 1 ˝ x/p D x Œp ˝ 1 C 1 ˝ x Œp x p ˝ 1 1 ˝ x p D x Œp x p ˝ 1 C 1 ˝ x Œp x p ; " x Œp x p D " x Œp " .x p / D " x Œp ".x/p D 0; S x Œp x p D S x Œp S .x p / D x Œp .x/p D x Œp x p : Therefore, we obtain that .I.L// I.L/˝TK .L/CTK .L/˝I.L/, ".I.L// D 0, S.I.L// I.L/, and so I.L/ is a Hopf ideal of TK .L/. Then u.L/ D TK .L/=I.L/ admits a unique Hopf algebra structure such that the canonical epimorphism of Kalgebras W TK .L/ ! u.L/ is a homomorphism of K-Hopf algebras. The following famous theorem, named for H. Poincaré, G. Birkhoff and E. Witt, shows that the restricted enveloping algebras of restricted Lie algebras are finite dimensional. Theorem 2.23. Let L D L; Œ; ; Œp be a finite dimensional restricted Lie algebra over a field K of characteristic p > 0. Let e1 ; e2 ; : : : ; en be an ordered basis of L over K and x1 ; x2 ; : : : ; xn be the cosets of e1 ; e2 ; : : : ; en in u.L/ D TK .L/=I.L/, respectively. Then the elements xir11 xir22 : : : xirmm ; with i1 < i2 < < im in f1; : : : ; ng, m > 0, 0 < rk p 1 for k 2 f1; : : : ; ng, together with 1, form a basis of u.L/ over K. In particular, we have dimK u.L/ D p n , where n D dimK L. The proof of the above theorem follows (see [SF], Theorem 5.1) from the Poincaré–Birkhoff–Witt theorem, describing the basis of the usual enveloping algebra U.L/ of a Lie algebra L. Since the proof of this theorem is technical and long, we refer to the book [Hum] for its nice presentation. Corollary 2.24. Let L D L; Œ; ; Œp be a finite dimensional restricted Lie algebra over a field K of characteristic p > 0. Then the restricted enveloping algebra u.L/ is a finite dimensional cocommutative Hopf algebra over K.

3 The Larson–Sweedler theorems Let .A; m; / be a K-algebra. Then a right A-module M may be viewed as a pair .M; /, where M is a (right) K-vector space and W M ˝ A ! M is a K-linear

3. The Larson–Sweedler theorems

585

map such that the following diagrams are commutative:

M ˝A˝A

˝idA

M ˝A q8

idM ˝qqq

qq qqq M ˝ KL LLL LL M LLL L& M,

/M ˝A

idM ˝m

M ˝A

/M,

where M .m ˝ / D m for m 2 M and 2 K. Further, a homomorphism of right A-modules .M; M / ! .N; N / is a K-linear map f W M ! N such that the following diagram is commutative: M ˝A

M

f ˝idA

N ˝A

/M f

N

/N.

Dually, let C D .C; ; "/ be a K-coalgebra. Then a right C -comodule is a pair .M; %/, where M is a K-vector space and % W M ! M ˝ C is a K-linear map such that the following diagrams are commutative:

M

%

/M ˝C idM ˝

%

M ˝C

M MM MMM MMMM MM& % M ˝K, q8 q q qqq qqq idM ˝" M ˝C

%˝idC

/ M ˝ C ˝ C,

where M .m/ D m ˝ 1 for m 2 M . Moreover, a homomorphism of right C comodules .M; %M / ! .N; %N / is a K-linear map g W M ! N such that the following diagram is commutative: M

%M

g

N

/M ˝C g˝idC

%N

/ N ˝ C:

For a K-coalgebra C and a right C -comodule .M; %/, the value of the structure map % W M ! M ˝C on an element x 2 M is usually written in the sigma notation as X %.x/ D x.0/ ˝ x.1/ :

586


Then the conditions imposed on % may be written in the sigma notation as X X x.0/ .0/ ˝ x.0/ .1/ ˝ x.1/ D x.0/ ˝ x.1/ 1 ˝ x.1/ 2 ; X x.0/ ".x.1/ / D x; for any x 2 M . Similarly, the condition for a K-linear map g W M ! N to be a homomorphism of right C -comodules .M; %M / ! .N; %N / may be written as X X g.x/.0/ ˝ g.x/.1/ D g x.0/ ˝ x.1/ for any x 2 M . Let H D .H; m; ; ; "; S/ be a Hopf algebra over K. A K-vector space M is called a right H -Hopf module if M is a right H -module, with the action of an element h 2 H on an element x 2 M denotedP by xh, and a right H -comodule via the K-linear map % W M ! M ˝ H , %.x/ D x.0/ ˝ x.1/ for x 2 M , such that, for any x 2 M , h 2 H , the following equality holds: X %.xh/ D x.0/ h1 ˝ x.1/ h2 : A homomorphism of right H -Hopf modules M ! N is a K-linear homomorphism f W M ! N which is a homomorphism of right H -modules and a homomorphism of right H -comodules. Example 3.1. Let H D .H; m; ; ; "; S/ be a Hopf algebra over K and V be a K-vector space. Then we may define on V ˝ H a right H -module structure given by .v ˝ g/h D v ˝ gh, for v 2 V , g; h 2 H , and a right H -comodule P structure via the map % D idV ˝ W V ˝H ! V ˝H ˝H given by %.v ˝h/ D v ˝h1 ˝h2 , for v 2 V , h 2 H . Moreover, we have, for v 2 V and g; h 2 H , the equalities X X %..v ˝ g/h/ D %.v ˝ gh/ D v ˝ .gh/1 ˝ .gh/2 D v ˝ g1 h1 ˝ g2 h2 X X ..v ˝ g1 / h1 / ˝ g2 h2 D .v ˝ g/.0/ h1 ˝ .v ˝ g/.1/ h2 : D Therefore, V ˝H is a right H -Hopf module, called the right tensor H -Hopf module given by the K-vector space V . Let H D .H; m; ; ; "; S/ be a Hopf algebra over K. Then, for a right H comodule M D .M; %/, we may consider the K-vector subspace M coH D fx 2 M j %.x/ D x ˝ 1H g of M , called the subspace of coinvariants of M , with respect to the right H comodule structure on M . Observe that H is a right H -comodule via W H ! P coH coH H ˝H , and then H D K1H . Indeed, if h 2 H , then h1 ˝h2 D .h/ D


587

P h ˝ 1, and hence we obtain the equalities h D ".h1 /h2 D ".h/1H . This shows that H coH K1H . Clearly, for h D 1H 2 K1H , we have .h/ D .1H / D .1H ˝ 1H / D .1H / ˝ 1H D h ˝ 1H , and so h 2 H coH . Therefore, indeed we have H coH D K1H . The following fundamental theorem on Hopf modules, proved by R. G. Larson and M. E. Sweedler in [LaSw], shows that every right Hopf module is in fact a right tensor Hopf module. Theorem 3.2. Let H D .H; m; ; ; "; S/ be a Hopf algebra over K and M a right H -Hopf module. Then the K-linear map f W M coH ˝ H ! M given by f .x ˝ h/ D xh, for x 2 M coH and h 2 H , is an isomorphism of right H -Hopf modules, where M coH ˝ H is considered with the right tensor H -Hopf module structure. Proof. Let % W M ! M ˝ H be the right H -comodule structure of M . ConP map sider the K-linear map g W M ! M given by g.x/ D x.0/ S x.1/ for any x 2 M . We claim that g.M / M coH . Indeed, for x 2 M , we have the equalities X X % x.0/ S x.1/ %.g.x// D % x.0/ S x.1/ D X D x.0/ .0/ S x.1/ 1 ˝ x.0/ .1/ S x.1/ 2 X D x.0/ .0/ S x.1/ 2 ˝ x.0/ .1/ S x.1/ 1 X D x.0/ S x.1/ 22 ˝ x.1/ 1 S x.1/ 21 X D x.0/ S x.1/ 2 ˝ x.1/ 11 S x.1/ 12 X X D x.0/ S x.1/ 2 ˝ x.1/ 11 S x.1/ 12 X D x.0/ S x.1/ 2 ˝ " x.1/ 1 1H X D x.0/ S x.1/ 2 " x.1/ 1 ˝ 1H X X X " x.1/ 1 x.1/ 2 ˝ 1H D x.0/ S x.1/ ˝ 1H D x.0/ S D g .x/ ˝ 1H : We note that in the above transformations we applied the equalities S D .S˝S/TH;H D TH;H .S˝S/ (Proposition 2.8 (iii)), .%˝idH /% D .idM ˝ /%, .idM ˝H ˝ /.idM ˝ / D idM ˝.idH ˝ / D idM ˝. ˝ idH / . Since g.M / M coH , we may define the K-linear map ' W M ! M coH ˝ H

588


P by '.x/ D g.x.0/ / ˝ x.1/ for any x 2 M . We will show that ' is inverse of f . Take x 2 M coH and h 2 H . Then we have the equalities X X '.f .x ˝ h// D '.xh/ D g .xh/.0/ ˝ .xh/.1/ D g x.0/ h1 ˝ x.1/ h2 X X .xh1 /.0/ S .xh1 /.1/ ˝ h2 D g .xh1 / ˝ 1H h2 D X X D x.0/ h11 S x.1/ h12 ˝ h2 D xh11 S .1H h12 / ˝ h2 X X D x .h11 S .h12 // ˝ h2 D x".h1 / ˝ h2 X .".h1 /h2 / D x ˝ h; Dx˝ P since x.0/ ˝ x.1/ D x ˝ 1H . This shows that 'f D idM coH ˝H . Take now x 2 M . Then we have the equalities X X f .'.x// D f g x.0/ ˝ x.1/ D f x.0/ .0/ S x.0/ .1/ ˝ x.1/ X X D x.0/ .0/ S x.0/ .1/ x.1/ D x.0/ S x.1/ 1 x.1/ 2 X X X S x.1/ 1 x.1/ 2 D x.0/ " x.1/ D x; D x.0/ since .% ˝ idH /% D .idM ˝ /%. This shows that f ' D idM . Therefore, f W M coH ˝ H ! M is a K-linear isomorphism. It remains to show that f is a homomorphism of right H -Hopf modules. For x 2 M coH and g; h 2 H , we have f ..x ˝ g/h/ D f .x ˝ gh/ D x.gh/ D .xg/h D f .x ˝ g/h; which shows that f is a homomorphism of right H -modules. Further, for x 2 M coH and h 2 H , we have the equalities X X .%f /.x ˝ h/ D %.f .x ˝ h// D %.xh/ D x.0/ h1 ˝ x.1/ h2 D xh1 ˝ h2 X X D f .x ˝ h1 / ˝ h2 D .f ˝ idH /.x ˝ h1 ˝ h2 / D .f ˝ idH /.idM coH ˝ /.x ˝ h/: This shows that %f D .f ˝ idH /.idM coH ˝ /, which means that f is a homomorphism of right H -comodules, because the structure of a right H -comodule on M coH ˝ H is given by idM coH ˝ W M coH ˝ H ! M coH ˝ H ˝ H . Let H D .H; mH ; H ; H ; "H ; SH / be a finite dimensional Hopf algebra over K. Then it follows from Proposition 2.5 that we have also the finite dimensional


589

dual Hopf algebra H D .H ; mH ; H ; H ; "H ; SH / over K, where H

H;H

"H

mH W H ˝ H ! .H ˝ H / ! H ; mH

H W K D K ! H ; H

!H;H

H W H ! .H ˝ H / ! H ˝ H ;

"H W H ! K D K;

SH

SH W H ! H : We will show now that the dual space H admits a canonical structure of a right H -Hopf module. We denote by H W H ˝ H ! H the K-linear map defined for u 2 H and g; h 2 H by H .u ˝ h/.g/ D u.gSH .h//. Further, we denote by ıH W H ! HomK .H ; H / the K-linear map defined for u; v 2 H by ıH .u/.v/ D vu D mH .v ˝ u/. Moreover, we consider also the K-linear map !H W H ˝ H ! HomK .H ; H / defined for u; v 2 H and h 2 H by !H .u ˝ h/.v/ D v.h/u. Observe Pthat !H is an isomorphism. Indeed, !H is a monomorphism, because, for 0 ¤ riD1 ui ˝ hi 2 H ˝ H we may assume that ui ¤ 0 for i 2 f1; : : : ; rg and h1 ; : : : ; hr are linearly independent over K, and hence there exists v 2 HP such that v.h1 / D 1 r and v.hi / D 0 for all i 2 f2; : : : ; rg. This ensures that !H iD1 ui ˝ hi .v/ D v.h1 /u1 D u1 ¤ 0. Then !H is an isomorphism because dimK .H ˝ H / D .dimK H /.dimK H / D .dimK H /.dimK H / D dimK HomK .H ; H /. This allows us to define the composed K-linear map

ıH

1 !H

%H W H ! HomK .H ; H / ! H ˝ H:

P We note that %H is defined as follows: for u 2H we have %H .u/ D u.0/ ˝ P u.1/ 2 H ˝ H if and only if vu D v u.1/ u.0/ for any v 2 H . Indeed, for u; v 2 H , we have the equalities X u.0/ ˝ u.1/ .v/ vu D ıH .u/.v/ D ..!H %H /.u// .v/ D !H X D v u.1/ u.0/ : Proposition 3.3. Let H D .H; mH ; H ; H ; "H ; SH / be a finite dimensional Hopf algebra over K. Then the dual K-vector space H is a right H -Hopf module via the K-linear maps H W H ˝ H ! H and %H W H ! H ˝ H:

590


Proof. Observe first that H defines a right H -module structure on H . Indeed, applying Proposition 2.8 (i) and (ii), we obtain, for u 2 H and g; h; h0 2 H , the equalities H .u ˝ hh0 /.g/ D u gSH .hh0 / D u gSH .h0 /SH .h/ D u gSH .h0 / SH .h/ D H .u ˝ h/ gSH .h0 / D H H .u ˝ h/ ˝ h0 .g/; H .u ˝ 1H /.g/ D u .gSH .1H // D u .g1H / D u.g/: We will show now that %H W H ! H ˝ H defines a right H -comodule structure on H . We have to show that the following diagrams are commutative:

%H

H

idH ˝H

%H

H ˝ H

/ H ˝ H

%H ˝idH

/ H ˝ H ˝ H ,

H NN NNN NNHN NNN ' %H H7 ˝ K: pp ppp p p ppp idH ˝"H H ˝ H

For u 2 H , we have X .%H ˝ idH /.%H .u// D .%H ˝ idH / u.0/ ˝ u.1/ X D %H u.0/ ˝ u.1/ X D u.0/ .0/ ˝ u.0/ .1/ ˝ u.1/ ; X .idH ˝ H /.%H .u// D .idH ˝ H / u.0/ ˝ u.1/ X D u.0/ ˝ H u.1/ X D u.0/ ˝ u.1/ 1 ˝ u.1/ 2 : We set z D .%H ˝ idH /.%H .u// .idH ˝ H /.%H .u// 2 H ˝ H ˝ H , and claim that z D 0. Take v; w 2 H D HomK .H; K/ and consider the composed K-linear map idH ˝v˝w

v;w

W H ˝ H ˝ H ! H ˝ K ˝ K ! H ;

where is the canonical isomorphism given by .u ˝ ˝ / D u for u 2 H ,


591

D 0. Indeed, we have the equalities X u.0/ .0/ ˝ u.0/ .1/ ˝ u.1/ v;w ..%H ˝ idH / %H .u// D v;w X D u.0/ .0/ v u.0/ .1/ w u.1/ X X X D v u.0/ .1/ u.0/ .0/ w u.1/ D vu.0/ w u.1/ X Dv w u.1/ u.0/ D v .wu/ ; X u.0/ ˝ u.1/ 1 ˝ u.1/ 2 v;w ..idH ˝ H / %H .u// D v;w X D u.0/ v u.1/ 1 w u.1/ 2 X X X D u.0/ v u.1/ 1 w u.1/ 2 D u.0/ .vw/ u.1/ X D .vw/ u.1/ u.0/ D .vw/ u;

; 2 K. We show that

v;w .z/

and consequently v;w .z/ D v.wu/ .vw/u D 0. Finally, we conclude that z D 0. Let e1 ; : : : ; en be a basis of H over K, and e1 ; : : : ; en the associated dual basis of H , so we have ei .ej / D ıij for i; j 2 f1; : : : ; ng. We may write X zD 'i;j ˝ ei ˝ ej 2 H ˝ H ˝ H: Then, for fixed i0 ; j0 2 f1; : : : ; ng, we obtain X 'i;j ˝ ei ˝ ej D 'i0 ;j0 : 0 D ei ;ej .z/ D ei ;ej 0

0

0

0

This shows that z D 0. Therefore, we have proved that .%H ˝ idH /%H D .idH ˝ H /%H . Further, for u 2 H , we have X X u.0/ ˝ "H u.1/ u.0/ ˝ u.1/ D .idH ˝"H /.%H .u// D .idH ˝"H / X X D u.0/ "H u.1/ ˝ 1K D u.0/ "H u.1/ ˝ 1K X D "H u.1/ u.0/ ˝ 1K D "H u ˝ 1K D u ˝ 1K D H .u/; because "H D H .1H / is the identity 1H of the K-algebra H . Hence the required equality .idH ˝"H /%H D H holds. Summing up, we have proved that %H W H ! H ˝ H defines a right H -comodule structure on H . It remains to show that, for u 2 H and h 2 H , the equality X u.0/ h1 ˝ u.1/ h2 %H .uh/ D

592


holds. This is equivalent to the equality X v.uh/ D v u.1/ h2 u.0/ h1 for all u; v 2 H and h 2 H . For u; v 2 H and g; h 2 H , we have the equalities .v.uh//.g/ D .mH .v ˝ uh// .g/ D . H H;H /.v ˝ uh/ .g/ X D H;H .v ˝ uh/ . H .g// D H;H .v ˝ uh/ g 1 ˝ g2 X X D v.g1 /.uh/.g2 / D v.g1 / .H .u ˝ h// .g2 / X X X D v.g1 /u .g2 SH .h// D h1 "H .h2 / v.g1 /u g2 SH X X D v.g1 /"H .h2 /u .g2 SH .h1 // D v .g1 "H .h2 // u .g2 SH .h1 //; X X v u.1/ h2 u.0/ h1 .g/ v u.1/ h2 u.0/ h1 .g/ D X X D v u.1/ h2 H u.0/ ˝ h1 .g/ D v u.1/ h2 u.0/ .gSH .h1 // X X .h2 v/ u.1/ u.0/ .gSH .h1 // D ..h2 v/ u/ .gSH .h1 // D X D H;H ..h2 v/ ˝ u/ H .gSH .h1 // X D H;H ..h2 v/ ˝ u/ H .g/ H .SH .h1 // X X X D H;H ..h2 v/ ˝ u/ SH .h12 / ˝ SH .h11 / g1 ˝ g2 X X D H;H ..h2 v/ ˝ u/ g1 SH .h12 / ˝ g2 SH .h11 / X .h2 v/ .g1 SH .h12 // u .g2 SH .h11 // D X D v .g1 SH .h12 / h2 / u .g2 SH .h11 // X X X D H;H .v ˝ u/ g1 ˝ g2 SH .h12 / h2 ˝ SH .h11 / X X X D H;H .v ˝ u/ g1 ˝ g2 SH .h21 / h22 ˝ SH .h1 / X X X D H;H .v ˝ u/ g1 ˝ g2 SH .h21 / h22 ˝ SH .h1 / X D H;H .v ˝ u/ .g1 "H .h2 / ˝ g2 SH .h1 // X D v .g1 "H .h2 // u .g2 SH .h1 // ; because ˝ idH/ H D .idH ˝ H / H , and consequently v.uh/ D P . H v u.1/ h2 u.0/ h1 . We note that in the above transformations the left H -module structure on H given for g; h 2 H , u 2 H by .hu /.g/ D u.gh/ is involved.


593

Let H D .H; m; ; ; "; S/ be a finite dimensional Hopf algebra over K. Then the space H D HomK .H; K/ admits the canonical structure of a right H -Hopf module described in Proposition 3.3. In particular, we may consider the subspace of coinvariants ˚ .H /coH D u 2 H j %H .u/ D u ˝ 1H : Observe that ˚ .H /coH D u 2 H j vu D v.1H /u for any v 2 H ˚ D u 2 H j vu D "H .v/u for any v 2 H : The following theorem proved in [LaSw] by R. G. Larson and M. E. Sweedler is crucial for the representation theory of finite dimensional Hopf algebras. Theorem 3.4. Let H D .H; m; ; ; "; S/ be a finite dimensional Hopf algebra over K. Then the following statements hold. (i) dimK .H /coH D 1. (ii) There is a canonical isomorphism H ! H of right H -Hopf modules. (iii) The antipode S W H ! H is a K-linear isomorphism. Proof. It follows from Theorem 3.2 that there is a canonical isomorphism f W .H /coH ˝ H ! H of right H -Hopf modules given by f .u˝h/ D uh D H .u˝h/ for u 2 .H /coH and h 2 H . In particular, we obtain the equalities dimK .H /coH .dimK H / D dimK .H /coH ˝ H D dimK H ;

and consequently dimK .H /coH D 1, because dimK H D dimK H . Further, for a nonzero element ' 2 .H /coH , the K-linear map f' W H ! H given by f' .h/ D f .' ˝ h/, for h 2 H , is an isomorphism of right H -Hopf modules (see also Example 3.1). This shows that (i) and (ii) hold. We show now that the antipode S W H ! H is an isomorphism. It is enough to show that S is a monomorphism. Suppose S.h/ D 0 for a nonzero element h 2 H . Then, for a nonzero element u 2 .H /coH , we obtain that .f .u ˝ h//.g/ D .uh/.g/ D u.gS.h// D 0 for any g 2 H , and hence f .u ˝ h/ D 0. This is a contradiction, because u ˝ h ¤ 0 and f is a monomorphism. Hence, indeed S is a monomorphism, and consequently an isomorphism.

594


The space .H /coH is called the space of left integrals for H and denoted by R` the elements u of H are called left integrals of H . H (see [Swe2]). Moreover, Rr The space of right integrals H for H is defined as Z r ˚ D u 2 H j uv D v.1H /u for any v 2 H H ˚ D u 2 H j uv D "H .v/u for any v 2 H R`

Rr and the elements u of H are called right integrals of H . Rr R` R` Rr H D .H cop / and H D .H cop / .

Observe that

Lemma 3.5. Let H D .H; mH ; H ; H ; "H ; SH / be a finite dimensional Hopf algebra over K. Then, for the antipode SH D SH of the dual Hopf algebra H , we have

Z Z

Z Z `

SH

r

H

D

H

r

and SH

H

`

D

H

:

Proof. It follows from Theorem 3.4 (iii) that SH W H ! H is a K-linear isoR` morphism. Take u 2 H nf0g and v 2 H . Moreover, let w 2 H be such that v D SH .w/. Then, applying Proposition 2.8, we obtain that SH .u/v D SH .u/SH .w/ D SH .wu/ D SH ."H .w/u/ D "H .w/SH .u/ D "H .SH .w// SH .u/ D "H .v/SH .u/; Rr R` H and H D .H cop / are of dimension 1, R` Rr Rr by Theorem 3.4, we conclude that SH H D H . The proof that SH H D R` H is similar.

and hence SH .u/ 2

Rr

H

nf0g. Since

R`

The following consequence of Theorem 3.4 shows that the finite dimensional Hopf algebras form a class of Frobenius algebras. Theorem 3.6. Let H D .H; m; ; ; "; S/ be a finite dimensional Hopf algebra over K. Then H is a Frobenius algebra over K. Proof. Choose a nonzero element ' 2 .H /coH and consider the K-bilinear form .; /' W H H ! K given by .x; y/' D '.xy/ for all elements x; y 2 H . Clearly, .; /' is associative, because, for x; y; z 2 H , we have .xy; z/' D '..xy/z/ D '.x.yz// D .x; yz/' . We prove now that the form .; /' is nondegenerate. Let y be a nonzero element of H . Since S W H ! H is a K-linear isomorphism by Theorem 3.4,

4. The Radford theorem

595

we conclude that y D S.z/ for some z 2 H . Observe that, for x 2 H , we have .x; S.z//' D '.xS.z// D .'z/.x/, because the right H -module structure on the H -Hopf module H is given by the map H W H ˝ H ! H . Then we have the following equalities in H : .; y/' D .; S.z//' D 'z D f .' ˝ z/ ¤ 0: Similarly, as in the proof of Theorem IV.2.1, we show that .x; /' D 0 implies .; x/' D 0, and hence x D 0, by the above arguments.

4 The Radford theorem Let H D .H; m; ; ; "; S/ be a finite dimensional Hopf algebra over K. Our main task in this section is to show that the antipode S has finite order. We will prove first a series of preparatory facts. In particular, we will consider the following module structures: (1) the left H -module structure on H , given by .h * u/.g/ D u.gh/, for g; h 2 H and u 2 H ; (2) the right H -module structure on H , given by .u ( h/.g/ D u.hg/, for g; h 2 H and u 2 H ; P (3) the left H -module structure on H , given by u * h D u.h2 /h1 , for u 2 H and h 2 H ; P (4) the right H -module structure on H , given by h ( u D u.h1 /h2 , for u 2 H and h 2 H . We note the following useful lemma. Lemma 4.1. Let H D .H; m; ; ; "; S/ be a finite dimensional Hopf algebra over K. Then the following statements hold. (i) H is an H -bimodule, with the left H -module structure .1/ and the right H -module structure .2/. (ii) H is an H -bimodule, with the left H -module structure .3/ and the right H -module structure .4/. Proof. (i) For u 2 H and g; h; x 2 H , we have the equalities .h * .u ( g//.x/ D .u ( g/.xh/ D u.g.xh// D u..gx/h/ D .h * u/.gx/ D ..h * u/ ( g/.x/; and hence h * .u ( g/ D .h * u/ ( g.

596


(ii) For u; v 2 H and h 2 H , we have the equalities X X u * .h ( v/ D u * v.h1 /h2 D v.h1 / .u * h2 / X X D v.h1 /u.h22 /h21 D v.h11 /u.h2 /h12 X X D u.h2 /.h1 ( v/ D u.h2 /h1 ( v D .u * h/ ( v; P P because h1 ˝ h21 ˝ h22 D h11 ˝ h12 ˝ h2 .

Recall that we agreed to identify the category H -mod of finite dimensional left H -modules with the category mod H op of finite dimensional right H op -modules, where H op is the opposite algebra of H . In a similar way, the category H -mod of finite dimensional left H -modules is identified with the category mod.H /op of finite dimensional right .H /op -modules. Moreover, the functor D D HomK .; K/ W mod K ! mod K op D mod K induces the dualities mod H o

D D

/

mod H op

with 1mod H Š D B D and 1mod H op Š D B D, and the dualities mod H o

D

/ mod.H /op

D

with 1mod H Š D B D and 1mod.H /op Š D B D. Then observe that the left H -module structure on H D D.H / given in (1) is induced from the canonical right H -module structure on H , while the right H module structure on H D D.H / given in (2) is induced from the canonical left H -module structure on H . Similarly, the left H -module structure on H given in (3) induces the canonical right H -module structure on H D D.H /, while the right H -module structure on H given in (4) induces the canonical left H -module structure on H D D.H /. We note also the following useful lemma. Lemma 4.2. Let H D .H; m; ; ; "; S/ be a finite dimensional Hopf algebra over K and u 2 H . Then the following equalities hold. R` (i) u 2 H if and only if u * h D u.h/1H for any h 2 H . Rr (ii) u 2 H if and only if h ( u D u.h/1H for any h 2 H .


597

Proof. (i) Let h 2 H . Then the equality u * h D u.h/1H holds if and only if v.u * h/ D v .u.h/1H / for any element v 2 H . Moreover, we have X X X v.u * h/ D v u.h2 /h1 D u.h2 /v.h1 / D v.h1 /u.h2 / D .vu/.h/; v .u.h/1H / D u.h/v .1H / D v .1H / u.h/ D .v.1H /u/ .h/: Obviously, vu D v.1H /u if and only if .vu/.h/ D .v.1H /u/ .h/ for any h 2 H . This shows that the equivalence (i) holds. The proof that the equivalence (ii) holds is similar. We will need also the following lemma. Lemma 4.3. Let H D .H; mH ; H ; H ; "H ; SH / be a finite dimensional Hopf algebra over K. Consider H with the left H -module structure given in (3) and H with the canonical left H -module structure given by the multiplication mH ! H of left H -modules. in H . Then there is an isomorphism H Proof. It follows from Theorems 3.2 and 3.4 (and Proposition 3.3) that, for any Rl nonzero element u 2 .H /coH D H , the map fu W H ! H given by fu .h/ D f .u ˝ h/ D uh D H .u ˝ h/ for h 2 H is an isomorphism of right H -Hopf modules. In particular, fu W H ! H is a homomorphism of right H -comodules, that is, %H fu D .fu ˝ idH / H . Then, for any h 2 H , we obtain the equalities X %H .f .u ˝ h// D .f ˝ idH /.u ˝ H .h// D f .u ˝ h1 / ˝ h2 : Consider now the K-linear isomorphism !H W H ˝ H ! HomK .H ; H / given for v; w 2 H and h 2 H by !H .w ˝ h/.v/ D v.h/w. Then !H %H D ıH W H ! HomK .H ; H /, where ıH is defined by ıH .w/.v/ D vw D mH .v ˝ w/ for v; w 2 H . Hence, we obtain the equalities v .fu .h// D ıH .fu .h//.v/ D ıH .f .u ˝ h//.v/ X D !H .%H .f .u ˝ h/// .v/ D !H f .u ˝ h1 / ˝ h2 .v/ X X D v.h2 /f .u ˝ h1 / D v.h2 /fu .h1 / X D fu v.h2 /h1 D fu .v * h/; for any v 2 H and h 2 H . This shows that the K-linear isomorphism fu W H ! H is a required isomorphism of left H -modules. For an element g 2 G.H /, we consider the K-vector spaces ˚ Lg .H / D u 2 H j vu D v.g/u for any v 2 H ; ˚ Rg .H / D u 2 H j uv D v.g/u for any v 2 H :

598


R` Observe that 1 D 1H is a grouplike element of H , and we have L1 .H / D H Rr and R1 .H / D H . The space Lg .H / is called the space of left g-integrals of H . Similarly, the space Rg .H / is called the space of right g-integrals of H . We also note that an element g 2 G.H / induces on the one-dimensional Kvector space K a left H -module structure g K and a right H -module structure Kg given by u D u.g/ and u D u.g/; for u 2 H and 2 K, because, for v; w 2 H , we have .vw/.g/ D . H;H .v ˝ w/ /.g/ D H;H .v ˝ w/.g ˝ g/ D v.g/w.g/ D w.g/v.g/: Lemma 4.4. For a grouplike element g of a finite dimensional Hopf algebra H D .H; m; ; ; "; S/ over K, there are canonical isomorphisms of K-vector spaces Lg .H / Š Hom.H /op .g K; H / and Rg .H / Š HomH .Kg ; H /: Proof. Consider the K-linear map 'g W Lg .H / ! Hom.H /op .g K; H / which assigns to an element u 2 Lg .H / the K-linear map 'g .u/ W g K ! H given by 'g .u/./ D u for 2 g K. Observe that 'g .u/ is a homomorphism of left H -modules. Indeed, for v 2 H and 2 g K, we have 'g .u/.v/ D .v/u D .v.g//u D v.g/u D .vu/ D v.u/ D v'g .u/./, since u 2 Lg .H /. Conversely, for any homomorphism f W g K ! H of left H -modules, we have u D f .1K / 2 Lg .H / and f D 'g .u/, because vu D vf .1K / D f .v1K / D f .v.g/1K / D v.g/f .1K / D v.g/u for any v 2 H and f ./ D f .1K / D f .1K / D u for 2 g K. This shows that 'g is a K-linear isomorphism. Similarly, the K-linear map g

W Rg .H / ! HomH .Kg ; H /

which assigns to an element u 2 Rg .H / the K-linear map g .u/ W Kg ! H , given by g .u/./ D u for 2 Kg , is an isomorphism of K-vector spaces. Proposition 4.5. Let H D .H; mH ; H ; H ; "H ; SH / be a finite dimensional Hopf algebra over K and H D .H ; mH ; H ; H ; "H ; SH / the associated dual Hopf algebra over K. Then the following statements hold. (i) Lg .H /, g 2 G.H /, is the family of all pairwise different one-dimensional two-sided ideals of the K-algebra H . (ii) Rg .H /, g 2 G.H /, is the family of all pairwise different one-dimensional two-sided ideals of the K-algebra H .


599

Proof. (i) Fix g 2 G.H /. Observe first that Lg .H / is a two-sided ideal of H . Indeed, for u 2 Lg .H /, and v; w 2 H , we have the equalities w.vu/ D w.v.g/u/ D v.g/.wu/ D v.g/.w.g/u/ D w.g/.v.g/u/ D w.g/.vu/; w.uv/ D .wu/v D .w.g/u/v D w.g/.uv/; and hence vu and uv belong to Lg .H /. Further, by Lemma 4.3, there is an isomorphism H ! H of left H -modules, where the left H -module structure on H is given by (3) and the left H -module structure on H is given by the multiplication in H . Moreover, as pointed above, the left H -module structure on H induces on H D D.H / D HomK .H; K/ the canonical right H -module structure given by the multiplication in H . Applying now Lemma I.8.7, Lemma 4.4 and the duality functors D between mod H and mod.H /op D H -mod, we obtain isomorphisms of K-vector spaces Lg .H / Š Hom.H /op .g K; H / Š Hom.H /op .g K; H / Š HomH .D.H /; D.g K// Š HomH .H ; Kg / Š Kg ; and consequently dimK Lg .H / D 1. Assume now that I is a one-dimensional two-sided ideal of the K-algebra H . Take 0 ¤ x 2 I , so I D Kx D xK. Since I is a left H -submodule of H , for every v 2 H , there exists a unique scalar .v/ 2 K such that vx D .v/x. Observe that, for v; w 2 H , we have .wv/x D .wv/x D w.vx/ D w..v/x/ D .v/.wx/ D .v/.w/x D .w/.v/x, and so .wv/ D .w/.v/. Moreover, .1H / x D 1H x D x D 1K x, which gives .1H / D 1K . Therefore, we conclude that W H ! K is a homomorphism of K-algebras. Applying Lemma 2.16, we then obtain that is a grouplike element of the double dual Hopf algebra H D .H ; mH ; H ; H ; "H ; SH / of H over K. Further, by Proposition 2.7, we have the canonical isomorphism eH W H ! H D .H / of Hopf algebras over K which assigns to an element h 2 H the element eH .h/ 2 H D .H / such that eH .h/.v/ D v.h/ for any v 2 H . In particular, eH induces an isomorphism G.H / ! G.H / of the groups of grouplike elements of H and H . Hence there exists g 2 G.H / such that D eH .g/. But then we obtain vx D .v/x D eH .g/.v/x D v.g/x for any v 2 H . This shows that x 2 Lg .H /, and consequently I D Kx Lg .H /. Since dimK Lg .H / D 1 D dimK I , we obtain that I D Lg .H /. We also note that, for g; h 2 G.H /, the equality Lg .H / D Lh .H / implies that there is a nonzero u 2 H such that v.g/u D v.h/u for all v 2 H , and then g D h. The final conclusion follows from the fact that G.H / consists of linearly independent elements (Proposition 2.14), and hence for g ¤ h there exists v 2 H such that v.g/ D 1 and v.h/ D 0. (ii) We know from Theorem 3.4 that the antipode SH is a K-linear isomorphism, so we may consider also the finite dimensional co-opposite Hopf algebra H cop D cop cop 1 / of H , where H D TH;H H . Clearly, we have .H; mH ; H ; H ; "H ; SH

600


G.H / D G.H cop /. Take g 2 G.H /. Then, for u; v 2 H D HomK .H; K/ D .H cop / , we have H;H .u ˝ v/ H D uv in H and H;H .v ˝ u/ H cop D vu in .H cop / . Moreover, .v ˝ u/ H cop D .v ˝ u/TH;H H D .u ˝ v/ H . This shows that Rg .H / D Lg .H cop / : Observe also that .H cop / D .H /op as K-algebras. Applying (i) to H cop , we conclude that Rg .H / is a one-dimensional two-sided ideal of .H op / and hence of H , because every two-sided ideal of .H op / is a two-sided ideal of .H /op D .H op / . Conversely, every two-sided ideal of H is a two-sided ideal of .H op / D .H /op D .H cop / , and hence, applying (i) again, we conclude that, for a onedimensional two-sided ideal I of H , there exists an element g 2 G.H cop / D G.H / such that I D Lg ..H cop / / D Rg .H /. Moreover, as above we conclude that, if g; h 2 G.H / and Rg .H / D Rh .H /, then g D h. We note that it follows from the proof of (i) that every one-dimensional left (respectively, right) ideal of H is a two-sided ideal of H . As a direct consequence of Proposition 4.5 we obtain the following fact. Corollary 4.6. Let H D .H; m; ; ; "; S/ be a finite dimensional Hopf algebra over K. Then there exists a unique element a 2 G.H / such that Ra .H / D R` L1 .H / D H . The element a 2 G.H / such that Ra .H / D L1 .H / is called the distinguished grouplike element of H . Let H D .H; mH ; H ; H ; "H ; SH / be a finite dimensional Hopf algebra over K, and H D .H ; mH ; H ; H ; "H ; SH / the associated dual Hopf algebra over K. For an element g 2 G.H /, consider the K-linear homomorphisms lg W H ! H

and

rg W H ! H:

given by lg .x/ D gx and rg .x/ D xg for any x 2 H . Since g is an invertible element of H with the inverse g 1 D SH .g/, lg and rg are K-linear isomorphisms. We claim that lg and rg are isomorphisms of K-coalgebras. Indeed, for h 2 H , we have the equalities X H .lg .h// D H .gh/ D H .g/ H .h/ D .g ˝ g/ h1 ˝ h2 X X D gh1 ˝ gh2 D .lg ˝ lg / h1 ˝ h2 D .lg ˝ lg / . H .h// ; "H .lg .h// D "H .gh/ D "H .g/"H .h/ D "H .h/; and hence H lg D .lg ˝ lg / H and "H lg D "H . Similarly, we have H rg D .rg ˝ rg / H and "H rg D "H . Applying Lemma 1.5, we conclude that the dual homomorphisms lg W H ! H

and

rg W H ! H :


601

are isomorphism of K-algebras. Observe that lg .u/.h/ D u.lg .h// D u.gh/ and rg .u/.h/ D u.rg .h// D u.hg/, for h 2 H and u 2 H . Then we have the following lemma. Lemma 4.7. Let H D .H; mH ; H ; H ; "H ; SH / be a finite dimensional Hopf algebra over K, and g; h 2 G.H /. Then the following statements hold. (i) Lh1 g .H / D lh .Lg .H // D Lg .H / ( h. (ii) Lgh1 .H / D rh .Lg .H // D h * Lg .H /. (iii) Rh1 g .H / D lh .Rg .H // D Rg .H / ( h. (iv) Rgh1 .H / D rh .Rg .H // D h * Rg .H /. Proof. (i) Let u 2 Lg .H / and v 2 H . Then v D lh .v 0 / for some v 0 2 H , since lh is an isomorphism. We have the equalities vlh .u/ D lh .v 0 /lh .u/ D lh .v 0 u/ D lh .v 0 .g/u/ D v 0 .g/lh .u/ D v 0 .h.h1 g//lh .u/ D lh .v 0 /.h1 g/lh .u/ D v.h1 g/lh .u/: This shows that lh Lg .H / Lh1 g .H /. Then we obtain lh Lg .H / D Lh1 g .H /, since both sides of the equality are of dimension 1. Moreover, for u 2 H and x 2 H , we have the equalities lh .u/.x/ D u .lh .x// D u.hx/ D .u ( h/.x/; and hence lh .u/ D u ( h. Thus lh Lg .H / D Lg .H / ( h also holds. The proofs of the equalities in (ii), (iii), (iv) are similar.

Corollary 4.8. Let H D .H; mH ; H ; H ; "H ; SH / be a finite dimensional Hopf algebra over K. Then, for any g 2 G.H /, we have the equalities g * Lg .H / D L1 .H / D Lg .H / ( g: Let H D .H; mH ; H ; H ; "H ; SH / be a finite dimensional Hopf algebra over K. Recall that, by Lemma 2.16, the set G .H / of grouplike elements in the dual Hopf algebra H D .H ; mH ; H ; H ; "H ; SH / is the set AlgK .H; K/ of all K-algebra homomorphisms from H to K. For any element u 2 G.H /, we may define the K-vector subspaces of H Lu .H / D fx 2 H j hx D u.h/x for any h 2 H g ; Ru .H / D fx 2 H j xh D u.h/x for any h 2 H g : The following proposition describes the distinguished properties of these Kvector spaces.

602


Proposition 4.9. Let H D .H; mH ; H ; H ; "H ; SH / be a finite dimensional Hopf algebra over K. Then the following statements hold. (i) Lu .H /, u 2 G.H /, is the family of all pairwise different one-dimensional two-sided ideals of the K-algebra H . (ii) Ru .H /, u 2 G.H /, is the family of all pairwise different one-dimensional two-sided ideals of the K-algebra H . Proof. Consider the canonical isomorphism of Hopf algebras over K eH W .H; mH ; H ; H ; "H ; SH / ! .H ; mH ; H ; H ; "H ; SH / induced by the canonical isomorphism eH W H ! H of K-vector spaces (see Proposition 2.7). Let u 2 G.H /. Then we have the equalities of K-vector spaces eH .Lu .H // D Lu .H / D Lu .H / ; eH .Ru .H // D Ru .H / D Ru .H / : Hence the required statements (i) and (ii) follow from the statements (i) and (ii) of Proposition 4.5 applied to the dual Hopf algebra H D .H ; mH ; H ; H ; "H ; SH /. As a direct consequence of the above result we obtain the following fact. Corollary 4.10. Let H D .H; mH ; H ; H ; "H ; SH / be a finite dimensional Hopf algebra over K. Then there exists a unique element ˛ 2 G.H / such that R˛ .H / D L"H .H /. The element ˛ 2 G.H / such that R˛ .H / D L"H .H / is called the distinguished grouplike element of H . Moreover, we have the following version of Corollary 4.8. Corollary 4.11. Let H D .H; mH ; H ; H ; "H ; SH / be a finite dimensional Hopf algebra over K. Then, for any u 2 G.H /, we have the equalities u * Lu .H / D L"H .H / D Lu .H / ( u: In the next technical lemmas we assume that H D .H; mH ; H ; H ; "H ; SH / is a finite dimensional Hopf algebra over K and H D .H ; mH ; H ; H ; "H ; SH / is the dual Hopf algebra of H over K. Lemma 4.12. Let u 2 G.H /, g 2 G.H /, v; w 2 H , x 2 Lu .H /, and assume that v * x D g D x ( w. Then v 2 Lg .H / and w 2 Rg .H /.


603

Proof. Let r; t 2 H . Then we have the equalities .rt v/.x/ D .rt /.v * x/ D .rt /.g/ D r.g/t .g/ D r.v * x/t .g/ D .rt .g//.v * x/ D .rt .g/v/.x/; and hence .r.t v t .g/v//.x/ D 0, or equivalently, .t v t .g/v/.x ( r/ D 0. Thus we obtain that .t v t .g/v/.x ( H / D 0. Observe that, by Corollary 4.11, we have L"H .H / ( H D Lu .H / ( uH Lu .H / ( H D x ( H , because x ¤ 0 and dimK Lu .H / D 1. This gives the equality L"H .H / D x ( H . On the other hand, by Proposition 3.3, the dual Hopf algebra H is a right H -Hopf module. Applying now Theorem 3.2 we infer that the K-linear map f W .H /coH ˝ H ! H given by f .y ˝ n/ D y n for y 2 .H /coH and n 2 H , is an isomorphism of right H -Hopf modules. Invoking now the canonical isomorphism eH W H ! H of Hopf algebras and the fact that eH L"H .H / D .H /coH , we conclude that the canonical K-linear homomorphism fH W L"H .H / ˝ H ! H , given by fH .z ˝ n/ D z ( n for z 2 L"H .H / and n 2 H , is an isomorphism of right H -Hopf modules. In particular, we obtain that L"H .H / ( H D H , and consequently x ( H D H . This proves that tv D t.g/v and so v 2 Lg .H /. The fact that w 2 Rg .H / can be proved in a similar way. Corollary 4.13. Let v 2 H and x 2 L"H .H / be such that v * x D 1 D 1H . Then v 2 L1 .H / and x ( v D a (the distinguished grouplike element of H ). Proof. The proof that v 2 L1 .H / follows immediately from Lemma 4.12. Further, for u 2 H , we have the equalities u.x ( v/ D u

X

X v.x1 /u.x2 / D .vu/.x/ D .u.a/v/.x/ v.x1 /x2 D

D u.a/v.x/ D u.v.x/a/; because vP2 L1 .H / D Ra .H /. Moreover, by our assumption v * x D 1H , we have v.x2 /x1PD 1H . Applying "HP, we obtain that 1K D "H .1H / D P "H . v.x2 /x1 / D v.x2 /"H .x1 / D v . x2 "H .x1 // D v.x/. Therefore, we get u.x ( v/ D u.a/ for any u 2 H . This implies x ( v D a. Lemma 4.14. Let g 2 G.H /, u 2 G.H /, x 2 Lu .H /, v 2 H , and assume that v * x D g. Then for any w 2 H we have u.g/w.1H / D

X

w.x1 /v.gx2 /:

604


Proof. We have the equalities u.g/w.1H /

D u.g/w g 1 g D u.g/w mH g 1 ˝ g D u.g/mH .w/ g 1 ˝ g D u.g/ H;H !H;H mH .w/ g 1 ˝ g D u.g/ . H;H H / .w/ g 1 ˝ g X X D u.g/ H;H w1 ˝ w2 g 1 ˝ g D u.g/w1 g 1 w2 .g/ X X X D u.g/w1 g 1 w2 .v * x/ D v.x2 /x1 u.g/ w1 g 1 w2 X X D w1 g 1 w2 v.x2 /x1 u.g/ X X X D w1 g 1 w2 v.gx2 /gx1 D w1 g 1 w2 .v.gx2 /gx1 / X X D w1 g 1 w2 .gx1 / v.gx2 / X . H;H H .w// g 1 ˝ gx1 v.gx2 / D X X D mH .w/ g 1 ˝ gx1 v.gx2 / D w mH g 1 ˝ gx1 v.gx2 / X X D w g 1 gx1 v.gx2 / D w.x1 /v.gx2 /;

because x 2 Lu .H /, and hence we have the equalities X u.g/ x1 ˝ x2 D u.g/ H .x/ D H .u.g/x/ D H .gx/ D H .g/ H .x/ X X D .g ˝ g/ x1 ˝ x2 D gx1 ˝ gx2 : Lemma 4.15. Let g 2 G.H /, u 2 G.H /, x 2 Lu .H /, v 2 H , and assume that v * x D g. Then for any h 2 H we have SH g 1 .u * h/ D .v ( h/ * x: Proof. Let w 2 H . Then, applying Lemma 4.14, we obtain the equalities X w SH g 1 .u * h/ D w SH g 1 u.h2 /h1 X X D u.h2 /w SH g 1 h1 D u.h2 /w SH .h1 / SH g 1 X X D u.h2 /w .SH .h1 / g/ D u gg 1 h2 w .SH .h1 / g/ X u.g/w .SH .h1 / g/ u g 1 h2 D X D u.g/ .wSH / g 1 h1 u g 1 h2 X D u.g/ ..wSH / u/ g 1 h D u.g/ w1 "H .w2 / SH u g 1 h X D u.g/ ...w1 w2 .1H // SH / u/ g 1 h


605

X

..w1 SH / u/ g 1 h u.g/w2 .1H / X ..w1 SH / u/ g 1 h w2 .x1 /v.gx2 / D X ..w1 SH / u/ g 1 h w2 .v.gx2 /x1 / D X .w1 SH / g 1 h1 u g 1 h2 w2 .v.gx2 /x1 / D X D w1 SH g 1 h1 w2 v.gx2 /u g 1 h2 x1 X D w1 SH .h1 / SH g 1 w2 v gg 1 h22 x2 g 1 h21 x1 X D w1 .SH .h1 / g/ w2 v .h22 x2 / g 1 h21 x1 X D w SH .h1 / gg 1 h21 x1 v .h22 x2 / X X D w .SH .h1 / h21 x1 v .h22 x2 // D w .SH .h11 / h12 x1 v .h2 x2 // X X X D w ."H .h1 /x1 v .h2 x2 // D "H .h1 /h2 x2 w x1 v X D w .x1 v .hx2 // D w ..v ( h/ * x/ ; D

P P because h1 ˝ h21 ˝ h22 DP h11 ˝ h12 ˝ h2 , where we have used the equalP 1 ity u g h2 x1 ˝ x2 D g 1 h21 x1 ˝ g 1 h22 x2 induced by the equality u g1 h2 x D g 1 h2 x, since x 2 Lu .H /. Therefore, the required equality SH g 1 .u * h/ D .v ( h/ * x holds. Corollary 4.16. Let H D .H; mH ; H ; H ; "H ; SH / be a finite dimensional Hopf algebra over K, a 2 G.H / and ˛ 2 G.H / the distinguished grouplike elements of H and H , h 2 H , and v; w 2 H . Then the following statements hold. (i) If x 2 L"H .H / and v * x D 1H , then SH .h/ D .v ( h/ * x. 1 .˛ * h/ D .h * v/ * x. (ii) If x 2 R˛ .H / and v * x D 1H , then SH (iii) If x 2 R˛ .H / and x ( w D a, then SH .h ( ˛/a1 D x ( .h * w/. 1 1 (iv) If x 2 L"H .H / and x ( w D g, then SH g h D x ( .w ( h/.

Proof. The above statements (i), (ii), (iii) and (iv) are special cases of Lemma 4.15 for the Hopf algebras H , H cop , H op;cop and H op , respectively. We note that, for a finite dimensional Hopf algebra H D .H; mH ; H ; H ; "H ; SH / over K and elements h 2 H , and u; v 2 H , by Lemma 4.1 we have the well-defined element u * h ( v D .u * h/ ( v D u * .h ( v/:

606


Theorem 4.17. Let H D .H; mH ; H ; H ; "H ; SH / be a finite dimensional Hopf algebra over K and a 2 G.H /, ˛ 2 G.H / the distinguished grouplike elements of H and H . Then for any h 2 H we have 4 SH .h/ D a1 ˛ * h ( ˛ 1 a: Proof. Recall that L1H .H / D Ra .H / and L"H .H / D R˛ .H /. Take a nonzero element x in L"H .H /. Since mH D mH cop , H D H cop and "H D "H cop , we have L"H .H / D L"H cop .H cop /. We know that the K-linear map fH cop W L"H cop .H cop / ˝ .H cop / ! H cop given by fH cop .z ˝ w/ D z ( w, for z 2 L"H cop .H cop / and w 2 .H cop / , is an isomorphism of right .H cop / -Hopf modules, and hence an isomorphism of K-vector spaces. In particular, we obtain the following K-linear cop cop P isomorphism fHx cop W P .H cop / ! H cop given for w 2 .H cop / by fHx cop .w/ D w x1 x2 , cop cop where x1 ˝ x2 D H cop .x/. Since H cop D TH;H H , we obtain that P P P cop cop w .x2 / x1 , where H .x/ D x1 ˝ x2 . Therefore, the Kw x1 x2 D P x x Q Q linear map fH W H ! H given for w 2 H by fH .w/ D w.x2 /x1 D w * x is a K-linear isomorphism. In particular, we may choose v 2 H such that v * x D 1H . Then, by Corollary 4.13, we get v 2 L1 .H / and x ( v D a. 4 2 We claim now that SH .h/ * v D v ( SH .˛h/. Since ˛ 2 G.H /, we have 4 2 1 D ˛SH D ˛. Moreover, by ˛SH D SH .˛/ D SH .˛/ D ˛ , and so ˛SH 4 Proposition 2.8 (iii), we have H SH D .SH ˝ SH /TH;H H , and hence H SH D 4 4 .SH ˝ SH / H . Applying the relations (i) and (ii) in Corollary 4.16, we obtain the equalities 4 1 4 SH .h/ * v * x D SH ˛ * SH .h/ X 4 1 4 D SH ˛ SH .h/2 SH .h/1 X 4 1 4 D SH ˛ SH .h2 / SH .h1 / X 1 4 4 D SH ˛SH .h1 / .h2 /SH X 1 4 D SH ˛.h2 /SH .h1 / X 1 4 D SH SH ˛.h2 /h1 2 .˛ * h/ D SH SH 2 .˛ * h/ * x: D v ( SH 4 .h/ * v D v ( Since fQHx W H ! H is an isomorphism, we obtain that SH 2 SH .˛ * h/. On the other hand, applying the relations (iii) and (iv) in Corol-


607

lary 4.16, we obtain the equalities 1 2 2 1 .˛ * h/ D SH a SH .˛ * h/ x ( v ( SH X 1 2 D SH a1 SH ˛.h2 /h1 X 1 2 D SH a1 ˛.h2 /SH .h1 / X 1 2 2 D SH SH .a1 / ˛.h2 /SH .h1 / X 1 2 D SH SH a1 ˛.h2 /h1 2 1 1 D SH SH a .˛ * h/ D SH a1 .˛ * h/ D SH a1 .˛ * h/ aa1 1 D SH a ˛ * h ( ˛ 1 a ( ˛ a1 D x ( a1 ˛ * h ( ˛ 1 a * v ; because a ( ˛ D ˛.a/a. Recall also that the K-linear map fHx W H ! H given for w 2 H by fHx .w/ D x ( w is an isomorphism. Therefore, we obtain the equality 2 x ( SH .˛ * h/ D a1 ˛ * h ( ˛ 1 a * v; and hence the equality 4 SH .h/ * v D a1 ˛ * h ( ˛ 1 a * v: Finally, observe that the map uv W H ! H , given for g; h 2 H by uv .h/.g/ D .h * v/.g/ D v.gh/, is a K-linear isomorphism. In fact, the K-linear homomorphism h * v D .; h/v W H ! H is associated with the nondegenerate K-linear form.; /v W H H ! H . Hence, we obtain the required equality 4 SH .h/ D a1 ˛ * h ( ˛ 1 a for any h 2 H . We need also the following lemma. Lemma 4.18. Let H D .H; mH ; H ; H ; "H ; SH / be a finite dimensional Hopf algebra over K, g 2 G.H /, h 2 H , and u 2 G.H /. Then the following equalities hold. (i) u * g 1 hg D g 1 .u * h/g. (ii) g 1 hg ( u D g 1 .h ( u/g.

608


Proof. (i) Since H .g/ D g ˝g, H .g 1 / D g 1 ˝g 1 , and H W H ! H ˝H and u W H ! K are homomorphisms of K-algebras, we obtain the equalities X u g 1 hg 2 g 1 hg 1 u * g 1 hg D X D u g 1 h2 g g 1 h1 g X D u g 1 u .h2 / u .g/ g 1 h1 g X D g 1 u.g/1 u.g/u.h2 /h1 g X D g 1 u.h2 /h1 g D g 1 .u * h/g: The proof of (ii) is similar.

We are now in position to exhibit the theorem proved in 1976 by D. Radford [Rad]. Theorem 4.19. Let H D .H; mH ; H ; H ; "H ; SH / be a finite dimensional Hopf algebra over K. Then the antipode SH has finite order, smaller than or equal to 4 dimK H . Proof. Let h 2 H and n be a positive integer. Then, applying Lemmas 4.1, 4.18 and Theorem 4.17, we conclude, by induction on n, that 4n .h/ D an .˛ n * h ( ˛ n / an : SH

Further, by Corollary 2.15, G.H / and G.H / are finite groups whose orders are bounded by dimK H . Since a 2 G.H / and ˛ 2 G.H /, we have ak D 1H and ˛ k D "H for some k dimK H . Moreover, by the counity property, we have 4k "H * h D h D h ( "H . Therefore, we conclude that SH D idH . We end this section with a result showing that the finite dimensional Hopf algebras with the antipode of odd order are very special. Proposition 4.20. Let H D .H; mH ; H ; H ; "H ; SH / be a finite dimensional Hopf algebra over K with SH of odd order. Then H is commutative, cocommutative and SH D idH . Proof. Let r D 2k C 1, for some k 0, be the order of SH . Then, applying Proposition 2.8 (i), we obtain the equalities 2k 2k r 2k xy D SH .xy/ D SH SH .xy/ D SH SH .x/SH .y/ 2k 2k 2kC1 2kC1 D SH SH .y/ SH SH .x/ D SH .y/SH .x/ r r D SH .y/SH .x/ D yx;

5. The Fischman–Montgomery–Schneider formula

609

for any elements x and y in H . Hence H is a commutative K-algebra. But then 2 SH D idH , by Corollary 2.12. Therefore, r D 1 and SH D idH . Moreover, by Proposition 2.8 (iii), we obtain that X X H .h/ D H .SH .h// D SH .h2 / ˝ SH .h1 / D h2 ˝ h1 D TH;H H .h/; for any h 2 H , and so H is also cocommutative.

Example 4.21. Let G be a finite group with the property g D g 1 for any element g 2 G, or equivalently, G is isomorphic to an elementary 2-group .Z=2Z/ .Z=2Z/. Then, for any field K, the group algebra KG of G is a commutative and cocommutative Hopf algebra whose antipode is the identity idKG .

5 The Fischman–Montgomery–Schneider formula Let H D .H; m; ; ; "; S/ be a finite dimensional Hopf algebra over K. We know from Theorem 3.6 that H is a Frobenius algebra over K with nondegenerate associative K-bilinear form .; /' W H H ! K given by .x; y/' D '.xy/, R` where ' is a nonzero element (left integral) from .H /coH D H . We note that ' R` is determined uniquely up to a nonzero scalar from K because dimK H D 1. Then the Nakayama automorphism H of H associated to .; /' is a K-algebra automorphism of H such that .H .x/; y/' D .y; x/' for all x; y 2 H (see Proposition IV.3.1). The next theorem, proved by D. Fischman, S. Montgomery and H.-J. Schneider in [FMS], provides the formula for the Nakayama automorphism of a finite dimensional Hopf algebra over a field. Theorem 5.1. Let H D .H; mH ; H ; H ; "H ; SH / be a finite dimensional Hopf algebra over K, ˛ 2 G.H / the distinguished grouplike element of H , and .; /' the nondegenerate associative K-bilinear form on H given by a nonzero left integral ' of H . Then the Nakayama automorphism H of H associated to .; /' is given by the formula 2 2 .h/ D SH .˛ * h/ H .h/ D ˛ * SH

for any h 2 H . Proof. It follows from Theorem 3.2 and Proposition 3.3 that there is a canonical isomorphism of right H -Hopf modules Z fW

`

H

˝H ! H

610


R` given by f .u ˝ h/ D H .u ˝ h/ for u 2 H and h 2 H . Moreover, f .u ˝ h/.g/ D H .u ˝ h/.g/ D u.gSH .h// for any g 2 H . Therefore, f .u ˝ h/ D R` R` SH .h/ * u for u 2 H and h 2 H . Choose now a nonzero element ' 2 H . Then there exists exactly one element x 2 H such that f .' ˝ x/ D "H . Take t D SH .x/. Then we obtain that t * ' D "H , or equivalently, '.gt / D .t * '/.g/ D "H .g/ for any g 2 H . In particular, we conclude that '.t / D '.1H t / D R` "H .1H / D 1K . Since ' 2 H , by Lemma 4.2, we have also X '.h2 /h1 D ' * h D '.h/1H for any h 2 H . We claim now that t 2 L"H .H /. For g; h 2 H , we have the equalities .ht * '/.g/ D '.g.ht// D '..gh/t / D "H .gh/ D "H .g/"H .h/ D "H .h/"H .g/ D "H .h/'.gt / D ' ."H .h/gt/ D ' .g ."H .h/t // D ."H .h/t * '/ .g/: Hence, for any h 2 H , we have ht * ' D "H .h/t * ', and then ht D "H .h/t , because the K-linear map f' W H ! H which assigns to z 2 H the K-linear form f' .z/ D z * ' is an isomorphism. Therefore, we have t 2 L"H .H /. Since L"H .H / D R˛ .H /, we have also t 2 R˛ .H /, and consequently t h D ˛.h/t for any h 2 H . Our next aim is to prove that X 1 .t1 / D h '.ht2 /SH for any h 2 H . Indeed, for h 2 H , we have the equalities X X X 1 1 1 .'.ht2 /1H / SH .' * ht2 / SH .t1 / D .t1 / D .t1 / '.ht2 /SH X X 1 1 D ' ..ht2 /2 / .ht2 /1 SH .t1 / D ' .h2 t22 / h1 t21 SH .t1 / X 1 D ' .h2 t2 / h1 t12 SH .t11 / X X X 1 D ' .h2 t2 / h1 t12 SH .t11 / D ' .h2 t2 / h1 "H .t1 / X X X D ' h2 "H .t1 /t2 h1 D ' .h2 t / h1 X X D ' ."H .h2 /t/ h1 D '.t / h1 "H .h2 / D '.t /h D h; since '.t/ D 1K . Replacing h by H .h/, we obtain the equalities X X 1 1 .H .h/; t2 /' SH ' .H .h/t2 / SH .t1 / D .t1 / H .h/ D X X X 1 1 1 .t2 ; h/' SH D .t1 / D ' .t2 h/ SH .t1 / D SH .t1 /' .t2 h/ ;


611

2 , we obtain, for any h 2 H , the equalities for any h 2 H . Further, applying SH X X 2 2 1 .H .h// D SH SH SH .t1 /' .t2 h/ D SH .t1 / .' .t2 h/ 1H / X X D SH .t1 / .' * t2 h/ D SH .t1 /' ..t2 h/2 / .t2 h/1 X X D SH .t1 /' .t22 h2 / t21 h1 D SH .t11 /' .t2 h2 / t12 h1 X X X D SH .t11 /t12 ' .t2 h2 / h1 D "H .t1 /' .t2 h2 / h1 X X X D ' "H .t1 /t2 h2 h1 D ' .t h2 / h1 X X D ' .˛.h2 /t/ h1 D '.t / ˛.h2 /h1 D ˛ * h;

since '.t/ D 1K and t 2 R˛ .H /. Therefore, we obtain that 2 .˛ * h/ H .h/ D SH

for any h 2 H . Moreover, since ˛ 2 G.H /, we have the equalities ˛SH D 2 2 SH .˛/ D SH .˛/ D ˛ 1 , and hence ˛SH D ˛ and ˛SH D ˛. Then, applying Proposition 2.8 (iii), we obtain that X X 2 2 2 2 2 ˛ * SH .h/ D ˛ SH .h/2 SH .h/1 D ˛ SH .h2 / SH .h1 / X X 2 2 2 D ˛SH .h1 / D ˛.h2 /SH .h1 / .h2 /SH X 2 2 .˛ * h/ ; D SH ˛.h2 /h1 D SH for any h 2 H . Summing up, we have proved that the required formula 2 2 H .h/ D ˛ * SH .h/ D SH .˛ * h/ for h 2 H;

holds.

Observe that the formula on the Nakayama automorphism H of a finite dimensional Hopf algebra H does not depend on the choice of the nonzero left integral ' of H creating the K-bilinear form .; /' on H . Theorem 5.2. Let H D .H; mH ; H ; H ; "H ; SH / be a finite dimensional Hopf algebra over K and H the Nakayama automorphism associated to the nondegenerate associative K-bilinear form .; /' given by a nonzero left integral ' of H . Then H has finite order, smaller than or equal to 2 dimK H . 2 2 .˛ * h/ Proof. It follows from Theorem 5.1 that H .h/ D ˛ * SH .h/ D SH for any h 2 H , where ˛ is the distinguished grouplike element of H . Then for any positive integer m and h 2 H , we have 2m m .h/ D ˛ m * SH .h/: H

612


2.m1/ m1 Indeed, assume by induction that H .h/ D ˛ m1 * SH .h/ for some m 2 and any h 2 H . Then we obtain the equalities m1 2.m1/ m H .h/ D H H .h/ D H ˛ m1 * SH .h/ m1 2.m1/ 2 2m D ˛ * SH * SH .h/ D ˛ * ˛ m1 * SH .h/ ˛ 2m D ˛ m * SH .h/: 4k D idH for a positive integer We know from Theorem 4.19 (and its proof) that SH k k dimK H with the property ˛ D "H . Clearly, then we have 2k 4k H .h/ D ˛ 2k * SH .h/ D "H * h D h 2k for any h 2 H . Therefore H D idH for k dimK H .

In fact, we have a better information on the order of the Nakayama automorphism of a finite dimensional Hopf algebra over a field. The following Hopf algebra freeness theorem was proved by W. D. Nichols and M. B. Zoeller in [NZ]. Theorem 5.3. Let H be a finite dimensional Hopf algebra over K and B a K-Hopf subalgebra of H . Then H is a free left and right B-module. In particular, dimK B divides dimK H . Corollary 5.4. Let H be a finite dimensional Hopf algebra over K. Then order of G.H / divides dimK H . Proof. It follows from Corollary 2.15 that the group algebra KG.H / of G.H / is a K-Hopf subalgebra of H . Then, applying Theorem 5.3, we conclude that jG.H /j D dimK KG.H / divides dimK H . Corollary 5.5. Let H D .H; mH ; H ; H ; "H ; SH / be a finite dimensional Hopf algebra over K. Then the following statements hold. (i) The order of the antipode SH divides 4 dimK H . (ii) The order of the Nakayama automorphism H of H divides 2 dimK H . We have also the following criterion for a finite dimensional Hopf algebra to be a symmetric algebra. Corollary 5.6. Let H D .H; mH ; H ; H ; "H ; SH / be a finite dimensional Hopf algebra over K. Then the following statements are equivalent. (i) H is a symmetric algebra. 2 .h/ D ˛ * chc 1 , for an invertible element c of H , the distinguished (ii) SH grouplike element ˛ of H and any h 2 H .


613

Proof. It follows from Theorem 5.1 that the Nakayama automorphism H of H , associated to a K-bilinear form .; /' on H given by a nonzero left integral ' of H , 2 is given by the formula H .h/ D SH .˛ * h/. Further, by Corollary IV.3.4, we conclude that H is a symmetric algebra if and only if H is an inner automorphism of H . Hence H is a symmetric algebra if and only if there exists an invertible element 2 2 b 2 H such that SH .˛ * h/ D bhb 1 , or equivalently, ˛ * h D SH .bhb 1 /, 2 1 for any h 2 H . Observe that the equality ˛ * h D SH .bhb / is equivalent to 2 .h/ D ˛ * chc 1 , for c D b 1 . Therefore the statements (i) and the equality SH (ii) are equivalent. Corollary 5.7. Let H D .H; mH ; H ; H ; "H ; SH / be a finite dimensional Hopf algebra over K which is commutative or cocommutative, and ˛ be the distinguished grouplike element of H . Then the following statements are equivalent. (i) H is a symmetric algebra. (ii) bhb 1 D ˛ * h for an invertible element b of H and any element h 2 H . 2 Proof. Since H is either commutative or cocommutative, we have SH D idH , by Corollary 2.12. Then the equivalence of (i) and (ii) follows from Corollary 5.6.

Example 5.8. Let G be a finite group and KG the group algebra of G over K. Then KG is a Hopf algebra over K with the comultiplication , the counit ", and the antipode S given by .g/ D g ˝ g; ".g/ D 1K ; S.g/ D g 1 ; for any g 2 G: (see Examples 1.1 (d) and 2.4 (c)). We claim that L" .KG/ D Kt D R" .KG/; P

where t D g2G g. Since L" .KG/ and R" .KG/ are K-vector subspaces of KG of dimension 1, it is enough to show that t 2 L" .KG/ and t 2 R" .KG/. This is obvious because for any element h 2 G we have the equalities X hg D t D ".h/t; ht D g2G

th D

X

gh D t D ".h/t:

g2G

In particular, we conclude that the distinguished grouplike element ˛ of .KG/ coincides with the counity for any element x 2 KG, we have P " of KG. Hence, P ˛ * x D " * x D ".x2 /x1 D x1 ".x2 / D x. Moreover, S2 D idKG . Hence, we obtain that KG .x/ D ˛ * S2 .x/ D ˛ * x D x, for any x 2 KG, which shows that KG D idKG and confirms that KG is a symmetric K-algebra.

614


R` We describe now the space L1 ..KG/ / D Le ..KG/ / D .KG/ , where 1 D 1KG D e is the identity of KG and G, of left integrals of .KG/ . Consider the K-linear map ' W KG ! K P defined for x D g2G g g in KG by '.x/ D e (the coefficient ofPx at the unit element e of G). Then, for arbitrary elements u 2 .KG/ and x D g2G g g 2 KG, we have the equalities X .u'/.x/ D KG;KG .u ˝ '/ .x/ D KG;KG .u ˝ '/ g .g ˝ g/ D

X

g u.g/'.g/ D e u.e/ D u.e/'

X

g2G

g2G

g g D u.e/'.x/;

g2G

R` and hence u' D u.e/'. This shows that ' 2 .KG/ . Similarly, we show that Rr 'u D u.e/' for any u 2 .KG/ , and hence also ' 2 .KG/ . Since ' is nonzero R` Rr and .KG/ , .KG/ are one-dimensional K-vector spaces, we obtain that Le ..KG/ / D

Z

Z

`

.KG/

D K' D

r

.KG/

D Re ..KG/ /:

In particular, we conclude that e is the distinguished grouplike element of KG. Finally, observe that the nondegenerate associative K-bilinear form .; /' W KG KG ! K P P associated to ' is given, for x D g2G g g and y D g2G g g in KG, by .x; y/' D '.xy/ D

X

g g 1 ;

g2G

and hence coincides with the K-bilinear form .; / defined in Example IV.2.6. Example 5.9. Let n 2 be an integer and be a primitive n-th root of unity in K. Consider the n2 -dimensional Taft algebra Hn2 ./ D KhC; Xi=.C n 1; X n ; XC CX / defined in Example 2.4 (f). Denote by c the coset of C and by x the coset of X in Hn2 ./. Then the elements c i x j , i; j 2 f0; 1; : : : ; n 1g, form a basis of Hn2 ./ over K, and we have the relations c n D 1;

x n D 0;

xc D cx:


615

Further, the comultiplication , the counit ", and the antipode S of Hn2 ./ are defined on the K-algebra generators c and x as .c/ D c ˝ c; ".c/ D 1;

.x/ D c ˝ x C x ˝ 1;

".x/ D 0;

S.c/ D c n1 ;

S.x/ D c n1 x:

Moreover, the Hopf algebra Hn2 ./ is neither commutative nor cocommutative. Since S.c/ D c n1 D c 1 and S.x/ D c n1 x D c 1 x, we obtain that S2 .c/ D S.c 1 / D c and S2 .x/ D S.c 1 x/ D S.c 1 x/ D S.x/S.c 1 / D c n1 xc D c n1 cx D c n x D x: Then we conclude that the antipode S of Hn2 ./ has order 2n, because is a primitive n-th root of unity in K. Consider the element tD

n1 X

c m x n1

mD0

of Hn2 ./. We claim that

L" .Hn2 .// D Kt:

Indeed, for a basis element h D c i x j , i; j 2 f0; 1; : : : ; n 1g, we have ".h/ D ".c i x j / D ".c/i ".x/j equals 1 if j D 0 and 0 if j ¤ 0. Hence, we have two cases to consider. For j D 0, h D c i and then ht D c i

n1 X

n1 X c m x n1 D c mCi x n1 D t D ".c i /t D ".h/t:

mD0

mD0

For j 1, we obtain that ht D c x t D .c x i

j

D .c x i

i

j 1

j 1

n1 X

/

/.xt / D .c x i

m m n

c x

j 1

n1 X

/

xc m x n1

mD0

D 0 D ".h/t:

mD0

Hence t 2 L" .Hn2 .//, and consequently L" .Hn2 .// D Kt , because t ¤ 0 and L" .Hn2 .// is of dimension 1. Observe also that ".t / D

n1 X mD0

and hence L" .Hn2 .// Ker ".

".c/m ".x/n1 D 0;

616


Our next aim is to describe the distinguished grouplike element ˛ of Hn2 ./ , that is, an element ˛ 2 G.Hn2 ./ / such that R˛ .Hn2 .// D L" .Hn2 .// D Kt . Hence, we are looking for a K-algebra homomorphism ˛ W Hn2 ./ ! K (see Lemma 2.16) such that t h D ˛.h/t for any element h 2 Hn2 ./. Since ˛.c i x j / D ˛.c/i ˛.x j / for any basis element c i x j of Hn2 ./, it is enough to determine ˛.c/ and ˛.x/. We have the equalities tx D

n1 X

m n1

xD

c x

mD0

tc D

n1 X

c m x n D 0;

mD0

c m x n1 c D

mD0

D n1

n1 X

n1 X

n1 X c m x n1 c D c m n1 cx n1

mD0

n1 X

mD0

c mC1 x n1 D n1 t;

mD0

and hence ˛.c/ D n1 D 1 and ˛.x/ D 0. Moreover, we have the equalities ˛ * c D ˛.c/c D n1 c; ˛ * x D ˛.x/c C ˛.1/x D x: Hence, applying Theorem 5.1, we obtain that the Nakayama automorphism associated to a nondegenerate associative K-bilinear form .; /' W Hn2 ./Hn2 ./ ! R` K, given by a nonzero element ' 2 L1 .Hn2 ./ / D H 2 ./ , is defined on the n algebra generators c and x of Hn2 ./ as .c/ D ˛ * S2 .c/ D ˛ * c D n1 c; .x/ D ˛ * S2 .x/ D ˛ * 1 x D 1 .˛ * x/ D n1 x: Therefore, for an arbitrary basis vector c i x j , i; j 2 f0; 1; : : : ; n 1g of Hn2 ./, is given by .c i x j / D .c/i .x/j D .n1/.i Cj / c i x j : Observe also that has order n, again because is a primitive n-th root of unity in K.

6 The module category In this section we exhibit some properties of the category mod H of finite dimensional right H -modules over a finite dimensional Hopf algebra H over a field K. We know from Theorem 3.6 that every such an algebra H is a Frobenius algebra, hence a selfinjective algebra (see Proposition IV.3.8), and consequently the full

6. The module category

617

subcategory proj H of mod H consisting of the projective modules coincides with the full subcategory inj H of mod H consisting of the injective modules. The following semisimplicity criterion of finite dimensional Hopf algebras has been established by R. G. Larson and M. E. Sweedler in [LaSw]. Theorem 6.1. Let H D .H; mH ; H ; H ; "H ; SH / be a finite dimensional Hopf algebra over K. The following statements are equivalent. (i) H is a semisimple K-algebra. (ii) "H .L"H .H // ¤ 0. (iii) H D Ker "H ˚ L"H .H / as H -bimodules. Proof. Assume that H is a semisimple K-algebra. Consider the K-algebra epimorphism "H W H ! K. Then Ker "H is a two-sided ideal of H with dimK Ker "H D dimK H 1. Clearly, Ker "H is a left H -submodule of H . Since H is a semisimple K-algebra, H is a semisimple left H -module, so it follows from Lemma I.5.4 that there exists a left H -submodule X of H such that H D Ker "H ˚ X . Observe that dimK X D dimK H dimK Ker "H D 1. Let 1H D y C x with y 2 Ker "H and x 2 X. Then x ¤ 0, because 1H … Ker "H , and so X D Kx. For any h 2 H , we have hx 2 X. On the other hand, we have h D .h "H .h/1H / C "H .h/1H , and then hx D .h "H .h/1H /x C "H .h/x D .hx "H .h/x/ C "H .h/x. Observe that "H .hx "H .h/x/ D "H .hx/ "H ."H .h/x/ D "H .h/"H .x/ "H .h/"H .x/ D 0, and so hx "H .h/x 2 Ker "H . Since H D Ker "H ˚ X and hx D 0 C hx, with hx 2 X, we conclude that hx D "H .h/x. Therefore, x 2 L"H .H /, and clearly "H .x/ ¤ 0, because x … Ker "H . This shows that "H .L"H .H // ¤ 0. Thus (i) implies (ii). Assume now that "H .L"H .H // ¤ 0. Then there exists an element z 2 L"H .H / such that "H .z/ ¤ 0. Take t D "H .z/1 z 2 L"H .H /, so "H .t / D 1. Let M be a nonzero module in H -mod D mod H op . We will show that M is a semisimple left H -module. Take an arbitrary left H -submodule N of M , and consider the inclusion homomorphism ! W N ! M of left H -modules. Since ! is a section in K-mod D mod K, there exists a K-linear map p W M ! N such that p! D idN . We will show that ! is a section in H -mod D mod H op . We define the K-linear map W M ! N by X .m/ D t1 p .SH .t2 /m/ for any m 2 M . We show that is a homomorphism of left H -modules. Indeed, for h 2 H and m 2 M , we have the equalities X XX h1 "H .h2 / t1 p .SH .t2 /m/ h.m/ D ht1 p .SH .t2 /m/ D X X D h1 t1 p .SH .t2 /"H .h2 /m/ D h1 t1 p .SH .t2 /SH .h21 /h22 m/

618


D

X

h1 t1 p .SH .h21 t2 /h22 m/ D

X

h11 t1 p .SH .h12 t2 /h2 m/ X D .h1 t /1 p .SH ..h1 t /2 / h2 m/ D "H .h1 /t1 p .SH .t2 /h2 m/ X X X D t1 p SH .t2 / t1 p .SH .t2 /hm/ D .hm/; "H .h1 /h2 m D X

P P because h1 ˝ h21 ˝ h22 D h11 ˝ h12 ˝ h2 and h1 t D "H .h1P /t implies P .h1 t/1 ˝.h1 t /2 D .h1 t / D ."H .h1 /t / D "H .h1 / .t / D "H .h1 / . t1 ˝ t2 /. Moreover, for any n 2 N , we have X X X !.n/ D t1 p .SH .t2 /n/ D t1 SH .t2 / n t1 SH .t2 /n D D ."H .t /1H / n D n; because "H .t / D 1 and SH .t2 /n 2 N . Hence is a homomorphism in H mod D mod H op with ! D idN . Applying now Lemma I.4.2, we conclude that M D Im ! ˚ Ker D N ˚ Ker . Therefore, M is a semisimple left H -module, by Lemma I.5.4. Summing up, we have proved that every nonzero module in H -mod D mod H op is semisimple. Then, by the Wedderburn structure Theorem I.6.3, we conclude that H op , and hence H , is a semisimple K-algebra. Thus (ii) implies (i). The equivalence of the statements (ii) and (iii) follows from the fact that Ker "H and L"H .H / are two-sided ideals of H of dimensions dimK Ker "H D dimK H 1 and dimK L"H .H / D 1. We note that the above theorem is an extension of the classical Maschke’s theorem (Theorem I.6.18). Indeed, let G be a finite group and H D KG the group Hopf algebraP of G over a field K. It follows from P Example 5.8 that L"H .H / D Kt , where t D g2G g. Moreover, "H .t / D g2G "H .g/ D jGj1K . Hence, we obtain that "H L"H .H / ¤ 0 if and only if the characteristic of K does not divide the order jGj of the group G. We also point out that every Taft algebra Hn2 ./ is a nonsemisimple finite dimensional algebra (see Example 5.9). In particular, in contrast to the group algebras case, there are many nonsemisimple finite dimensional Hopf algebras over fields of characteristic 0 (for example, the field C of complex numbers). Let H D .H; mH ; H ; H ; "H ; SH / be a finite dimensional Hopf algebra over K. The comultiplication H W H ! H ˝ H allows us to consider the covariant K-linear functor ˝ D ˝K W mod H mod H ! mod H defined as follows. For two modules M and N in mod H with the right H -module structures given by K-linear homomorphisms M W M ˝ H ! M and N W N ˝ H ! N , the right H -module structure on the K-vector space M ˝ N D M ˝K N

619


is given by the composition M ˝N W M ˝ N ˝ H ! M ˝ N of the K-linear homomorphisms M ˝N ˝H

idM ˝ idN ˝H

/ M ˝N ˝H ˝H

idM ˝TN;H ˝idH

/ M ˝H ˝N ˝H M ˝ N

M ˝N .

Hence, in the sigma notation, we have X X .m˝n/h D M ˝N .m˝n˝h/ D M .m˝h1 /˝N .n˝h2 / D mh1 ˝nh2 ; P for m 2 M , n 2 N , h 2 H , where H .h/ D h1 ˝ h2 . Observe that, for m 2 M , n 2 N , and h; g 2 H , we have the equalities X X .m ˝ n/.hg/ D m.hg/1 ˝ n.hg/2 D m.h1 g1 / ˝ n.h2 g2 / X X D .mh1 /g1 ˝ .nh2 /g2 D .mh1 ˝ nh2 /g D ..m ˝ n/h/g; .m ˝ n/1H D m1H ˝ n1H D m ˝ n; because H W H ! H ˝ H is a homomorphism of K-algebras. Therefore, indeed M ˝N is a module in mod H of dimension dimK .M ˝N / D .dimK M /.dimK N /. Finally, for homomorphisms u W M ! M 0 and v W N ! N 0 in mod H , the K-linear map u ˝ v W M ˝ N ! M 0 ˝ N 0 is a homomorphism of right H -modules, because X X .u ˝ v/..m ˝ n/h/ D .u ˝ v/ mh1 ˝ nh2 D u.mh1 / ˝ v.nh2 / X D u.m/h1 ˝ v.n/h2 D .u.m/ ˝ v.n// h D .u ˝ v/.m ˝ n/h: Further, the counit "H W H ! K allows us to consider K as a right H -module via idK ˝"H

˛K

the composed K-linear homomorphism K W K ˝ H ! K ˝ K ! K, that is, h D K . ˝ h/ D "H .h/ for 2 K and h 2 H . Clearly, we have .hg/ D "H .hg/ D ."H .h/"H .g// D ."H .h//"H .g/ D .h/g and 1H D "H .1H / D 1K D , for 2 K, and h; g 2 H , because "H W H ! K is a homomorphism of K-algebras. The right H -module K defined above is called the trivial H -module. We note the following property of the trivial module. Lemma 6.2. Let H D .H; mH ; H ; H ; "H ; SH / be a finite dimensional Hopf algebra over K. Then for any module M in mod H the K-linear isomorphisms ˛M W K ˝ M ! M

and M W M ˝ K ! M;

620


given by ˛M . ˝ m/ D m and M .m ˝ / D m, for 2 K, m 2 M , are isomorphisms of right H -modules. Proof. For 2 K, m 2 M and h 2 H , we have the equalities X X ˛M .. ˝ m/h/ D ˛M h1 ˝ mh2 D ˛M "H .h1 / ˝ mh2 X X "H .h1 /h2 D "H .h1 /mh2 D .m/

D .m/h D ˛M . ˝ m/h; X X mh1 ˝ h2 D M mh1 ˝ "H .h2 / M ..m ˝ /h/ D M X X h1 "H .h2 / D mh1 "H .h2 / D .m/ D .m/h D M .m ˝ /h; by the counity property of "H . Hence ˛M and M are isomorphisms of right H modules. Let H D .H; mH ; H ; H ; "H ; SH / be a finite dimensional Hopf algebra over K. For two modules M and N in mod H , the K-vector space HomK .M; N / has the natural structure of a right H -module given by .uh/.m/ D

X

u .mSH .h1 // h2

P for u 2 HomK .M; N /, m 2 M , and h 2 H , with H .h/ D h1 ˝ h2 . Indeed, for u 2 HomK .M; N /, m 2 M , and g; h 2 H , we have the equalities ..ug/h/.m/ D D

X X

.ug/ .mSH .h1 // h2 D

X

u .mSH .h1 /SH .g1 // g2 h2 X u .mSH ..gh/1 // .gh/2 u .mSH .g1 h1 // g2 h2 D

D .u.gh//.m/; .u1H /.m/ D u .mSH .1H // 1H D u.m/; and hence .ug/h D u.gh/ and u1H D u. Further, for homomorphisms v W X ! M and w W N ! Y in mod H , the induced K-linear homomorphisms HomK .v; N / W HomK .M; N / ! HomK .X; N /; HomK .M; w/ W HomK .M; N / ! HomK .M; Y /


621

are homomorphisms of right H -modules. Indeed, for u 2 HomK .M; N /, m 2 M , x 2 X, and h 2 H , we have the equalities X HomK .v; N /.uh/.x/ D .uh/.v.x// D u .v.x/SH .h1 // h2 X X D u .v .xSH .h1 /// h2 D .uv/ .xSH .h1 // h2 D ..uv/h/.x/ D .HomK .v; N /.u/h/ .x/; HomK .M; w/.uh/.m/ D .w.uh//.m/ D w ..uh/.m// X X Dw u mSH .h1 / h2 D w .u .mSH .h1 /// h2 D ..wu/h/.m/ D .HomK .M; w/.u/h/ .m/; and hence HomK .v; N /.uh/ D HomK .v; N /.u/h and HomK .M; w/.uh/ D HomK .M; w/.u/h. Summing up, we obtain the covariant functor HomK .M; / W mod H ! mod H; and the contravariant functor HomK .; N / W mod H ! mod H: The next lemma exhibits another useful property of the trivial modules of Hopf algebras. Lemma 6.3. Let H D .H; mH ; H ; H ; "H ; SH / be a finite dimensional Hopf algebra over K and M and N be modules in mod H . Then there is a canonical K-linear isomorphism .M; N / W HomH .K; HomK .M; N // ! HomH .M; N / which is natural in M and N . Proof. Observe that there is a canonical K-linear isomorphism .M; N / W HomK .K; HomK .M; N // ! HomK .M; N / of K-vector spaces which assigns to a K-linear map f W K ! HomK .M; N / the homomorphism .M; N /.f / D f .1K / 2 HomK .M; N / and .M; N / is natural in M and N . We will show that it restricts to a K-linear isomorphism .M; N / W HomH .K; HomK .M; N // ! HomH .M; N /:

622


Take f 2 HomH .K; HomK .M; N // and abbreviate for any h 2 H we have the equalities f

f

D

.M; N /.f /. Then

h D f .1K /h D f .1K h/ D f ."H .h/1K / D "H .h/f .1K / D "H .h/

We claim that equalities f

f

f

:

2 HomH .M; N /. Indeed, for m 2 M and h 2 H , we have the X

X D f ..mh1 /"H .h2 // X X D ."H .h2 / f /.mh1 / D f h2 .mh1 / X X D f ..mh1 /SH .h21 // h22 D f ..mh11 /SH .h12 // h2 X X X h11 SH .h12 / h2 D D f m f .m"H .h1 // h2 X "H .h1 /h2 D f .m/

.mh/ D

f

D

f

m

h1 "H .h2 /

.m/h;

P because h1 ˝h21 ˝h22 D h11 ˝h12 ˝h2 . Assume now that a homomorphism f 2 HomK .K; HomK .M; N // has the property f D f .1K / 2 HomH .M; N /. Then, for any m 2 M and h 2 H , we have the equalities X X . f h/.m/ D f .mSH .h1 // h2 D f .m/SH .h1 /h2 X SH .h1 /h2 D f .m/"H .h/ D f .m/ D "H .h/ f .m/; P

and hence f h D "H .h/ f . This gives f .1K /h D "H .h/f .1K / D f .1K "H .h// D f .1K h/, and so f belongs to HomH .K; HomK .M; N //. Let H D .H; mH ; H ; H ; "H ; SH / be a finite dimensional Hopf algebra over K and M be a module M in mod H . Then we have the covariant functors ˝K M and HomK .M; / from mod H to mod H . The following theorem asserts that ˝K M is left adjoint to HomK .M; /. Theorem 6.4. Let H D .H; mH ; H ; H ; "H ; SH / be a finite dimensional Hopf algebra over K. For any modules M , X , Y in mod H there is a K-linear isomorphism M .X; Y / W

HomH .X ˝K M; Y / ! HomH .X; HomK .M; Y //

which is natural in M , X and Y .


623

Proof. It follows from the adjoint theorem Theorem II.4.3 that there is a K-linear isomorphism 'M .X; Y / W HomK .X ˝K M; Y / ! HomK .X; HomK .M; Y //; natural in M , X and Y , given by ..'M .X; Y /.f // .x// .m/ D f .x ˝ m/ for f 2 HomK .X ˝K M; Y /, x 2 X , m 2 M . Observe also that X ˝K M , Y , X , HomK .M; Y / are modules in mod H , and consequently HomK .X ˝K M; Y / and HomK .X; HomK .M; Y // are modules in mod H . Applying Lemma 6.3, we obtain the commutative diagram of K-linear isomorphisms HomH .K;'M .X;Y //

/ HomH .K; HomK .X; HomK .M; Y ///

HomH .K; HomK .X ˝K M; Y // .X ˝K M;Y /

.X;HomK .M;Y //

HomH .X ˝K M; Y /

M .X;Y /

Clearly, the required isomorphism

M .X; Y /

/ HomH .X; HomK .M; Y // .

is natural in M , X and Y .

Proposition 6.5. Let H D .H; mH ; H ; H ; "H ; SH / be a finite dimensional Hopf algebra over K, M a module in mod H , and P a projective module in mod H . Then P ˝ M is a projective module in mod H . Proof. Consider the covariant functor HomK .P ˝ M; / W mod H ! mod K: It follows from Theorem 6.4 that HomH .P ˝ M; / is isomorphic to the composition HomH .P; HomK .M; // of the functors HomK .M; / W mod H ! mod H and HomH .P; / W mod H ! mod K. Since P is a projective module, we conclude from Proposition II.2.6 that the functor HomH .P; / is exact. Obviously the functor HomK .M; / is exact. Therefore, the functor HomH .P ˝ M; / is exact, and, applying Proposition II.2.6 again, we conclude that P ˝ M is a projective module in mod H . The following theorem shows that the periodicity of the module category mod H of a finite dimensional Hopf algebra H over K reduces to the periodicity of its trivial module K. Theorem 6.6. Let H D .H; mH ; H ; H ; "H ; SH / be a finite dimensional Hopf algebra over K. The following statements are equivalent.

624


(i) mod H is periodic. (ii) The trivial H -module K is periodic. Proof. Obviously (i) implies (ii). Assume that K is an H -periodic module, say n H .K/ Š K in mod H for some n 1. Then there exists a long exact sequence in mod H of the form 0 ! K ! Pn1 ! ! P1 ! P0 ! K ! 0 with P0 ; P1 ; : : : ; Pn1 projective modules. Let M be an indecomposable nonprojective module in mod H . Applying the functor M ˝K W mod H ! mod H to the above long exact sequence, we obtain the long exact sequence in mod H of the form 0 ! K˝K M ! Pn1 ˝K M ! ! P1 ˝K M ! P0 ˝K M ! K˝K M ! 0; where K ˝K M Š M in mod H and Pn1 ˝K M; : : : ; P1 ˝K M; P0 ˝K M are projective right H -modules, by Lemma 6.2 and Proposition 6.5. Hence, it follows from Proposition IV.8.1 (v) that we have in mod H an isomorphism M Š n n H .M / ˚ P for some projective right H -module P . Moreover, H .M / is an indecomposable nonprojective module in mod H , by Propositions IV.8.1 and IV.8.3. n This leads to P D 0, and an isomorphism M Š H .M / in mod H . Therefore, the category mod H is periodic. This shows that (ii) implies (i). Example 6.7. Let n 2 be an integer, a primitive n-th root of unity in K, and H D Hn2 ./ D KhC; Xi=.C n 1; X n ; XC CX / the associated Taft algebra. Denote by c the coset of C and by x the coset of X in H . Then we have the relations c n D 1;

x n D 0;

xc D cx;

and the elements c i x j , i; j 2 f0; 1; : : : ; n 1g, form a basis of H over K. Recall also that H is a Hopf algebra with the comultiplication , counit " and the antipode S, defined on the K-algebra generators c and x as .c/ D c ˝ c; ".c/ D 1;

".x/ D 0;

.x/ D c ˝ x C x ˝ 1; S.c/ D c n1 ;

S.x/ D c n1 x:

In particular, H is neither commutative nor cocommutative. Moreover, H is a Frobenius algebra whose Nakayama automorphism H has order n (see Example 5.9). We will show that H is a Nakayama algebra having n pairwise nonisomorphic indecomposable projective right H -modules, each of them of dimension n.


625

In fact, we want to show that H is isomorphic to the bound quiver algebra KQ=I , where Q D Q.n/ is the quiver 1 n

˛1

n |

˛2

2\ ˛3

˛n

n1

M 3

˛n1

˛4

:

- i

˛i C1

˛i

and I D I.n/ is the admissible ideal in the path algebra KQ of Q generated by the paths ˛rCn1 : : : ˛rC1 ˛r , r 2 f1; : : : ; ng, where ˛nCs D ˛s for s 2 f1; : : : ; n 1g. Observe that I is generated by all paths of Q of length n. Consider the elements 1 X ri i er D c ; n n1

r 2 f1; : : : ; ng;

iD0

of H . Note that the fact that is a primitive n-th root of unity forces that the characteristic of K does not divide n, and so we have in K the element n1 . Moreover, ; 2 ; : : : ; n1 ; n D 1 D 1K are pairwise different n-th roots of 1K . We claim that e1 ; e2 ; : : : ; en are pairwise orthogonal primitive idempotents of H such that 1H D e1 C e2 C C en . Observe first that n X

1 X X ri i 1 X X ri i c D c n rD1 n rD1 n n1

er D

rD1

n1

iD0

D

1 n1H C n

n

iD0

n1 X n X iD1

ri c i

rD1

D 1H ;

P because nrD1 ri D 0 for any fixed index i 2 f1; : : : ; n 1g. For any r 2 f1; : : : ; ng, we have also the equalities er2 D

n1 n1 n1 1 X r.iCj / iCj 1 X ri i X rj j D c c c n2 n2 iD0

j D0

i;j D0

n1 n1 1 X rk k 1 X rk k D 2 n c D c D er : n n kD0

kD0

626


Further, for any r ¤ s in f1; : : : ; ng, we have er es D

n1 n1 n1 1 X riCsj iCj 1 X ri i X sj j D c c c n2 n2 j D0

iD0

1 D 2 n

n1 X

k

n1 X

kD0

lD0

c

l

i;j D0

1 n X l D 2 c D 0: n .1 / n1

lD0

In order to show that the idempotents e1 ; e2 ; : : : ; en are primitive, it is enough to prove, by Corollary I.5.8, that e1 H; e2 H; : : : ; en H are indecomposable right H -modules. Since e1 ; e2 ; : : : ; en are pairwise orthogonal idempotents of H with 1H D e1 C e2 C C en , applying Corollary I.5.10, we conclude that there is in mod H a decomposition HH D e1 H ˚ e2 H ˚ ˚ en H: We claim now that e1 H; e2 H; : : : ; en H are uniserial right H -modules of dimension n over K. Observe first that .H xH /n D 0 forces x 2 rad H , by Lemma I.3.5. For any r 2 f1; : : : ; ng, the elements 1 X ri i j c x ; n n1

er x j D

j 2 f0; 1; : : : ; n 1g;

iD0

are linearly independent (over K) elements of the right H -module er H , and hence dimK er H n. Since n2 D dimK H D dimK .e1 H ˚ e2 H ˚ ˚ en H / D dimK e1 H C dimK e2 H C C dimK en H , we conclude that dimK er H D n and the elements er ; er x; : : : ; er x n1 form a basis of er H over K, for any r 2 f1; : : : ; ng. Recall also that radk .er H / D er .rad H /k for any positive integer k (see Proposition I.5.13). Therefore, we conclude that, for any r 2 f1; : : : ; ng, the right H -module er H is of Loewy length n, the radical series er H rad er H rad2 er H radn1 er H radn er H D 0 of er H is the unique composition series of er H , and radp er H , for p 2 f1; : : : ; n 1g, is the K-vector space with the basis er x p ; er x pC1 ; : : : ; er x n1 (see also Proposition I.10.1). Hence, indeed e1 H; e2 H; : : : ; en H are uniserial projective right H -modules. Since HH D e1 H ˚ e2 H ˚ ˚ en H in mod H , applying Proposition I.8.2, we conclude that every indecomposable projective module in mod H is uniserial. Invoking now the fact that H is a Frobenius (hence selfinjective) algebra, we obtain from Propositions I.8.19 and I.10.1 that H is a Nakayama algebra. Obviously, the uniseriality of the modules e1 H; e2 H; : : : ; en H also implies that the idempotents e1 ; e2 ; : : : ; en are primitive. Furthermore, the indecomposable projective right H -modules e1 H; e2 H; : : : ; en H are pairwise nonisomorphic.


627

Indeed, take r ¤ s in f1; : : : ; ng. Observe that every element z in er H can be written in the form z D er C y with 2 K and y 2 er .rad H /, and hence zes D er es C yes D yes belongs to er .rad H /es . This shows that er Hes D er .rad H /es , and hence er H and es H are nonisomorphic right H modules, by Lemma I.8.12. Summing up, we have proved that H is a basic Nakayama algebra and e1 ; : : : ; en form a complete set of pairwise orthogonal primitive idempotents of H with 1H D e1 C C en . For each r 2 f1; : : : ; ng, set ar D er x, and observe that ar D D

1 X ri i 1 X .r1/i i i c xD c x n n 1 n

n1

n1

iD0

iD0

n1 X

.r1/i xc i D xer1

iD0

because xc D cx, where e0 D en . In particular, we have ar D er ar er1 D er xer1 , for any r 2 f1; : : : ; ng. Then we obtain, for any r 2 f1; : : : ; ng, the equalities arCp : : : arC1 ar D erCp x pC1 er1

for p 2 f0; 1; : : : ; n 1g:

Recall also that H D e1 H ˚ e2 H ˚ ˚ en H , where each er H is the K-vector space with the basis er x i with i 2 f0; 1; : : : ; n 1g. Therefore, we conclude that the K-linear map ' W KQ=I ! H defined by '."r C I / D er '.˛rCp : : : ˛rC1 ˛r C I / D arCp : : : arC1 ar

for r 2 f1; : : : ; ng; for p 2 f0; 1; : : : ; n 1g; r 2 f1; : : : ; ng;

is an isomorphism of K-algebras. In particular, it follows from Theorem I.2.10 that there exists a K-linear equivalence of categories F W mod H ! repK .Q; I /; where repK .Q; I / is the category of finite dimensional K-linear representations of the quiver Q bound by the relations generating the ideal I . On the other hand, we know from Theorems I.10.5 and III.8.7 that every indecomposable module in mod H is isomorphic to a module of the form P = radm P for some indecomposable projective module P and m 2 f1; : : : ; n 1g, and we have in mod H an almost split sequence 0 ! rad P = radmC1 P ! .rad P = radm P /˚.P = radmC1 P / ! P = radm P ! 0:

628


Therefore, the modules Mr;k D er H=er radk H , r; k 2 f1; : : : ; ng, form a complete family of pairwise nonisomorphic indecomposable modules in mod H . Observe that Pr D Mr;n D er H are the indecomposable projective modules and Sr D Mr;1 D er H=er rad H are the simple modules in mod H , for r 2 f1; : : : ; ng. In particular, the Auslander–Reiten quiver H of H consists of n2 indecomposable modules lying on the n sectional paths Pr ! Mr;n1 ! Mr;n2 ! ! Mr;2 ! Sr ; r 2 f1; : : : ; ng, as well as on the n sectional paths SrC1 ! MrC1;2 ! MrC2;3 ! ! Mr1;n1 ! Pr ; with SrC1 Š radn1 Pr D soc Pr , MrC1;2 Š radn2 Pr , MrC2;3 Š radn3 Pr , : : : , Mr1;n1 Š rad Pr , for r 2 f1; : : : ; ng. Then we conclude that the stable s of H is of the form ZAn1 =. n /, H has the upper Auslander–Reiten quiver H part of the form P1 ?

? P3 ? ? P2 ? ?? ?? ???? ?? M1;n1 M2;n1 ? ?? ? ?? ?? ?? Mn;n2 M1;n2 M2;n2 ? ?? ? ?? ?? :: :

::: :::

P? n1? ? P1 ? Pn ? ???? ???? Mn;n1 Mn1;n1 ? ?? ? ?? ?? ?? ?? Mn1;n2 Mn;n2 Mn2;n2 ? ?? ? ?? ? :: :

:: :

:: :

and the lower part of the form :: :

:: : ? ?? ? M2;3 M3;3 M4;3 ?? ?? ?? ? ? ?? ?? M3;2 M2;2 ?? ?? ? ? ? ?? ? S1 S2 S3

::: :::

:: :: : : ?? ? ?? ? Mn;3 M1;3 M2;3 ? ?? ?? ? ? ?? ?? M1;2 Mn;2 ?? ?? ? ? ?? ? ?? ? Sn1 Sn S1 :

n m In particular, we obtain that H M Š M and H M © M if m 2 f1; : : : ; n 1g, for any indecomposable nonprojective module M in mod H , and hence M has period n with respect to the Auslander–Reiten operator H D D Tr. Observe also that the Nakayama permutation of f1; : : : ; ng D f1; : : : ; nH g is given by .1/ D n and .i / D i 1 for i 2 f2; : : : ; ng. Clearly, has order n, which is also the order of the Nakayama automorphism H of H (see Example 5.9).


629

We also note that, for r 2 f1; : : : ; ng, the simple H -module Sr D er H=er rad H is isomorphic to the one-dimensional right H -module Kr with Kr D K as K-vector space and c D r , x D 0 for 2 K. In particular, the trivial right H -module K coincides with Kn , because n D 1, "H .c/ D 1, "H .x/ D 0, and consequently is isomorphic to the simple module Sn D en H=en rad H . It follows from Proposition IV.3.13 that the Nakayama functor NH D D HomH .; H / W mod H ! mod H is equivalent to the functor 0 NH D ./ 1 W mod H ! mod H H

1 of the Nakayama automorphism H . Then we coninduced by the inverse H n clude that NH Š 1mod H . Moreover, by Theorem IV.8.5, for any indecomposable nonprojective module M in mod H , there are isomorphisms 2 2 H .NH .M // Š H M Š NH .H .M //; 2n and consequently H .M / Š M in mod H . In fact, for any indecomposable nonprojective module M in mod H , we have 2 H .M / Š M and H .M / © M , which shows that M has period 2 with respect to the syzygy operator H . Indeed, for M D Mr;k D er H=er radk .er H /, r; k 2 f1; : : : ; ng, we have short exact sequences u

0 ! Mrk;nk ! Pr ! Mr;k ! 0; v

0 ! Mr;k ! Prk ! Mrk;nk ! 0; in mod H , with u and v the canonical epimorphisms (projective covers) Pr ! Pr = radk Pr and Prk ! Prk = radnk Prk , respectively, because Ker u D radk Pr Š Mrk;nk and Ker v D radnk Prk Š Mr;k , where l D n jlj for l 2 f0; 1; 2; : : : ; n C 1g. Observe also that Mr;k has the simple composition factors of the form Sr ; Sr1 ; : : : ; SrkC1 while Mrk;nk has the simple composition factors of the form Srk ; Srk1 ; : : : ; SrC1 , and consequently 2 Mr;k and Mrk;nk are nonisomorphic. This shows that H .Mr;k / Š Mr;k but H .Mr;k / D Mrk;nk © Mr;k . Summing up, we have proved that mod H is periodic with H -period n and H -period 2.

630


7 Exercises In all exercises below K will denote a field. 1. Let n 2 be a natural number and a primitive n-th root of unity in K. Consider the K-algebra Hn2 ./ D KhC; Xi=.C n 1; X n ; XC CX /; and denote by c the coset of C and by x the coset of X in Hn2 ./. Show that Hn2 ./ is a Hopf algebra with the comultiplication , counit " and the antipode S satisfying the relations .c/ D c ˝ c; ".c/ D 1;

".x/ D 0;

.x/ D c ˝ x C x ˝ 1; S.c/ D c n1 ;

S.x/ D c n1 x:

2. Assume K is of characteristic different from 2. Prove that any 2-dimensional Hopf algebra over K is isomorphic to the group algebra KC2 of the cyclic group C2 of order 2. 3. Assume K is of characteristic 2 and C2 the cyclic group of order 2. Prove the following statements. (a) The group algebra KC2 and its dual algebra .KC2 / are nonisomorphic K-Hopf algebras. (b) The Hopf algebra H2 D KŒX =.X 2 / with the comultiplication , counit " and the antipode S given by .x/ D x ˝ 1 C 1 ˝ x, ".x/ D 0, S.x/ D x, where x is the coset of X in H2 , is neither isomorphic to KC2 nor to .KC2 / . (c) Every 2-dimensional Hopf algebra H over K is isomorphic to KC2 , .KC2 / , or H2 . 4. Assume K is algebraically closed. Describe the 3-dimensional Hopf algebras over K. 5. Let H be the K-vector space with infinite basis cn , n 2 N. Consider the K-linear homomorphisms W H ! H ˝ H

and m W H ˝ H ! H

given on the basis vectors of H by .cn / D

n X iD0

ci ˝ cni ;

m.cn ˝ cp / D

nCp cnCp ; n

7. Exercises

631

for n 2 N. Prove that there exists K-linear homomorphisms

W K ! H;

" W H ! K;

S W H ! H;

such that .H; m; ; ; "; S/ is a K-Hopf algebra. 6. Let H D .H; mH ; H ; H ; "H ; SH / ! B D .B; mB ; B ; B ; "B ; SB / be a homomorphism of K-Hopf algebras. Prove the following assertions. (a) Ker f is a Hopf ideal of H . (b) Im f is a K-Hopf subalgebra of B. (c) There is a canonical isomorphism of K-Hopf algebras H= Ker f ! Im f induced by f . 7. Let H D .H; mH ; H ; H ; "H ; SH / be a finite dimensional Hopf algebra over K and H D .H ; mH ; H ; H ; "H ; SH / the double dual Hopf algebra of H over K. Prove that the evaluation K-linear isomorphism eH W H ! H is an isomorphism of K-Hopf algebras. 8. Let C D .C; C ; "C / be a nonzero finite dimensional K-coalgebra and let .C ; mC ; C / be the associated dual K-algebra, where mC D C C;C and

C D "C . Prove that C is a simple K-coalgebra (the only K-subcoalgebras of C are 0 and C ) if and only if C is a simple K-algebra. 9. Let C D .C; C ; "C / be a nonzero finite dimensional K-coalgebra. Prove that: (a) C contains a nonzero simple K-subcoalgebra. (b) The sum corad C of all simple K-subcoalgebras of C (coradical of C ) is a K-subcoalgebra of C . 10. Let C D .C; C ; "C / be a nonzero finite dimensional K-coalgebra and let .C ; mC ; C / be its dual K-algebra. Prove that C D corad C (C is cosemisimple) if and only if rad C D 0 (C is a semisimple K-algebra). 11. Let C D .C; C ; "C / be a nonzero finite dimensional K-coalgebra and G.C / the set of all grouplike elements of C . Prove that Cg D Kg , g 2 G.C /, is the family of all pairwise different one-dimensional K-subcoalgebras of C . 12. Let C D .C; C ; "C / be a nonzero finite dimensional K-coalgebra and let .C ; mC ; C / be its dual K-algebra. Prove that the following statements are equivalent: (a) Every simple K-subcoalgebra of C is one-dimensional (C is a pointed coalgebra).

632


(b) corad C D KG.C /. (c) C is a basic K-algebra. 13. Let C D .C; C ; "C / be a nonzero finite dimensional K-coalgebra, M a finite dimensional left H -module via a K-linear homomorphism M W M ! C ˝M , and N a finite dimensional right H -module via a K-linear homomorphism %N W N ! N ˝ C . Moreover, let .C ; mC ; C / be the dual K-algebra of C . Consider the K-linear homomorphisms TM;C

M;C

M

M W M ˝ C ! .M ˝ C / ! .C ˝ M / ! M ; TC;N

C;N

% N

%N W C ˝ N ! .C ˝ N / ! .N ˝ C / ! N : Prove that (a) M is a right C -module via M ; (b) N is a left C -module via %N . 14. Let A D .A; mA ; A / be a finite dimensional K-algebra and let further A D .A ; A ; "A / be the associated dual K-coalgebra, where A D !A;A mA and "A D A . Moreover, let M be a finite dimensional left A-module via a Klinear homomorphism M W A ˝ M ! M and N a finite dimensional right Amodule via a K-linear homomorphism "N W N ˝ A ! N . Consider the K-linear homomorphisms M

1

A;M

'N

1

N;A

TA ;M

M W M ! .A ˝ M / ! A ˝ M ! M ˝ A ; TN ;A

'N W N ! .N ˝ A/ ! N ˝ A ! A ˝ N : Prove that (a) M is a right A -comodule via M ; (b) N is a left A -comodule via 'N . 15. Assume K is algebraically closed of characteristic 0. Let H be a K-Hopf algebra of dimension 4. Prove that (a) H is pointed as a K-coalgebra; (b) dimK G.H / > 1. 16. Assume K is algebraically closed of characteristic 0. Let H be a K-Hopf algebra of dimension 4. Prove that H is isomorphic to one of the following KHopf algebras:

7. Exercises

633

(a) the group algebra KC4 of the cyclic group C4 of order 4, (b) the group algebra K.C2 C2 / of the Klein 4-group C2 C2 , (c) the Sweedler algebra H4 (Example 2.4 (e)). 17. Assume K is of characteristic 0. Let H be a finite dimensional commutative Hopf algebra over K. Prove that H is isomorphic (as a K-Hopf algebra) to the dual algebra .KG/ of the group algebra KG of a finite group G. 18. Assume K is of characteristic 0. Let H be a finite dimensional cocommutative Hopf algebra over K. Prove that H is isomorphic (as a K-Hopf algebra) to the group algebra KG of a finite group G. 19. Let K be of characteristic p > 0. Let L D K 2 with the canonical basis e1 D .1; 0/, e2 D .0; 1/ over K. Consider the K-bilinear map Œ; W L L ! L given by Œe1 ; e1 D 0;

Œe2 ; e2 D 0;

Œe1 ; e2 D e2 ;

Œe2 ; e1 D e2 ;

p Œp Œp and the map WŒpL ! L given by x D 1 e1 for x D 1 e1 C 2 e2 2 L. Show that L; Œ; ; is a restricted Lie algebra over K and describe the restricted enveloping algebra u.L/ of L.

20. Let K be of characteristic p > 0 and n be a positive integer. Consider the .2n C 1/-dimensional K-vector space Hn with the basis x1 ; : : : ; xn ; y1 ; : : : ; yn ; z and the K-bilinear map Œ; W Hn Hn ! Hn given by Œxi ; yj D ıij z; Œyj ; xi D ıij z; Œxi ; xj D 0; Œyi ; yj D 0; Œxi ; z D 0; Œz; xi D 0; Œyi ; z D 0; Œz; yi D 0; for all i; j 2 f1; : : : ; ng. Prove the following assertions. (a) .Hn ; Œ; / is a Lie algebra over K. (b) There exists a p-map Œp W Hn ! Hn such that xiŒp D 0, yiŒp D 0, z Œp D 0, for i 2 f1; : : : ; ng, and describe the restricted enveloping algebra u.Hn / of the restricted Lie algebra Hn D Hn ; Œ; ; Œp . (c) There exists a p-map Œp W Hn ! Hn such that xiŒp D 0, yiŒp D 0, z Œp D z, for i 2 f1; : : : ; ng, and describe the restricted enveloping algebra u.Hn / of the restricted Lie algebra Hn D Hn ; Œ; ; Œp . 21. Let H be a finite dimensional cocommutative Hopf algebra over K and M and N modules in mod H . Prove that the right H -modules M ˝ N and N ˝ M are isomorphic.

634


22. Let H be a finite dimensional Hopf algebra over K, M a module in mod H , and P a projective module in mod H . Prove that M ˝ P is a projective module in mod H . 23. Let H be a finite dimensional Hopf algebra over K and M a module in mod H . Prove that the right H -modules M and M D .M / are isomorphic. 24. Let H be a finite dimensional Hopf algebra over K and M and N modules in mod H . Prove that the right H -modules HomK .M; N / and M ˝ N are isomorphic. 25. Let H be a finite dimensional Hopf algebra over K and M and N modules in mod H . Prove that there is a K-linear isomorphism HomH .K; M ˝ N / Š HomH .N ; M /. 26. Let H be a finite dimensional Hopf algebra over K and M a module in mod H . Prove that M is a direct summand of M ˝ M ˝ M in mod H . 27. Let H be a finite dimensional Hopf algebra over K and M a module in mod H . Prove that the following conditions are equivalent: (a) M is projective in mod H . (b) M ˝ M is projective in mod H . (c) M ˝ M is projective in mod H . (d) M is projective in mod H . 28. Let K be a field of characteristic p > 0, G a p-group, and H D KG the group algebra of G over K. Prove that the trivial module K is a unique simple right H -module (up to isomorphism). 29. Let K be a field of characteristic ¤ 2 and H D K hC; Xi =.C 2 1; X 2 ; CX C XC /; Sweedler’s Hopf algebra. Prove the following statements. (a) For any simple right H -modules S and T , S ˝ T is a simple right H -module. (b) For any indecomposable projective right H -modules P and Q, P ˝ Q is a direct sum of two indecomposable projective right H -modules. 30. Let K be an algebraically closed field of characteristic 3, SL2 .F3 / the group of 2 2 matrices of determinant 1 over the field F3 with three elements (regarded as the prime subfield of K) and H D K SL2 .F3 / the group algebra of SL2 .F3 / over

7. Exercises

635

K. Let KŒX; Y be the polynomial algebra in two commuting variables X and Y , considered as the right H -modules, with the right actions of elements a b c d of SL2 .F3 / on KŒX; Y as polynomial automorphisms given on X and Y by a b a b X D aX C bY; Y D cX C d Y: c d c d Let S1 D K be the trivial right H -module, S2 the right H -submodule of KŒX; Y formed by the linear homogeneous polynomials, and S3 the right H -submodule of KŒX; Y formed by the homogeneous polynomials of degree 2. Show that (a) S2 is a simple nonprojective right H -module; (b) S3 is a simple projective right H -module; (c) S2 ˝ S2 Š S1 ˚ S3 in mod H .

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Index

A-bilinear map, 134 A-bimodule, 125 acyclic quiver, 6 additive function of Cartan matrix, 448 adjoint pair of functors, 141 adjoint theorem, 144 adjunction of categories, 141 adjunction of functors, 141 admissible group, 293 admissible ideal, 9 admissible sequence, 516 algebraically closed field, 2 almost split sequence, 270 ˛-camp, 359 A-module homomorphism, 16 A-module isomorphism, 16 antipode, 554 arrow ideal, 9 arrows of quiver, 6 associative bilinear form, 336 A-submodule, 18 Auslander–Reiten quiver, 282 Auslander–Reiten theorem, 273 Auslander–Reiten translation, 244 Baer sum, 213 basic algebra, 172, 173 basic idempotent, 376 basic primitive idempotent, 375 basis of root system, 496 ˇ-camp, 359 bialgebra, 554 bimodule, 125 block, 40 bound quiver, 9 bound quiver algebra, 9 bound representation, 26

Brauer quiver, 359 Brauer tree, 359 Brauer tree algebra, 360 canonical decomposition, 375 canonical mesh algebra, 454 Carlson theorem, 422 Cartan matrix of algebra, 106 Cartan matrix on set, 447 category, 123 center of algebra, 368 central algebra, 370 central idempotent, 35 class of morphisms, 123 class of objects, 123 co-opposite coalgebra, 547 co-opposite Hopf algebra, 571 coalgebra, 541 cocommutative, 541 cogenerator of category, 175 coideal, 552 cokernel, 19 commutative ring, 1 comodule, 585 component, 291 composite of functors, 141 composition multiplicity, 70 composition of functors, 123 composition series, 68 composition vector, 106 comultiplication of coalgebra, 541 connected quiver, 6 contravariant functor, 124 convolution product, 552 coradical of coalgebra, 631 cosemisimple coalgebra, 631 cosyzygy module, 392 counit of adjunction of functors, 142

646

Index

counit of coalgebra, 541 covariant functor, 124 covering morphism, 292 Coxeter graph, 501 Coxeter group, 490 cycle, 6 degree of the representation, 12 dense functor, 157 density theorem, 61 dihedral group, 491 dimension, 13 direct product, 199 direct sum, 19 direct sum of modules, 19, 199 direct summand, 44 distinguished grouplike element of dual Hopf algebra, 602 distinguished grouplike element of Hopf algebra, 600 division ring, 2 double centralizer property, 155 double quiver, 452 dualities of categories, 175 duality module, 183 duality of categories, 175 Dynkin graph, 448 Dynkin quiver, 457 Dynkin type, 463 effective subgroup, 490 enveloping algebra, 369 equivalence of categories, 157 equivalences of categories, 157 equivalent categories, 157 equivalent extensions of modules, 209 equivalent representations of algebras, 13 equivalent representations of groups, 13 essential submodule, 81 Euclidean graph, 449 evaluation map, 26

exact functor, 132 exact sequence of modules, 88 exceptional cycle, 359 Ext-algebra, 414 extension, 209 factor representation, 21 factor right module, 18 faithful functor, 157 faithful module, 154 fibered product, 114 fibered sum, 115 field, 2 finite dimensional algebra, 2 finite dimensional module, 16 finite dimensional representation, 21 finite quiver, 6 finite representation type, 104 finitely generated module, 18 first (left) regular representation of an algebra, 15 first Brauer–Thrall conjecture, 304 Fitting lemma, 41 free module, 73 Frobenius algebra, 336 full functor, 157 full subcategory, 124 fundamental reflection, 496 fundamental root of Coxeter group, 496 generalized quaternion algebra, 121 generator of category, 151 generators of module, 18 geometrically isomorphic groups, 497 Grothendieck group, 105 group algebra, 3 grouplike element of Hopf algebra, 572 Hecke algebra, 516 Heisenberg algebra, 581 hereditary algebra, 93 hom functor, 131 homomorphism of bialgebras, 554

Index

homomorphism of coalgebras, 542 homomorphism of Hopf algebras, 555 homomorphism of right comodules, 585 homomorphism of right Hopf modules, 586 Hopf algebra, 555 Hopf ideal, 572 Hopf module, 586 Hopf subalgebra, 571 ideal, 2 idempotent, 35 identity functor, 141 identity map, 19 identity matrix, 3 identity morphism, 123 identity of ring, 1 image, 19 indecomposable algebra, 37 indecomposable Cartan matrix, 448 indecomposable module, 19 indecomposable representation, 22 infinite radical, 207 injective dimension, 90 injective envelope, 82 injective module, 79 injective representation, 86 injective resolution, 88 injective vertex, 283, 292 injectively stable category, 234 inner automorphism, 184 inner automorphism group, 184 integral marked graph, 501 irreducible Coxeter group, 498 irreducible homomorphism, 260 isomorphic algebras, 2 isomorphic modules, 16 isomorphic sequences of modules, 269 isomorphism of representations, 21 isomorphism of translation quivers, 292 Jacobi identity, 580

647

Jacobson radical of algebra, 30 Jacobson radical of module, 50 Jacobson radical of module category, 204 Jacobson radical of the homomorphism space, 203 Jordan–Hölder theorem, 68 K-algebra, 2 K-algebra homomorphism, 2 K-category, 124 kernel, 19 K-Hopf algebra, 555 K-linear functor, 131 K-linear representation of a quiver, 21 Kronecker quiver, 7 K-subalgebra, 2 left adjoint functor, 141 left almost split homomorphism, 257 left annihilator, 348 left exact contravariant functor, 132 left exact covariant functor, 131 left hereditary algebra, 93 left ideal, 2 left integral, 594 left minimal almost split homomorphism, 258 left minimal homomorphism, 257 left module, 16 left perpendicular set, 349 length function, 293 length of admissible sequence, 516 length of an element of the Coxeter group, 509 length of composition series of module, 70 length of path, 6 Lie algebra, 579 Lie bracket, 579 local algebra, 32 locally finite quiver, 292 Loewy length of algebra, 54 Loewy length of module, 54

648

Index

Loewy series, 54 loop, 6 marked graph, 497, 499 marked subgraph, 500 Maschke theorem, 65 matrix coalgebra, 545 maximal ideal, 30 minimal epimorphism, 75 minimal injective cogenerator, 177 minimal injective copresentation, 249 minimal injective resolution, 89 minimal monomorphism, 81 minimal progenerator, 154 minimal projective presentation, 235 minimal projective resolution, 88 Morita equivalence of module categories, 160 Morita equivalence theorem, 165, 166 Morita equivalent algebras, 160 Morita invariant, 161 Morita–Azumaya duality, 180 Morita–Azumaya duality theorem, 180, 182 morphism in category, 123 morphism of representations, 21 mouth of tube, 294 multiplication of algebra, 540 mutually inverse equivalences, 157

Noether–Skolem theorem, 373 nondegenerate bilinear form, 336 n-th extension space, 402 n-th tensor algebra, 139 A -periodic module, 414 opposite algebra, 4 opposite category, 124 opposite Hopf algebra, 570 opposite quiver, 12 opposite-co-opposite Hopf algebra, 562 ordinary cycle, 359 orthogonal idempotents, 35, 47 outer automorphism, 184 outer automorphism group, 184

parastrophic matrix, 333 path algebra, 6 path of irreducible homomorphisms of length t , 301 path of quiver, 6 perfect field, 435 period of algebra, 439 period of module, 414 periodic algebra, 439 periodic module, 414 periodic module category, 414 p-map, 580 pointed coalgebra, 631 positive root, 496 Nakayama algebra, 101 preprojective algebra, 454 Nakayama automorphism, 346 primitive element of Hopf algebra, 574 Nakayama functor, 247 primitive idempotent, 47 Nakayama lemma, 31 product of algebras, 3 Nakayama permutation, 377 progenerator of module category, 153 natural equivalence of functors, 157 natural isomorphism of functors, 141, projective cover, 76 projective dimension, 90 157 projective module, 73 natural linear isomorphism, 141 natural transformation of functors, 140 projective representation, 86 projective resolution, 88 negative root, 496 projective vertex, 283, 292 nilpotent element, 31 projectively stable category, 233 nilpotent ideal, 31

Index

proper ideal, 2 proper submodule, 18 pull-back, 114 push-out, 115

649

ring, 1 ring homomorphism, 2 root of Coxeter group, 490 root system of Coxeter group, 490

second (right) regular representation of an algebra, 15 section, 41 sectional path in translation quiver, 312 radical of algebra, 30 sectional path of irreducible homomorphisms, 312 radical of module, 50 selfduality of module category, 183 radical of module category, 204 radical of the homomorphism space, 203 selfinjective algebra, 178 semisimple algebra, 57 radical series of module, 54 semisimple module, 45 reduced presentation of an element of the Coxeter group, 509 separable algebra, 432 reflection along the vector, 489 separable extension of field, 435 reflection of the space, 489 set of morphisms, 123 reflection through the hyperplane, 489 short exact sequence of modules, 88 regular bimodule, 369 sigma notation, 542 relation in path algebra, 10 simple algebra, 58 representation of algebra, 13 simple composition factor, 68 representation of group, 12 simple module, 45 representation of quiver, 21 simple representation, 86 restricted commutator algebra, 582 simple root of Coxeter group, 496 restricted enveloping algebra, 583 sink in quiver, 108 restricted Lie algebra, 580 skew field, 2 retraction, 41 socle equivalent algebras, 383 right adjoint functor, 141 socle of algebra, 383 right almost split homomorphism, 258 socle of module, 53 right annihilator, 348 source of quiver, 6 right exact contravariant functor, 132 space of irreducible homomorphisms, 284 right exact covariant functor, 132 space of left g-integrals, 598 right hereditary algebra, 93 space of left integrals, 594 right ideal, 2 space of right g-integrals, 598 right integral, 594 space of right integrals, 594 right minimal almost split homomorphism, 258 special biserial algebra, 365 right minimal homomorphism, 257 splittable exact sequence of modules, 88 right module, 16 splittable extension, 210 right perpendicular set, 349 stable Auslander–Reiten quiver, 455 right tensor Hopf module, 586 stable translation quiver, 293 quasi-inverse functor, 157 quaternion group, 122 quiver, 6

650

Index

stable tube of rank r, 294 standard K-duality of module categories, 20 strictly smaller, 451 structure constants of algebra, 4 subadditive function of Cartan matrix, 448 subcategory, 124 subcoalgebra, 552 subrepresentation, 21 subspace of coinvariants, 586 summand of idempotent, 48 superfluous submodule, 75 Sweedler’s algebra, 558 symmetric algebra, 336 symmetric algebra of bimodule, 139 syzygy module, 392 Taft algebra, 559 target of quiver, 6 A -periodic module, 415 A -periodic module category, 415 t-basis of root system, 491 tensor algebra of bimodule, 139 tensor functor, 136 tensor Hopf algebra, 560 tensor Hopf algebra of vector space, 575 tensor product, 134 tensor product coalgebra, 546

t -negative root, 491 top of module, 53 t-positive root, 491 trace map, 342 trace of matrix, 342 translation, 292 transpose, 236 trivial extension algebra, 343 trivial idempotent, 35, 47 trivial module, 619 trivial path of quiver, 6 two-sided ideal, 2 underlying graph, 6 uniserial module, 100 unit of adjunction of functors, 141 unit of algebra, 540 unit vector, 489 universal property of fibered product, 114 universal property of fibered sum, 115 universal property of tensor product, 135 valued translation quiver, 292 valued translation quiver morphism, 292 vertices of quiver, 6 weakly symmetric algebra, 378 Wedderburn theorem, 57 zero matrix, 3

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Introduction to Lie Algebras and Representation Theory

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Elements of the representation theory of associative algebras. Representation-infinite tilted algebras

Elements of the Representation Theory of Associative Algebras: Representation-Infinite Tilted Algebras

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