Equivalence and Duality for Module Categories with Tilting and Cotilting for Rings

EQUIVALENCE AND DUALITY FOR MODULE CATEGORIES (with Tilting and Cotilting for Rings) This book provides a unified appr...

Author: Robert R. Colby | Kent R. Fuller

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EQUIVALENCE AND DUALITY FOR MODULE CATEGORIES (with Tilting and Cotilting for Rings)

This book provides a unified approach to many of the theories of equivalence and duality between categories of modules that have transpired over the last 45 years. In particular, during the past dozen or so years many authors (including the authors of this book) have investigated relationships between categories of modules over a pair of rings that are induced by both covariant and contravariant representable functors, in particular by tilting and cotilting theories. By here collecting and unifying the basic results of these investigations with innovative and easily understandable proofs, the authors’ aim is to provide an aid to further research in this central topic in abstract algebra and a reference for all whose research lies in this field. Robert R. Colby is Professor Emeritus at the University of Hawaii and Independent Scholar at the University of Iowa. Kent R. Fuller is a professor of mathematics at the University of Iowa.

CAMBRIDGE TRACTS IN MATHEMATICS General Editors

` W. FULTON, A. KATOK, F. KIRWAN, B. BOLLOBAS, P. SARNAK, B. SIMON

161

Equivalence and Duality for Module Categories (with Tilting and Cotilting for Rings)

EQUIVALENCE AND DUALITY FOR MODULE CATEGORIES (with Tilting and Cotilting for Rings)

ROBERT R. COLBY University of Hawaii and University of Iowa

KENT R. FULLER University of Iowa

cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge cb2 2ru, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521838214 © Robert R. Colby and Kent R. Fuller 2004 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format isbn-13 isbn-10

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isbn-13 isbn-10

978-0-521-83821-4 hardback 0-521-83821-5 hardback

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Contents

Preface Acknowledgment 1

2

3

4

5

page vii ix

Some Module Theoretic Observations 1.1 The Kernel of Ext1R (V, ) 1.2 Gen(V ) and Finiteness 1.3 Add(VR ) and Prod(VR ) 1.4 Torsion Theory Representable Equivalences 2.1 Adjointness of Hom R (V, ) and ⊗ S V 2.2 Weak ∗-modules 2.3 ∗-modules 2.4 Three Special Kinds of ∗-modules Tilting Modules 3.1 Generalized Tilting Modules 3.2 Tilting Modules 3.3 Tilting Torsion Theories 3.4 Partial Tilting Modules 3.5 The Tilting Theorem 3.6 Global Dimension and Splitting 3.7 Grothendieck Groups 3.8 Torsion Theory Counter Equivalence Representable Dualities 4.1 The U -dual 4.2 Costar Modules 4.3 Quasi-Duality Modules 4.4 Morita Duality Cotilting 5.1 Cotilting Theorem 5.2 Cotilting Modules v

1 1 2 8 9 12 12 14 18 22 28 28 31 34 36 40 45 49 57 66 67 69 75 84 86 86 91

vi

A

B

Contents

5.3 Cotilting Bimodules 5.4 Cotilting via Tilting and Morita Duality 5.5 Weak Morita Duality 5.6 Finitistic Cotilting Modules and Bimodules 5.7 U -torsionless Linear Compactness 5.8 Examples and Questions Adjoints and Category Equivalence A.1 The Yoneda–Grothendieck Lemma A.2 Adjoint Covariant Functors A.3 Equivalence of Categories Noetherian Serial Rings B.1 Finitely Generated Modules B.2 Injective Modules

Bibliography Index

97 103 108 115 119 126 131 131 132 135 139 139 143 147 151

Preface

Approximately forty-five years ago K. Morita presented the first major results on equivalences and dualities between categories of modules over a pair of rings. These results, which characterized an equivalence between the entire categories of right (or left) modules over two rings as being represented by the covariant Hom and tensor functors induced by a balanced bimodule that is a so-called progenerator on either side, and which characterized a duality between reasonably large subcategories of right and left modules over two rings as being represented by the contravariant Hom functors induced by a balanced bimodule that is an injective cogenerator on both sides, have come to be known as the Morita theorems. Morita’s theorems on equivalence are exemplified by the equivalence of the categories of right modules over a simple artinian ring and the right vector spaces over its underlying division ring. More than a dozen years later the second author, expanding on the relationship between the category of modules generated by a simple module and the vector spaces over its endomorphism ring, introduced the concept of a quasi-progenerator to characterize the equivalences between a subcategory of right modules over one ring that is closed under submodules, epimorphic images, and direct sums and the category of all right modules over another ring. In the interim, employing the notion of linear compactness, B. J. Müller had characterized the reflexive modules under Morita duality and given a one-sided characterization of the bimodules inducing these dualities. These results led to several investigations of the dual notion of a quasi-progenerator that we refer to as a quasi-duality module. In the early 1980s the notion of tilting modules and the tilting theorem for finitely generated modules over artin algebras was introduced and polished in papers by S. Brenner and M. C. R. Butler, D. Happel and C. M. Ringel, and K. Bongartz to provide new insight and examples in the representation theory of artin algebras. This has proved particularly effective in the study of algebras of various representation types. The definition of a tilting module also applies to modules over arbitrary rings and, in this setting, induces torsion theories in vii

viii

Preface

the categories of modules over two rings and a pair of equivalences between the torsion and torsion-free parts of the torsion theories. These tilting modules and equivalences are among the principal topics of this book. Near the end of the 1980s C. Menini and A. Orsatti introduced a type of module, which has come to be known as a ∗-module, that generalizes both quasi-progenerators and tilting modules by inducing an equivalence between a subcategory of modules over one ring that is closed under direct sums and epimorphic images and a subcategory of modules over a second ring that is closed under direct products and submodules. At about the same time, the first author introduced a generalization of Morita duality and the notions of cotilting modules and a cotilting theorem for noetherian rings. This inspired several versions of cotilting modules and cotilting theorems that are further principal topics of this book. Just as progenerators, quasi-progenerators, and tilting modules are all ∗-modules, so are Morita duality modules, quasi-duality modules, and the various flavors of cotilting modules, all dual versions of ∗-modules, a kind of module that we call costar modules. As is so often the case in mathematics, generalizations of a concept led to a better understanding of the concept. Here we provide unified proofs regarding the various types of equivalence and duality, and the modules that induce them, by approaching them via the general notions of ∗-modules and costar modules. We feel that this approach should yield improved accessibility to, and better understanding of, these concepts. En route we present much of the relatively little that is known about how properties of (the modules over) one of the rings in question are transformed to the other ring under these various equivalences and dualities. We hope that this exposition will inspire further research in this and related directions. This book contains some of the work of many authors that has inspired and, to a small extent been inspired by, our own research. We have made an attempt to give credit where it is due, but surely we have inadvertently omitted references to some works that should have been mentioned. To the authors of these works, we offer our sincere apologies, and note that they have undoubtedly been referred to in one or more of the papers and/or books in our bibliography.

Acknowledgment

Colby wishes to express his gratitude for the hospitality of the University of Iowa, where he is an independent scholar.

ix

1 Some Module Theoretic Observations

We begin with a chapter consisting of several general facts involving various closure properties of certain categories of modules. These results are part of the background necessary for our future chapters, and we believe that they are of interest in themselves. Throughout this book R denotes an associative ring with identity 1 ∈ R, and Mod-R and R-Mod represent the categories of right and left R-modules and homomorphisms, while mod-R and R-mod denote their subcategories of finitely generated modules. 1.1. The Kernel of Ext1R (V, ) For any R-module V we denote the kernel of Ext1R (V, ) by V ⊥ . Closure properties of V ⊥ are related to both homological and module-theoretic properties of V. We denote the projective dimension of a module M by proj . dim .M. Proposition 1.1.1. VR⊥ is closed under factors if and only if proj . dim .VR ≤ 1. Proof. If VR⊥ is closed under factors, M ∈ Mod-R, and E(M) is the injective envelope of M, then, since E(M)/M ∈ V ⊥ , the exactness of the sequence 0 = Ext1R (V, E(M)/M) → Ext2R (V, M) → Ext2R (V, E(M)) = 0 implies proj . dim .VR ≤ 1. Conversely, if proj . dim .VR ≤ 1 and M ∈ V ⊥ with K a submodule of M, we obtain M/K ∈ V ⊥ from the exactness of the sequence 0 = Ext1R (V, M) → Ext1R (V, M/K ) → Ext2R (V, K ) = 0. Proposition 1.1.2. If VR ∈ Mod-R is finitely presented, then Ext1R (V, ) commutes with direct sums, so VR⊥ is closed under direct sums. 1

2

Some Module Theoretic Observations

Proof. If {Mα }α∈A is a family of modules in Mod-R, the natural monomorphism φV : ⊕ A Hom R (V, Mα ) → Hom R (V, ⊕ A Mα ) is an isomorphism whenever V is finitely generated by [1, Exercise 16.3]. Moreover, φ induces natural homomorphisms θ M : ⊕ A Ext1R (M, Mα ) → Ext1R (M, ⊕ A Mα ). By hypothesis there is an exact sequence 0 → K → P → V → 0 with P, K finitely generated and P projective. We obtain the commutative diagram with exact rows ⊕ A Hom R (P, Mα )→⊕ A Hom R (K , Mα )→⊕ A Ext1R (V, Mα )→0 ↓ φK ↓ θV ↓ φP Hom R (P, ⊕ A Mα )→Hom R (K , ⊕ A Mα )→Ext1R (V, ⊕ A Mα )→0 from which the lemma follows. We note that a partial converse of this last result is found in the proof of Lemma 1.2 of [77]. Proposition 1.1.3. If VR is finitely generated and VR⊥ is closed under factors and direct sums, then VR is finitely presented. Proof. We have proj . dim .VR ≤ 1 by Proposition 1.1.1; thus, since VR is finitely generated, there is an exact sequence 0 → L → R n → V → 0, where L is projective. Hence, there is a split monomorphism j : L → R (X ) for some set X . By hypothesis E(R)(X ) ∈ V ⊥ , so the composition of j with the inclusion i of R (X ) into E(R)(X ) has an extension to an element f ∈ Hom(R n , E(R)(X ) ). Then f (R n ) ⊆ E(R)(F) ⊆ E(R)(X ) for some finite subset F of X . It follows that j(L) ⊆ R (F) ⊆ R (X ) ; therefore, since j is split monic, L is finitely generated.

1.2. Gen(V) and Finiteness We recall (see [1]) that for any collection V of R-modules, Gen(V) (gen(V)) denotes the full category of R-modules that are epimorphic images of (finite) direct sums of modules isomorphic to those in V, and we let TrV (M) denote the trace of V in M, the unique largest submodule of M that belongs to Gen(V). If V consists of a single module VR we simply write Gen(VR ), and if S = End(VR ), then TrV (M) is the image of the canonical mapping ν M : V ⊗ S Hom R (V, M) → M. In order to characterize when Gen(VR ) is closed under direct products, we employ the following notions and lemmas.

1.2. Gen(V ) and Finiteness

3

Given that {Mα }α∈A is a family in Mod-R, for each N ∈ R-Mod we let η A Mα ,N : ( A Mα ) ⊗ R N −→ A (Mα ⊗ R N ) denote the canonical mapping to obtain a natural transformation η A Mα : (( A Mα ) ⊗ R ) −→ A (Mα ⊗ R ). Lemma 1.2.1. Suppose that {Mα }α∈A is a family in Mod-R. If R N is finitely generated (finitely presented), then the canonical homomorphism η A Mα ,N : ( A Mα ) ⊗ R N −→ A (Mα ⊗ R N ) is an epimorphism (isomorphism). Proof. If {x1 , . . . , xn } generate R N , then any element of Mα ⊗ R N can be written in the form i m αi ⊗ xi . Now assume that N is finitely presented and let 0 → K → P → N → 0 be an exact sequence with P finitely generated and projective and K finitely generated. Then we have a commutative diagram ( A Mα ) ⊗ K → ( A Mα ) ⊗ P → ( A Mα ) ⊗ N → 0 η A Mα ,K ↓ η A Mα ,P ↓ η A Mα ,N ↓ A (Mα ⊗ K ) → A (Mα ⊗ P) → A (Mα ⊗ N ) → 0 with exact rows, in which η A Mα ,K is epic, and η A Mα ,P is easily seen to be an isomorphism by naturalness of η A Mα . Hence, by the Five Lemma, η A Mα ,N is an isomorphism. Identifying R ⊗ R N = N , we have the following result. Lemma 1.2.2. Let N ∈ R-Mod. Then the canonical homomorphism η R A ,N : (R A ) ⊗ R N −→ N A is an epimorphism (isomorphism) for all sets A if and only if N is finitely generated (finitely presented). Proof. The condition is sufficient in either case by Lemma 1.2.1. Conversely, letting A = N , if the diagonal element (n)n∈N is the image of some element m (ani )n∈N ⊗ R xi , then, for all n ∈ N , n = i ani xi. . Thus, R N is finitely i=1 generated whenever η R N ,N is epic. Now supposing that η R A ,N is an isomorphism for all sets A, there is an exact sequence 0 → K → P → N → 0

4


with P finitely generated and projective. Then both η R A ,P and η R A ,N are isomorphisms in the commutative diagram R A ⊗ K −→ R A ⊗ P −→ R A ⊗ N → 0 η R A ,K ↓ η R A ,P ↓ η R A ,N ↓ −→ PA −→ NA −→ 0 0→ KA with exact rows. Hence, by the Snake Lemma, η R A ,K is an epimorphism, and so K is finitely generated. Now we are in position to determine just when Gen(VR ) is closed under direct products. Proposition 1.2.3. The following statements about a module V R with S = End(VR ) are equivalent: (a) Gen(VR ) contains V A for all sets A; (b) Gen(VR ) is closed under direct products; (c) S V is finitely generated. Proof. A module M R is in Gen(VR ) if and only if the canonical trace mapping ν M : Hom R (V, M) ⊗ S V → M is epic. For any set A we have the commutative diagram ∼ =

Hom R (V, V A ) ⊗ S V → Hom R (V, V ) A ⊗ S V ↓= ↓ νV A VA

η S A ,V

←−

S A ⊗ S V,

so (a) ⇔ (c) follows from Lemma 1.2.2. (b) ⇒ (a) is clear. For (c) ⇒ (b), assume that S V is finitely generated and {Mα }α∈A belong to Gen(VR ). Then the composite of the canonical homomorphisms Hom R (V, A Mα ) ⊗ S V ∼ = ( A Hom R (V, Mα )) ⊗ S V η −→ A (Hom R (V, Mα ) ⊗ S V ) A ν Mα

−→ A Mα

is epic by Lemma 1.2.1 and this composite is νMα . Next we obtain a mapping, in addition to the trace map, that determines whether a module belongs to Gen(V A ).


5

Lemma 1.2.4. Let i : R R → VR be a homomorphism. If Hom R (i, M) : Hom R (V, M) → Hom R (R, M) is an epimorphism, then M ∈ Gen(VR ). Proof. Let ϕ : Hom R (R, M) → M be the canonical isomorphism. Then, by hypothesis, for each m ∈ M there is an f m ∈ Hom R (V, M) such that m = ϕ(Hom R (i, M)( f m )) = ϕ( f m ◦ i) = f m (i(1)), so m ∈ TrV (M). Let VR be a fixed module with S = End(VR ). Let {xα }α∈A be a generating set for S V and define i : R R → VRA via i(r ) = (xα r )α∈A . Plainly, Ker(i) = Ann R (V ). Thus we have the exact sequence i

0 → Ann R (V ) → R R → VRA → (V A /i(R)) R → 0. ∗ For any M ∈ Mod–R, denote by i M the composite of the homomorphisms

Hom R (V A , M)

Hom(i,M)

→

∼ =

Hom R (R, M) → M.

∗ is epic. In particular, if VR Proposition 1.2.5. M ∈ Gen(VRA ) if and only if i M is finitely generated over its endomorphism ring (and A is taken to be finite), ∗ is epic if and only if M ∈ Gen(VR ). then i M ∗ Proof. Denote the class of M ∈ Mod-R for which i M is epic by E. We first claim that VR ∈ E and E is closed under epimorphic images. If v ∈ V , let v = kj=1 sα j xα j . Define f ∈ Hom R (V A , V ) via f ((vα )) = kj=1 sα j vα j . Then η i V∗ ( f ) = ( f ◦ i)(1) = v. For the second assertion suppose that M → L is epic ∗ ∗ is also epic. Then, since η ◦ i M = i L∗ ◦ Hom(V A , η), i L∗ in R-Mod where i M is also epic and our claim is proved. Next we note that E is closed under arbitrary direct sums and products. Plainly, E is closed under arbitrary direct products and hence under finite direct sums. Let M = ⊕β∈B Mβ , where each Mβ ∈ E, and let m ∈ M. There is a finite subset B0 of B such that if κ is the canonical inclusion M0 = ⊕β∈B0 Mβ → M, m = κ(m ), with m ∈ M0 . But then there exists f ∈ Hom R (V A , M0 ) ∗ ∗ such that m = κ(m ) = κ(i M ( f )) = i M (Hom(V A , κ)( f )). 0 A Now, Gen(V ) ⊆ E follows from what we have proved thus far. The reverse inclusion follows immediately from Lemma 1.2.4. ∗ Since the class of modules M for which i M is epic is clearly closed under direct products, Proposition 1.2.5 implies that Gen(V A ) is closed under direct products; hence, we have the following corollary by Proposition 1.2.3.

6


Corollary 1.2.6. For any VR , if V is generated over its endomorphism ring by |A| elements, then V A is finitely generated over its endomorphism ring. A module VR is small if, as is the case for a finitely generated module, Hom R (V, ⊕ A Mα ) ∼ = ⊕ A Hom R (V, Mα ), canonically, for all {Mα } A in ModR. A module VR is self-small if Hom R (V, V (A) ) ∼ = Hom R (V, V )(A) , canonically, for all sets A. This notion is a key element in the proof of the following proposition due to J. Trlifaj [78]. Proposition 1.2.7. If Hom R (V, ) commutes with direct limits (with directed index sets) of modules in Gen(VR ), then VR is finitely generated. (F) Proof. First we note that, since V (A) = lim such that F is a finite − →V subset of A [69, pp. 44–45], we have, by hypothesis, Hom R (V, V (A) ) = (F) ∼ (F) ∼ (A) Hom R (V, lim − → Hom R (V, V ) = Hom R (V, V ) ; thus, V is self− →V ) = lim small. Let V = A xα R and let ια : V → V (A) (α ∈ A) be the canonical injections. Then, identifying xα = ιa xα , since V ∼ = (⊕ A ια xα R)/K , we have a monomorphism

ϕ : V → V (A) /K with K ≤ ⊕ A xα R and ϕ : xα → xα + K . Then, letting {K i }i∈I denotes the finitely generated submodules of K , with canonical epimorphisms γi : V (A) /K i → V (A) /K (i ∈ I ), (A) (V (A) /K , {γi } I ) = lim − →V /K i .

Now, by hypothesis, (A) (Hom R (V, V (A) /K ), {Hom R (V, γi )} I ) = lim − → Hom R (V, V /K i )

so that (see [69, Theorem 2.17]) Hom R (V, V (A) /K ) = ∪ I Im Hom R (V, γi ). Thus there is an i ∈ I and a ϕi ∈ Hom R (V, V (A) /K i ) with ϕ = γi ϕi . There is a finite set F ⊆ A such that K i ⊆ V (F) , and hence V (A) /K i = V (F) /K i ⊕ V (A\F) .


7

So since V is self-small Im ϕi ⊆ V (H ) /K i for some finite set H with F ⊆ H ⊆ A. Now we have ϕ(V ) ⊆ γi (V (H ) /K i ) = (V (H ) + K )/K , so, for each α ∈ A, there is a vα ∈ V (H ) such that vα + K = ϕ(xα ) = xα + K . But then vα ∈ V (H ) ∩ (⊕ A xα R) = ⊕ H xα R, and we have Im ϕ ⊆ (⊕ H xα R + K )/K ⊆ Im ϕ; thus, V ∼ = Im ϕ is finitely generated. Another closure property of Gen(VR ) forces V to be flat over its endomorphism ring. Proposition 1.2.8. Suppose V ∈ Mod-R and S = End(VR ). If Gen(VR ) is closed under submodules, then S V is flat. Proof. Recall [1, Lemma 19.19] that S V is flat if and only if for every relation m

si xi = 0

(si ∈ S, xi ∈ V )

i=1

there exist y j ∈ V, σi j ∈ S, i ≤ m and 1 ≤ j ≤ n n

1 ≤ i ≤ m, 1 ≤ j ≤ n, such that for all 1 ≤

σi j y j = xi and

j=1

m

si σi j = 0.

i=1

So suppose we do have m

si xi = 0

(si ∈ S, xi ∈ V );

i=1

let π j : V m → V, 1 ≤ j ≤ m, be the canonical projections and let K = Ker d

8


where d is the homomorphism d : V (m) → V defined by d : z →

m

si πi (z),

z ∈ V (m) .

i=1

Then x = (x1 , . . . , xm ) ∈ K and so, since V generates K , there exist f j : V → K,

and

y j ∈ V,

1≤ j ≤n

such that x=

n

f j yj.

j=1

Now let σi j = πi f j ∈ S,

1 ≤ i ≤ m, 1 ≤ j ≤ n,

to obtain n

σi j y j = π j (x) = xi ,

1 ≤ i ≤ m,

j=1

and for each u ∈ V m i=1

si σi j u =

m

si πi f j u = d( f j u) = 0,

1 ≤ j ≤ n.

i=1

1.3. Add(VR ) and Prod(VR ) We denote the subcategories of Mod-R consisting of all direct summands of a direct sum, respectively, a direct product, of copies of a module VR by Add(V R ), respectively, by Prod(VR ). According to [1, Theorems 19.20 and 28.4], if S is a left coherent right perfect ring, then every direct product of projective right S-modules is projective, that is, belongs to Add(SS ). (This result and its converse are due to S. Chase [12], who also proved that if every direct product of copies of SS is projective, then S is a left coherent right perfect ring.) On the other hand we have Lemma 1.3.1. If S is a left coherent right perfect ring, then every projective right S-module belongs to Prod(SS ). Proof. Letting J = J (S), suppose that PS is projective and P/P J = ⊕α∈A Tα with each Tα simple. Let Q = SSA . Then Q J ≤ J A and ⊕α∈A Tα is isomorphic

1.4. Torsion Theory

9

∼ S A /J A , and so there is an epimorphism to a direct summand of (S/J ) A = Q → P/P J. Thus by [1, Lemma 17.17] P, the projective cover of P/P J , is isomorphic to a direct summand of Q. This last lemma and the paragraph preceding it tell us that, if S is a left coherent right perfect ring, then Add(SS ) = Prod(SS ). Proposition 1.3.2. Let VR be a self-small module with End(VR ) = S. If S is left coherent and right perfect, and S V is finitely presented, then Prod(VR ) = Add(VR ). Proof. Since V is self-small Hom R (V, ) : Add(VR ) Add(SS ) : ( ⊗ S V ) is an equivalence of categories. But Hom R (V, ) commutes with direct products and, by Lemma 1.2.1, so does ( ⊗ S V ). Thus the proposition follows from the fact that Add(SS ) = Prod(SS ). A ring R is an artin algebra if its center K is an artinian ring and R is finitely generated as a K -module. Any finitely generated module over an artin algebra is finitely generated over its endomorphism ring, which is also an artin algebra. Thus we have Corollary 1.3.3. If VR is a finitely generated module over an artin algebra R, then Prod(VR ) = Add(VR ). Note that we have only used one implication of Chase’s theorem. Using his full theorem, H. Krause and M. Saor´ın showed in [53] that a self-small module VR with S = End(VR ) has Add(VR ) closed under direct products if and only if S is a left coherent right perfect ring and S V is finitely presented. In view of Proposition 1.3.2 this is equivalent to Prod(VR ) = Add(VR ).

1.4. Torsion Theory Definition 1.4.1. If C is an abelian category, a torsion theory in C is a pair of classes of objects (T , F) of C such that (1) T = {T ∈ C | HomC (T, F) = 0 for all F ∈ F}, (2) F = {F ∈ C | HomC (T, F) = 0 for all T ∈ T },

10


(3) for each X ∈ C there is a subobject T of X such that T ∈ T and X/T ∈ F. When this is the case, the objects in T are called torsion objects, the elements of F are called torsion-free objects, and if the object T of (3) is unique, we denote it by τ (X ) and call it the torsion subobject of X. Suppose that C is a full subcategory of Mod-R that is closed under submodules, epimorphic images, extensions, direct sums, and direct products. If (T , F ) is a torsion theory in C, then it follows that T is closed under epimorphic images and direct sums, F is closed under submodules and direct products, and both are closed under extensions. A class T (F ) of modules in C with these closure properties is called a torsion (torsion-free) class in C. Then one easily verifies Proposition 1.4.2. Let C be a full subcategory of Mod-R that is closed under submodules, epimorphic images, extensions, direct sums, and direct products. (1) If T is a torsion class in C, then (T , F ) is a torsion theory in C, where F = {F ∈ C | HomC (T, F) = 0 for all T ∈ T }. (2) If F is a torsion-free class in C, then (T , F) is a torsion theory in C, where T = {T ∈ C | HomC (T, F) = 0 for all F ∈ F. Dual to Gen(V), if V is a class of R-modules, Cogen (V) (cogen(V)) consists of the R-modules that embed in (finite) direct products of modules isomorphic to members of V, and the reject of V in M is RejV (M), the intersection of the kernels of all maps from M into members of V. Proposition 1.4.3. Let (T , F ) be a torsion theory in Mod-R and M a module in Mod-R. Then TrT (M) = RejF (M). Proof. That TrT (M) ⊆ RejF (M) follows from Hom R (T, F) = 0 whenever T ∈ T and F ∈ F. But since TrT (M) ∈ T and T is closed under extensions, M/ TrT (M) ∈ F and hence RejF (M) ⊆ TrT (M). If (T , F ) is a torsion theory in Mod-R, we let τT (M) = TrT (M) = RejF (M)

1.4. Torsion Theory

11

and call it the torsion submodule of M. Then every module in Mod-R admits an exact sequence 0 → τT (M) −→ M −→ M/τT (M) → 0 with τT (M) simultaneously the largest submodule of M belonging to T and the smallest submodule of M such that M/τT (M) belongs to F. We shall meet torsion theories like those in the following proposition in later sections. Proposition 1.4.4. If Gen(VR ) ⊆ V ⊥ , then (Gen(VR ), Ker Hom R (V, )) is a torsion theory in Mod-R. Proof. If 0 → M1 −→ X −→ M2 → 0 is exact with M1 , M2 ∈ Gen(VR ) ⊆ V ⊥ , letting S = End(VR ), we obtain a commutative diagram with exact rows Hom R (V, M1 ) ⊗ S V →Hom R (V, X ) ⊗ S V →Hom R (V, M2 ) ⊗ S V →0 νX ↓ ν M2 ↓ ν M1 ↓ 0→ M1 → X → M2 in which the trace maps ν M1 and ν M2 are epimorphisms. But then, by the Snake Lemma, so is ν X , and Proposition 1.4.2 applies.

2 Representable Equivalences

We are concerned with equivalences and dualities between subcategories of the categories of modules over rings. Henceforth, by “subcategory” we shall mean full subcategory that is closed under isomorphic images, and all functors between categories of modules are assumed to be additive. Suppose C and D are subcategories of Mod-R and Mod-S, respectively. A functor H : C → D is an equivalence if there is a functor T : D → C such that T ◦ H and H ◦ T are naturally isomorphic to the identity functors 1C and 1D , respectively. When this is the case we write C ≈ D. By Theorem A.3.4 these natural isomorphisms can be taken to be of the form µ : T H → 1C and θ : 1D → H T where H µ ◦ θ H = 1 H and µT ◦ T θ = 1T . That is, µ and θ, an arrow of adjunction and its quasi-inverse, establish T as a left adjoint of H (see Appendix A). If S VR is a bimodule, then we have functors Hom R (V, ) : Mod -R Mod -S : ( ⊗ S V ), and we say that the equivalence H : C D : T is representable by S VR if H and T are naturally isomorphic to the restrictions of these functors, that is, H∼ = Hom R (V, ) | C and T ∼ = ( ⊗ S V ) | D. In this case we shall make the identifications H = Hom R (V, ) and T = ( ⊗ S V ), and then by Theorem A.2.2 the canonical natural transformations ν and η defined below are natural isomorphisms when restricted to C and D, respectively. 2.1. Adjointness of Hom R (V, ) and ⊗ S V If S VR is a bimodule, then for any M ∈ Mod-R and N ∈ Mod-S there is an isomorphism α = α N ,M : Hom S (N , Hom R (V, M)) → Hom R ((N ⊗ S V ), M) 12

2.1. Adjointness of Hom R (V, ) and ⊗ S V

13

with, for δ ∈ Hom S (N , Hom R (V, M)), n ∈ N , v ∈ V, and γ ∈ Hom R ((N ⊗ S V ), M), α(δ) : n ⊗ v → δ(n)(v) and a −1 (γ )(n) : v → γ (n ⊗ v). The isomorphism α is natural in both M and N , that is, α defines two natural transformations α : Hom S (N , Hom R (V, )) → Hom R ((N ⊗ S V ), ) and α : Hom S ( , Hom R (V, M)) → Hom R (( ⊗ S V ), M). In this chapter, when S VR is a given bimodule, we shall write H M = Hom R (V, M) and T N = N ⊗ S V for M ∈ Mod-R and N ∈ Mod-S. Thus the functor T : Mod-S → Mod-R is a left adjoint of the functor H : Mod-R → Mod-S. Moreover, associated with this adjunction there are the canonical natural transformations ν : T H → 1Mod -R and η : 1Mod -S → H T with, for each ϕ ∈ H M, v ∈ V and n ∈ N , ν M : ϕ ⊗ v → ϕ(v) and η N (n) : v → n ⊗ v. Direct calculations verify that H (ν M ) ◦ η H M = 1 H M and νT N ◦ T (η N ) = 1T N .

(2.1)

We shall say that M is ν-reflexive if ν M is an isomorphism and that N is η-reflexive if η N is an isomorphism. The preceding display easily yields Proposition 2.1.1. For all M ∈ Mod-R, and all N ∈ Mod-S, (1) η H M is a split monomorphism and νT N is a split epimorphism; (2) If M is ν-reflexive, then H M is η-reflexive; if N is η-reflexive, then T N is ν-reflexive. Thus H and T yield an equivalence between the category of ν-reflexive modules in Mod-R and the category of η-reflexive modules in Mod-S. Note that Im ν M = TrV (M).

14

Representable Equivalences

On the other hand, given an injective cogenerator C R , let VS∗ = Hom R ( S VR , C R ). Then, according to [1, Exercise 19.20], Ker η N = Ann N ( S V ) = RejV ∗ (N ). Thus we have Lemma 2.1.2. If M ∈ Mod-R and N ∈ Mod-S, then (1) ν M is an epimorphism if and only if M ∈ Gen(VR ); (2) η N is a monomorphism if and only if N ∈ Cogen(VS∗ ). We let Pres(VR ) denote the class of R-modules that are presented by V in the sense that there is an exact sequence V (B) −→ V (A) −→ M → 0 for some sets A and B. Dually, M ∈ Copres(W R ) if it is copresented by W in the sense that there is an exact sequence 0 → M R −→ W A −→ W B . g

Applying T and H to exact sequences S (B) −→ S (A) −→ N S → 0 and 0 → M R −→ C A −→ C B , with C R an injective cogenerator, we obtain Lemma 2.1.3. If M ∈ Mod-R and N ∈ Mod-S, then (1) T N ∈ Pres(VR ) ⊆ Gen(VR ); (2) H M ∈ Copres(VS∗ ) ⊆ Cogen(VS∗ ). Also we note that Proposition 2.1.4. The module SS is η-reflexive if and only if S ∼ = End(VR ) canonically, and if so, then VR is ν-reflexive. 2.2. Weak ∗-modules Suppose that C and D are subcategories of Mod-R and Mod-S, respectively, with an equivalence H : C D : T. If SS ∈ D, we have natural isomorphisms H( ) ∼ = Hom S (S, H ( )) ∼ = Hom R (T (S), ).

2.2. Weak ∗-modules

15

Hence, letting S VR = T ( S SS ) be an S–R bimodule in the canonical way, H is isomorphic to Hom R (V, ). Suppose, in addition, that ⊗ S V : D → C. Then by [1, Exercise 20.7], T is naturally isomorphic to ⊗ S V , and thus the equivalence H : C D : T is representable by S VR . Also note that in this setting we have H T (S) = End(VR ). Throughout the remainder of this chapter, we shall assume that we are dealing with a given bimodule S VR and that H, T, ν, η and V ∗ are as in Section 2.1. As we shall see, most of the interesting equivalences between module categories are ones between Gen(VR ) and Cogen(VS∗ ), namely, those that are induced by a so-called ∗-module. En route, here we shall investigate equivalences between Pres(VR ) and Cogen(VS∗ ). f

Lemma 2.2.1. (1) Suppose that 0 → K −→ X −→ M → 0 is exact in Mod-R, that X is ν-reflexive, and H f is epic. Then M is ν-reflexive if and only if K ∈ Gen(VR ). g (2) Suppose that 0 → N −→ Y −→ L → 0 is exact in Mod-S, that Y is η-reflexive, and that T g is monic. Then N is η-reflexive if and only if L ∈ Cogen(VS∗ ). Proof. (1) Apply the Snake Lemma to the commutative diagram T HK → T HX → T HM → 0 ↓ ↓ ↓ 0 → K → X → M → 0. (2) Apply the Snake lemma to 0→

N → Y → L →0 ↓ ↓ ↓ 0 → HT N → HTY → HT L . The next lemma provides a connection between reflexivity and exactness of H and T. f

Lemma 2.2.2. (1) Suppose 0 → K −→ X −→ M → 0 is exact in Mod-R. If X is ν-reflexive and K ∈ Gen(VR ), then H f is epic if and only if Im H f is η-reflexive. g (2) Suppose 0 → N −→ Y −→ L → 0 is exact in Mod-S. If Y is η-reflexive and L ∈ Cogen(VS∗ ), then T g is monic if and only if Im T g is ν-reflexive.

16


Proof. (1) From the exact sequence p

0 → H K −→ H M −→ Im H f → 0 and the embedding i

0 → Im H f −→ H M, we obtain a commutative diagram Tp

T H K −→ T H X −→ T (Im H f ) → 0 νX ↓ α↓ νK ↓ 0→

−→

K

f

−→

X

M

→0

where α = ν M ◦ T (i). Here, by the Five Lemma, α, and hence H (α), is an isomorphism. Also i = 1H M ◦ i = H (ν M ) ◦ η H M ◦ i = H (ν M ) ◦ H T (i) ◦ ηIm H f = H (α) ◦ ηIm H f where the third equality is due to the naturalness of η : i

→

Im H f ηIm H f ↓

HM ηH M ↓

H T (i)

H T (Im H f ) −→ H T H M. Thus H f is epic if and only if i is an isomorphism if and only if ηIm H f is an isomorphism. (2) From the exact sequence j

0 → Im T g −→ T Y −→ T L → 0 and the epimorphism q

T N −→ Im T g −→ 0 we obtain a commutative diagram 0→

N β↓

g

−→ Hj

Y → L →0 ηY ↓ ηL ↓

0 → H (I mT g) −→ H T (Y ) → H T (L)

2.2. Weak ∗-modules

17

where β = H (q) ◦ η N . Here, by the Five Lemma, β, and hence T (β), is an isomorphism. Also q = q ◦ 1T N = q ◦ νT N ◦ T (η N ) = νIm T g ◦ T H (q) ◦ T (η N ) = νIm T g ◦ T (β) where the third equality is due to the naturalness of ν. Thus Im T g is νreflexive if and only if q is an isomorphism if and only if T g is monic. Recall that M is self-small if Hom R (M, M (A) ) ∼ = Hom R (M, M)(A) , canonically, for all sets A. Lemma 2.2.3. If S = End(VR ), then VR is self-small if and only if SS(A) is η-reflexive for all sets A. Moreover, if this is the case, then V (A) is ν-reflexive. Proof. Assuming that V is self-small, S (A) ∼ = H T (S (A) ), = Hom R (V, V (A) ) ∼ and these isomorphisms compose to η S (A) . Conversely, if SS(A) is η-reflexive, then Hom R (V, V (A) ) = H T (S (A) ) ∼ = S (A) = Hom R (V, V )(A) . (A) The last statement follows, since T (S (A) ) ∼ = VR .

Now we are prepared to prove the main result of this section. Proposition 2.2.4. If S = End(VR ), the following are equivalent: (a) H : Pres(VR ) Cogen(VS∗ ) : T is an equivalence; (b) N S is η-reflexive whenever η N is a monomorphism; (c) η N is epic for all N ∈ Mod-S; f (d) VR is self-small, and for each exact sequence 0 → K −→ V (A) −→ M → 0 in Mod-R, H f is epic if K ∈ Gen(VR ). Proof. (a) ⇒ (b). This is obvious, in view of Lemma 2.1.2. (b) ⇒ (d). Since SS(A) = (H V )(A) ∈ Cogen(VS∗ ), assuming (b), Lemmas 2.1.2 and 2.2.3 imply VR is self-small and V (A) is ν-reflexive. Also, by Lemma 2.1.3, Im H f ⊆ H M ∈ Cogen(VS∗ ), and so by Lemma 2.1.2 ηIm H f is monic. Thus by (b) and Lemma 2.2.2 H f is epic if K ∈ Gen(VR ).

18


(d) ⇒ (a). As noted above, H M ∈ Cogen(VS∗ ). Since VR is self-small, if M ∈ Pres(VR ), then, assuming K is in Gen(V ) in (d), M is ν-reflexive by Lemma 2.2.1. Suppose η N is monic. Since N ∈ Pres(SS ), we have an exact Tg sequence 0 → K −→ V (A) −→ T N → 0 with K ∈ Gen(VR ). Thus, by (d), H T g is epic, and we have a commutative diagram with exact rows g

−→

S (A) η S (A) ↓

N → 0 ηN ↓

HTg

H T (S (A) ) −→ H T N → 0. So since S (A) is η-reflexive by Lemma 2.2.3, η N is an isomorphism. (b) ⇔ (c). Since Ann N ( S V ) = RejVS∗ (N ) = Ker η N , we have an exact sequence p

0 → Ann N ( S V ) → N −→ N / RejVS∗ (N ) → 0. It follows that T p is an isomorphism. Thus, from the commutative diagram N ηN ↓

p

−→ HT p

H T (N ) −→

N / RejVS∗ (N ) η N / RejV ∗ (N ) ↓ S

HT N

we see that η N / RejV ∗ (N ) is an isomorphism if and only if η N is an epimorS phism. An R-module Q is quasi-projective if, whenever Q → L → 0 is exact, so is Hom R (Q, Q) → Hom R (Q, L) → 0. Example 2.2.5. A module VR induces an equivalence H : Pres(VR ) Cogen(VS∗ ) : T if (1) VR is finitely generated and (quasi-)projective, or (2) VR is self-small and Gen(VR ) ⊆ Ker Ext1R (V, ), since in either case V satisfies (d). 2.3. ∗-modules Suppose C and D are subcategories of Mod-R and Mod-S, respectively, and that C is closed under direct sums and epimorphisms and D is closed under direct products and submodules. If I = {a ∈ S | N a = 0 for all N ∈ D}, then I is an ideal of S, such that S/I ∈ D and D is a subcategory of Mod-S/I. Thus, supposing further that there is an equivalence C ≈ D, we may assume

2.3. ∗-modules

19

that S ∈ D. Then, as discussed at the beginning Section 2.2, we may also assume that the equivalence is of the form H : C D : T induced by a bimodule S VR = T (S) with S = End(VR ) because an epimorphism S (A) → N S yields an epimorphism V (A) → N ⊗ S VR . Now Gen(VR ) ⊆ C = T (D) ⊆ Gen(VR ) and D = H (C) ⊆ Cogen(VS∗ ) by Lemma 2.1.3. But also VS∗ = Hom R (V, C) = H (T r V (C)) ∈ H (C) = D. Thus C = Gen(VR ) and D = Cogen(VS∗ ), so our equivalence is induced by a ∗-module in the sense of the following definition. Definition 2.3.1. A module VR with S = End(VR ) is a ∗-module if H : Gen(VR ) Cogen(VS∗ ) : T is an equivalence. Although the term ∗-module was coined later, the notion was introduced by C. Menini and A. Orsatti [62] as a generalization of both quasi-progenerators and tilting modules. In addition to these authors, R. Colpi [24] and J. Trlifaj [78] are responsible for most of the results in this section. From the discussion preceding Definition 2.3.1, we have Proposition 2.3.2. Let C be a subcategory of Mod-R that is closed under direct sums and epimorphic images, and let D be a subcategory of Mod-S that is closed under direct products and submodules. Let I = r S (D), that is, I = {a ∈ S | N a = 0 for all N ∈ D}. Then any equivalence C ≈ D is representable by a ∗-module VR (corresponding to S/I S under the equivalence) with S/I ∼ = End(VR ), C = Gen(VR ) and D = Cogen(VS∗ ) If VR is a ∗-module with S = End(VR ), then VR acts as a projective object in Gen(VR ) and S V is “flat” in Cogen(VS∗ ). Proposition 2.3.3. If VR is a ∗-module with S = End(VR ), then H is exact on short exact sequences in Gen(VR ), and T is exact on short exact sequences in Cogen(VS∗ ). Proof. This follows at once from Lemma 2.2.2.

20


Lemma 2.3.4. If S W R is a bimodule and Q R is injective, then for any N S , ExtiS (N , Hom R (W, Q)) ∼ = Hom R (ToriS (N , W ), Q) for all i ≥ 0. Proof. This is [11, Page 120, Proposition 5.1]. Now we have the necessary tools to give homological characterizations of Cogen(VS∗ ) for a ∗-module VR . Proposition 2.3.5. If VR is a ∗-module with S = End(VR ), then Cogen(VS∗ ) = Ker Tor1S ( , V ) = Ker Ext1S ( , V ∗ ). g

Proof. Given a right S-module L , there is an exact sequence 0 → N −→ f

S (A) −→ L → 0 in which SS = H V ∈ Cogen(VS∗ ). Thus S (A) and N belong to Cogen(VS∗ ). Now we have an exact sequence Tg

Tf

0 = Tor1S (S (A) , V ) → Tor1S (L , V ) → T N −→ T S (A) −→ T L → 0. So if L ∈ Cogen(VS∗ ), then T g is monic by Proposition 2.3.3 and hence L ∈ Ker Tor1S ( , V ). On the other hand, if Tor1S (L , V ) = 0, then T g is monic and L ∈ Cogen(VS∗ ) by (2) of Lemma 2.2.1. The second equality is by Lemma 2.3.4. This last result and Trilfaj’s Proposition 1.2.7 lead to Theorem 2.3.6. Every ∗-module VR is finitely generated. Proof. Since TornS ( , V ) commutes with direct limits for all n ≥ 0 [69, Theorem 8.13], if ({Mi }, { f i j }) is a direct system with Mi ∈ Gen(VR ), then by ∗ Proposition 2.3.5, lim − → H (Mi ) ∈ Cogen(VS ). Thus ∼ ∼ ∼ H (lim − →T H (Mi )) = H T (lim − → H (Mi )) = lim − → H (Mi ), − → Mi ) = H (lim ∼ so − lim → Mi = T (lim − → H (Mi )) ∈ Gen(VR ), and VR is finitely generated by Proposition 1.2.7. The following lemma, an early version of which can be found in [8], and its dual version have been applied in several papers on equivalence and duality.

2.3. ∗-modules

21

Lemma 2.3.7. Suppose V, M ∈ Mod-R, S = End(VR ) and X is a set of generators for Hom R (V, M) S . Then there is an epimorphism µ

V (X ) → TrV (M) such that Hom R (V, µ) is epic. Thus if K = Ker(µ), there are exact sequences 0 → Hom R (V, K ) → Hom R (V, V (X ) )

Hom R (V,µ)

→

Hom R (V, T r V (M)) → 0

and 0 → Ext1R (V, K ) → Ext1R (V, V (X ) )

Ext1R (V,µ)

→

Ext1R (V, T r V (M)).

Proof. Since X generates Hom R (V, M) S , a straightforward argument shows µ (v f ) X → f ∈X f (v f ) defines an epimorphism V (X ) → TrV (M). To see that Hom R (V, µ) is epic, let g ∈ Hom R (V, TrV (M)) ⊆ Hom R (V, M). Then g = n (X ) denote the i=1 f i si where f i ∈ X and si ∈ S. Thus, letting ι f : V → V n (X ) ι fi si v, canonical injections and defining h ∈ Hom R (V, V ) via v → i=1 we obtain Hom R (V, µ)(h) = g. Now we are able to prove the following characterization of ∗-modules that is largely due to R. Colpi. Theorem 2.3.8. The following statements about a module VR with S = End(VR ) are equivalent: (a) VR is a ∗-module; (b) ν M is monic for all M ∈ Mod-R and η N is epic for all N ∈ Mod-S; (c) Gen(VR ) = Pres(VR ), VR is finitely generated, and H is exact on short exact sequences in Gen(VR ); (d) Gen(VR ) = Pres(VR ), VR is self-small, and for each exact sef quence 0 → K −→ V (A) −→ M → 0 in Mod-R, H f is epic if K ∈ Gen(VR ). f (e) VR is self-small, and for each exact sequence 0 → K −→ V (A) −→ M → 0 in Mod-R, H f is epic if and only if K ∈ Gen(VR ). Proof. (a) ⇔ (d). This follows at once from Proposition 2.2.4 and Lemma 2.1.3. (a) ⇔ (b). By Proposition 2.2.4 we need only show that ν M is monic for all M ∈ Mod-R if ν M is an isomorphism whenever M ∈ Gen(VR ). But since H M = H (TrV (M)), H i is an isomorphism where i : T r V (M) → M is the

22


inclusion map. Thus the equivalence follows from the commutative diagram T Hi

T H (TrV (M)) −→ T H M νM ↓ νTrV (M) ↓ 0→

TrV (M)

i

−→

M.

(a) ⇒ (c). From (a) and Lemma 2.1.3, Gen(VR ) = Pres(VR ). Also VR is finitely generated by Theorem 2.3.6, and H is exact on Gen(VR ) by Lemma 2.3.3. (c) ⇒ (d). This is clear. (a) and (d) ⇒ (e). Since V (A) is reflexive by (a), Lemma 2.2.1 shows that H f epic implies K ∈ Gen(VR ). (e) ⇒ (d). Here we need only show that Gen(VR ) ⊆ Pres(VR ). But if M ∈ Gen(VR ), then by Lemma 2.3.7 there is an epimorphism f

V (H M) −→ M → 0 such that H f is epic, and so by (e) Ker f ∈ Gen(VR ) and M ∈ Pres(VR ). 2.4. Three Special Kinds of ∗-modules In this section we consider the three principal types of ∗-modules. They induce equivalences between special categories of modules and historically were the inspiration for the concept. Definition 2.4.1. A progenerator is a finitely generated projective generator. As we shall prove shortly, progenerators are the modules that induce the so-called Morita equivalences between the full categories of modules over two rings, so they are ∗-modules. According to [1, Theorem 17.18], a progenerator VR with S = End(VR ) is also a progenerator in S-Mod and is faithfully balanced in the sense that R ∼ = End( S V ), canonically. A module U is projective relative to a module M if Hom R (U, f ) is an f epimorphism whenever M −→ L → 0 is exact. Hence if U is projective relative to itself, U is a quasi-projective module. According to [1, Proposition 16.12], the class of modules that U is projective relative to is closed under submodules, epimorphic images, and finite (arbitrary) direct sums (if U is finitely generated). Definition 2.4.2. A quasi-progenerator is a finitely generated quasi-projective module that generates all its submodules.

2.4. Three Special Kinds of ∗-modules

23

Of course the notion of quasi-progenerator is a generalization of that of progenerator. A simple module is always a quasi-progenerator, but most are neither projective nor generators. Definition 2.4.3. A finitely generated module VR is a tilting module if Gen(VR ) = VR⊥ . If VR is a progenerator Gen(VR ) = Mod-R = VR⊥ , so tilting modules also generalize progenerators. In Theorem 2.4.5 we shall see that progenerators, quasi-progenerators, and tilting modules are indeed ∗-modules, and we shall provide criteria for a ∗-module to be of each of these types. First we need Lemma 2.4.4. If V is projective relative to each Mi for i ∈ I and generates all submodules of each Mi , then V generates every submodule of ⊕ I Mi . Proof. It suffices to prove that V generates every cyclic submodule of ⊕ I Mi , so we may assume that I = {1, 2}. Suppose that L ⊆ M1 ⊕ M2 with projections π1 and π2 , and let x ∈ L. Since π1 (x) ∈ π1 (L) ⊆ M1 , there is an n ∈ N and an f : V n → π1 (L) with f (y) = π1 (x) for some y ∈ V n . By hypothesis, f factors through π1 | L , that is, f = π1 ◦ g where g : V n → L. Since π1 ◦ g(y) = f (y) = π1 (x), g(y) − x ∈ L ∩ M2 ⊆ M2 , so g(y), x − g(y) ∈ TrV (L); hence, x ∈ TrV (L) too. Note that it follows from the next theorem, from [24], [30], and [25], that the class of progenerators is the intersection of the classes of quasi-progenerators and tilting modules. Theorem 2.4.5. Let VR be an R-module. Then (1) VR is a progenerator if and only if VR is a ∗-module with Gen(VR ) = Mod-R; (2) VR is a quasi-progenerator if and only if VR is a ∗-module such that Gen(VR ) is closed under submodules; (3) VR is a tilting module if and only if V R is a ∗-module with E(R R ) ∈ Gen(VR ). Proof. (1) If VR is a progenerator, then Gen(VR ) = Mod-R and VR clearly satisfies condition (d) of Theorem 2.3.8. Conversely, a ∗-module VR is finitely generated by Theorem 2.3.6 and is projective when Gen(VR ) = Mod-R by Proposition 2.3.3.

24

Representable Equivalences f

(2) Suppose that 0 → K −→ V (A) −→ M → 0 is exact. If VR is a quasi-progenerator, then according to Lemma 2.4.4, Gen(VR ) is closed under submodules, so Gen(VR ) = Pres(VR ). Since V is quasi-projective and finitely generated, according to [1, Proposition 16.12] H f is epic. Thus by Theorem 2.3.8 (d), VR , is a ∗-module. Conversely, if VR is a ∗-module, K ∈ Gen(VR ) implies H f epic by Theorem 2.3.8, so VR is quasi-projective and generates its submodules when Gen(VR ) is closed under submodules. (3) If VR is a tilting module, then every injective R-module belongs to Gen(VR ). Since V (A) ∈ Gen(VR ) = V ⊥ , an exact sequence f

0 → K −→ V (A) −→ M → 0 yields an exact sequence Hf

H (V (A) ) −→ H M → Ext1R (V, K ) → 0. Thus H f is epic if and only if K ∈ V ⊥ = Gen(VR ), so Theorem 2.3.8(e) shows that VR is a ∗-module. Let VR be a ∗-module with E(R) ∈ Gen(VR ). One easily checks that E(R R ) generates every injective module. From the exact sequence f

0 → M −→ E(M) −→ E(M)/M → 0, in which both E(M) and E(M)/M belong to Gen(VR ), we obtain the exact sequence Hf

0 → H M −→ H (E(M)) −→ H (E(M)/M) → Ext1R (V, M) → 0. If M ∈ Gen(VR ), then by Proposition 2.3.3 H f is epic, and so M ∈ V ⊥ . If M ∈ V ⊥ , then H f is epic, so M ∈ Gen(VR ) by Proposition 2.2.1(1). Now, after noting that by Proposition 2.3.2 any equivalence Mod-R ≈ Mod-S is induced by a ∗-module, we have K. Morita’s seminal theorem on equivalence from 1958 [65]. Theorem 2.4.6 (Morita). A module VR with S = End(VR ) is a progenerator if and only if H : Mod-R Mod-S : T is an equivalence. Proof. If VR is a progenerator, it is a ∗-module with Gen(VR ) = Mod-R by Theorem 2.4.5(1), and since S V is projective, Cogen(VS∗ ) = Mod-S by Proposition 2.3.5. Conversely, if H : Mod-R Mod-S : T is an equivalence, then,


25

according to Lemma 2.1.3, Gen(VR ) = Mod-R and Cogen(VR∗ ) = Mod-S, and VR is a ∗-module. Thus Theorem 2.4.5(1) applies. Morita’s theorems on equivalence are discussed in detail in [1, Chapter 6]. In particular, if VR is a progenerator with S = End(VR ), then Hom S (V, ) : S -Mod R-Mod : (V ⊗ R ) is a category equivalence. Since, according to Lemma 2.4.4, Gen(VR ) is closed under submodules when V is a quasi-progenerator, it follows that (as in the case of Morita equivalence) the modules corresponding under the equivalence in the next theorem have isomorphic lattices of submodules. Theorem 2.4.7. A module VR with S = End(VR ) is a quasi-progenerator if and only if H : Gen(VR ) Mod-S : T is an equivalence. Proof. Assume H : Gen(VR ) Mod-S : T is an equivalence. Then VR is a ∗-module by Definition 2.3.1 and Lemma 2.1.3. Hence T is an exact functor by Proposition 2.3.3. We claim that Gen(VR ) is closed under submodules. f For, suppose that 0 −→ K −→ M −→ L −→ 0 is exact in Mod-R with M (and hence also L) in Gen(VR ). Applying H and letting Z = CokerH ( f ), we obtain exact sequences Hf

0 → HK → HM → Z → 0 and

T (ι)

0 → T Z → T HL

where ι : Z → H L is the inclusion. Hence ν L ◦ T (ι) is monic by Theorem 2.3.8(b). Thus we obtain the commutative diagram with exact rows 0 → T HK → T HM → T Z →0 ↓ νM ↓ ν L ◦T (ι) ↓ νK 0→ K → M → L →0 and conclude that ν K is epic by the Five Lemma. Hence K ∈ Gen(VR ) as claimed. Thus according to Theorem 2.4.5(2), VR is a quasi-progenerator. Conversely, if VR is a quasi-progenerator, then VR is a ∗-module with Gen(VR ) closed under submodules by Theorem 2.4.5(2), and by Proposition 1.2.8, S V is flat. Hence Mod-S = Cogen(VS∗ ) by Proposition 2.3.5. If V is a right R module and I = r R (V ), the right annihilator of V in R, then we may identify Mod-R/I with {M ∈ Mod-R | M I = 0}. Thus if S = End(VR ) and S VR/I induces an equivalence between subcategories C

26


and D of Mod-R/I and Mod-S, respectively, then so does S VR viewing C as a subcategory of Mod-R. Corollary 2.4.8. If R is a right artinian ring, then V R is a quasi-progenerator if and only if V is a progenerator in Mod-R/r R (V ). Proof. Since R is right artinian, any quasi-projective R-module is projective modulo its annihilator [1, Exercise 17.16 (2)], and R/r R (V ) embeds a finite direct sum of copies of any R-module V. Thus if VR is a quasi-progenerator, then VR/r R (V ) is a progenerator. Conversely, if VR/r R (V ) is a progenerator, then Gen(VR ) = Mod-R/r R (V ); thus, Theorem 2.4.7 applies. Jacobson’s density theorem [1, Theorem 14.4] suggests another method of obtaining examples of quasi-progenerators, as the next two results indicate. We say that a ring B is a ring extension of a ring R if there is a ring homomorphism ψ : R → B so that a B-module M B becomes an R-module M R with scalar multiplication mr = mψ(r ). Then, given VB , we say that R (or ψ(R)) is V -dense in B if, for each b ∈ B and each finite subset {v1 , . . . , vn } ⊆ V , there is an r ∈ R such that vi b = vi r for i = 1, . . . , n. Proposition 2.4.9. Let VR be a quasi-progenerator with S = End(VR ) and B = BiEnd(VR ) = End( S V ). Then VB is a quasi-progenerator and R is V -dense in B. Proof. Since VR is a quasi-progenerator, Gen(VR ) is closed under submodules. Thus, if X R ≤ V n , then X = TrV (X ), and if f : V → X, then there are s1 , . . . , sn ∈ S such that f (v) = (s1 v, . . . , sn v) for all v ∈ V. But then if b ∈ B, we have f (v)b = (s1 (vb), . . . , sn (vb)) = f (vb) ∈ X. Thus X B = X and Hom B (V, X ) = Hom R (V, X ). In particular, (v1 , . . . , vn )B = (v1 , . . . , vn )R B = (v1 , . . . , vn )R so that R is V -dense in B. Now if K ≤ VR and g : V → V /K is an Rhomomorphism, let g(v) = v + K . Then for each b ∈ B there is an r ∈ R such that vb = vr and v b = v r, and recalling that K ≤ VB , we have g(vb) = g(vr ) = g(v)r = v b + K = g(v)b. Thus it follows that VB is also a quasi-progenerator. Proposition 2.4.10. If VB is a quasi-progenerator with S = End(VB ) and R is V -dense in B, then VR is a quasi-progenerator with S = End(VR ).


27

Proof. Clearly density implies that R-submodules and B-submodules of V are one-and-the-same, and we have just shown in Proposition 2.4.9 that Hom R (V, V /K ) = Hom B (V, V /K ) for all K ≤ V ; thus, VR is quasiprojective. Similarly, Hom R (V, K ) = Hom B (V, K ), so VR generates its submodules. Example 2.4.11. Let S be an arbitrary ring, let B = RFMN (S), the ring of N × N row finite matrices over S, and let be the set of matrices in B with only finitely many nonzero entries. If e = e11 denotes the first diagonal idempotent, then eB is a (projective) quasi-progenerqator, and if R is any subring of B containing , then R is eB-dense in B, and so eB = e R is a quasi-progenerator over R with End(eB) ∼ = S and B = BiEnd(e R). In subsequent chapters we shall discuss tilting modules in detail. However, at this point we can employ part (3) of Theorem 2.4.5 to show that tilting modules and ∗-modules are closely related when certain finiteness conditions are satisfied by R. Corollary 2.4.12. If R R is finitely cogenerated (e.g., if R is right artinian), then any faithful ∗-module V is a tilting module. Proof. By hypothesis, R R → V n for some positive integer n, so V n , and hence V, generates every injective right R-module. If S = End(VR ) and S VR/I induces an equivalence between subcategories C and D of Mod-R/I and Mod-S, respectively, such that C is closed under direct sums and epimorphisms and D is closed under direct products and submodules, VR must be a ∗-module. On the other hand, M I = 0 for all M ∈ Gen(VR ). Thus we have Corollary 2.4.13. If R is a right artinian ring, then VR is a ∗-module if and only if V is a tilting module in Mod-R/r R (V ). R. Colpi and G. D’Este [27] have presented examples of faithful ∗-modules that are neither quasi-progenerators nor tilting modules.

3 Tilting Modules

Tilting modules were originally introduced in [8], [9], and [46] as a tool in the representation (or module) theory of finite dimensional algebras. They, and more general versions of tilting modules, have found many important applications to this theory. Our concern here is their application to the module theory of more general rings.

3.1. Generalized Tilting Modules We begin with a generalization of the notion of tilting modules that was introduced by R. Colpi and J. Trlifaj in [32]. Definition 3.1.1. A module VR is a generalized tilting module if Gen(VR ) = V ⊥. Important properties of tilting modules that hold in this more general setting are recorded in the next two propositions. Proposition 3.1.2. If VR is a generalized tilting module and S = End(VR ), then VR is faithful and S V is finitely generated. Proof. VR is faithful since E(R R ) ∈ Gen(VR ), and S V is finitely generated by Lemma 1.2.3 since V ⊥ is closed under direct products. Proposition 3.1.3. If VR is a generalized tilting module, then Gen(VR ) = Pres(VR ). Proof. Let M R ∈ Gen(VR ) (so M = TrV (M)) and apply Lemma 2.3.7. These propositions lead to the following characterizations of generalized tilting modules. 28

3.1. Generalized Tilting Modules

29

Theorem 3.1.4. A module V ∈ Mod-R is a generalized tilting module if and only if (i) proj . dim .(VR ) ≤ 1; (ii) Ext1R (V, V (A) ) = 0 for all sets A; (iii) there is an exact sequence 0 → R R → V0 → V1 → 0 where V0 , V1 ∈ Add(VR ). Proof. Assume that VR is a generalized tilting module. (i) follows from Proposition 1.1.1. (ii) is obvious from the definition. (iii) By Proposition 3.1.2 we have the exact sequence f

0 → R R → V n → L → 0, and by Proposition 3.1.3, Gen(VR ) = Pres(VR ); so there is an exact sequence g

0 → K → V (A) → L → 0 where K ∈ Gen(VR ) = V ⊥ . Now consider the pullback diagram 0 0 ↓ ↓ K = K ↓ ↓ 0→R→ X −→V (A) →0 ↓ g↓ f

0→R→V n −→ L →0 ↓ ↓ 0 0 Since K , V n ∈ V ⊥ , we have X ∈ V ⊥ = Pres(VR ), and an exact sequence 0 → N −→ V (B) −→ X → 0 with N ∈ V ⊥ . But this sequence splits since 0 = Ext1R (V (A) , N ) → Ext1R (X, N ) → Ext1R (R, N ) = 0 is exact. Thus X ∈ Add(VR ) and 0 →R→X →V (A) →0 verifies (iii).

30

Tilting Modules

Conversely, assume (i), (ii), and (iii). If M ∈ Gen(VR ), we have an exact sequence 0 → K → V (A) → M → 0, which in turn induces the exact sequence 0 = Ext1R (V, V (A) ) → Ext1R (V, M) → Ext2R (V, K ) = 0, and so we have Gen(VR ) ⊆ V ⊥ . (Note that this inclusion is implied by conditions (i) and (ii).) If M ∈ V ⊥ , then the exact sequence of (iii) induces the commutative diagram α Hom R (V0 , M) → Hom R (R, M) ∼ M = ↑∼ ↑β = Hom R (V0 , TrV0 (M)) → Hom R (R, TrV0 (M)) ∼ = TrV0 (M)

where β is monic and, since V1 ∈ Add(V ), α is epic. It follows that β is an isomorphism, and hence TrV0 (M) = M. Since V0 ∈ Add(V ), TrV (M) = M as well. Proposition 3.1.5. A module V ∈ Mod-R is a generalized tilting module if and only if (i) proj . dim .(VR ) ≤ 1; (ii) Ext1R (V, V (A) ) = 0 for all sets A; (iii) Ker(Hom R (V, )) ∩ V ⊥ = 0. Proof. If VR is generalized tilting and Hom R (V, M) = 0 = Ext1R (V, M), then M ∈ V ⊥ = Gen(VR ); therefore M = 0, proving (iii). (i) and (ii) follow from Proposition 3.1.4. Conversely, assume that (i), (ii), and (iii) hold. As noted in the proof of Proposition 3.1.4, (i) and (ii) imply that Gen(VR ) ⊆ V ⊥ . So assume M ∈ V ⊥ . Since Gen(VR ) ⊆ V ⊥ , TrV (M) ∈ V ⊥ . Hence, the exact sequence 0 → β Tr(M) → M → M/ Tr(M) → 0, where β is the inclusion, induces an exact sequence Hom(V,β)

0 → Hom R (V, TrV (M)) −→ Hom R (V, M) → Hom R (V, M/TrV (M)) → 0 where Hom(V, β) is epic by definition of trace, and so Hom R (V, M/ Tr(M)) = 0. But we also have the exact sequence 0 = Ext1R (V, M) → Ext1R (V, M/ TrV (M)) → 0; thus M/ Tr(M) = 0 by (iii), and we conclude that M ∈ Gen(VR ).

3.2. Tilting Modules

31

We conclude this section by showing that a generalized tilting module is a tilting module if it is self-small. Corollary 3.1.6. If V ∈ Mod-R is a generalized tilting module, then VR is a ∗-module if and only if it is self-small. Proof. The corollary follows from Proposition 3.1.5(ii) and Theorem 2.3.8(e). Corollary 3.1.7. If V ∈ Mod-R is a self-small generalized tilting module, then VR is a tilting module. Proof. A self-small generalized tilting module is finitely generated by Corollary 3.1.6 and Theorem 2.3.6.

3.2. Tilting Modules The results of the preceding section yield the basic characterizations of tilting modules in the following theorem due to R. Colpi [25]. There condition (b) is the classical definition of a tilting module. Theorem 3.2.1. The following are equivalent for a module V ∈ Mod-R: (a) VR is a tilting module; (b) VR is finitely presented and (i) proj . dim .(VR ) ≤ 1, (ii) Ext1R (V, V ) = 0, (iii) there is an exact sequence 0 → R R → V0 → V1 → 0 where V0 , V1 ∈ add(VR ); (c) VR is finitely presented and (i) proj . dim .(VR ) ≤ 1, (ii) Ext1R (V, V ) = 0, (iii) Ker(Hom R (V, )) ∩ V ⊥ = 0. Proof. (a) ⇒ (b). Assuming (a), VR is finitely presented by Proposition 1.1.3. Since VR is a generalized tilting module, (b)(i) and (b)(ii) are immediate from Theorem 3.1.4. It remains to prove (b)(iii). By Proposition 3.1.2 we have, as in Proposition 1.2.5, the exact sequence 0 → R R → V n → V n /R → 0,

32

Tilting Modules

and by Proposition 3.1.3 there is an exact sequence 0 → K → V (X ) → V n /R → 0 where K ∈ Gen(VR ). Thus we obtain the exact sequence i K∗

Hom R (V n , K ) → K → Ext1R (V n /R, K ) → Ext1R (V n , K ) = 0 where i K∗ is epic by Proposition 1.2.5; hence, we see by our hypothesis that Ext1R (V n /R, K ) = 0, and so we can conclude that, being finitely generated, V n /R ∈ add(VR ). (b) ⇒ (c). Since VR is finitely presented, Ext1R (V, ) commutes with direct sums by Proposition 1.1.2; thus, VR is a generalized tilting module by Proposition 3.1.4. Hence, (c)(iii) follows from Proposition 3.1.5. (c) ⇒ (a). As in the proof of (b) ⇒ (c), VR is a generalized tilting module by Proposition 3.1.5. Like Morita equivalence, tilting is a two-sided concept. Indeed, like progenerators, tilting modules are faithfully balanced and are also tilting modules over their endomorphism rings. Proposition 3.2.2. If VR is a tilting module with S = End(VR ), then S V is a tilting module with R ∼ = End( S V ), canonically. Proof. Let 0 → P1 −→ P0 −→ V → 0

(3.1)

be exact with the Pi finitely generated projective right R-modules, and let 0 → R R −→ V0 −→ V1 → 0

(3.2)

be exact with the Vi ∈ add(VR ). Since Ext1R (V, V ) = 0 0 → Hom R (V, V ) −→ Hom R (P0 , V ) −→ Hom R (P1 , V ) → 0 is exact, and so here we have an exact sequence in S-Mod 0 → S S −→ V0 −→ V1 → 0

(3.3)

with the Vi ∈ add( S V ). Similarly, applying Hom R ( , V ) to the exact sequence (3.2), we obtain an exact sequence 0 → P1 −→ P0 −→ S V → 0

(3.4)

3.2. Tilting Modules

33

with the Pi finitely generated projective left S-modules. Since S = End(VR ), the evaluation map δV : VR → Hom S (Hom R (V, V ), V ) is an isomorphism, that is, VR is S VR -reflexive. But then, according to [1, Proposition 20.13], so are V0 and V1 . Thus, applying Hom S ( , V ) to the exact sequence (3.4), we obtain a commutative diagram → V1 → 0 R → V0 δV0 ↓ δV1 ↓ δR ↓ 0 → Hom S (V, V ) → Hom S (P0 , V ) → Hom S (P1 , V ) → Ext1S (V, V ) → 0

0 →

with exact rows, in which δV0 and δV1 are isomorphisms. Now it follows that Ext1S (V, V ) = 0, and then that R ∼ = End( S V ), canonically. In view of this last result, if VR is a tilting module with S = End(VR ) we shall, on occasion, say that S VR is a tilting bimodule. Note that a faithfully balanced bimodule S VR that is finitely presented and satisfies conditions (i) and (ii) of Theorem 3.2.1 in both Mod-R and S-Mod is a tilting bimodule. If K is a commutative artinian ring with radical J (K ), then (see [1, Proposition 30.6]) Hom K ( , E(K /J (K )) induces a duality on the finitely generated K -modules. A ring R is an artin algebra if its center K is an artinian ring and R is finitely generated as a K -module; then D = Hom K ( , E(K /J (K ))) defines a duality D : mod -R R- mod : D called the artin algebra duality. Note, in particular, that, in this case D( R R R ) is a finitely generated two-sided injective cogenerator. For an artin algebra R, one can restrict the definition of tilting modules to mod-R. Proposition 3.2.3. If R is an artin algebra, then a module VR is a tilting module if and only if gen(VR ) = V ⊥ ∩ mod-R. Proof. One implication is obvious. So suppose that gen(VR ) = V ⊥ ∩ mod-R. We shall verify the conditions of Theorem 3.2.1(b). Clearly VR is finitely presented and Ext1R (V, V ) = 0. If M ∈ mod-R, then so is each term in the exact sequence 0 → M −→ E(M) −→ L → 0. Thus we have an exact sequence 0 = Ext1R (V, L) → Ext2R (V, M) → Ext2R (V, E(M)) = 0, and it follows that, since R is artinian, proj . dim .(VR ) ≤ 1. Finally, appealing directly to Lemma 2.3.7 (with X finite), rather than to Lemma 3.1.3, one can

34

Tilting Modules

verify condition (iii) of Theorem 3.2.1(b) by imitating the proof that (a) implies (b) (iii) of Theorem 3.2.1. 3.3. Tilting Torsion Theories If VR is a tilting module, then it follows at once from Definition 2.4.3 that Gen(VR ) is a torsion class with torsion-free class Ker Hom R (V, ). Thus it is of interest to determine which torsion theories so arise from a tilting module. Here we present an extension by Colpi and Trlifaj [32] of a characterization due to I. Assem [4] and S. Smalφ [71]. Definition 3.3.1. A torsion theory (T , F ) (torsion class T ) in Mod-R is called a tilting torsion theory (class) if there is a tilting module VR in Mod-R such that T = Gen(VR ). For arbitrary rings we have the following necessary and sufficient conditions for a finitely generated module U R with Gen(U R ) ⊆ U ⊥ (as in Proposition 1.4.4) to generate a tilting torsion class. Proposition 3.3.2. Suppose U R is a finitely generated module with Gen(U R ) ⊆ U R⊥ . Then Gen(U R ) is a tilting torsion class in Mod-R if and only if U R is faithful and finitely generated over End(U R ). In this case U R is a tilting module if and only if Gen(U R ) = Pres(U R ). Proof. Suppose Gen(U R ) = Gen(VR ) = V ⊥ where VR is a tilting module. Since VR is faithful and VR ∈ Gen(U R ), it is clear that U R is faithful. Also, since Gen(VR ) = Gen(U R ) is closed under direct products by Proposition 3.2.2 and Proposition 1.2.3, End(U R ) U is finitely generated by Proposition 1.2.3. Conversely, assume that U R is faithful, is finitely generated over both R and End(U R ) and that Gen(U R ) ⊆ U ⊥ . Referring to the discussion on page 5, assuming that End(U R ) U is generated by n elements, there is an exact sequence 0 → R → Un → U → 0 in mod-R. Then, for any M in Mod-R we have an exact sequence ∗ iM

Hom R (U , M) → Hom R (U n , M) → M → Ext1R (U , M) → Ext1R (U n , M) ∗ where i M is epic if and only if M ∈ Gen(U R ) by Proposition 1.2.5. Since Gen(U R ) ⊆ U ⊥ , it follows easily that Gen(U R ) = U ⊥ . Let VR = U R ⊕ U R . Then VR is finitely generated and Gen(VR ) = Gen(U R ) since U R is an

3.3. Tilting Torsion Theories

35

epimorphic image of U Rn . Thus we have ⊥

V ⊥ = U ⊥ ∩ U = U ⊥ ∩ Gen(U R ) = Gen(U R ) = Gen(VR ), so VR is a tilting module. Now assume in addition that Gen(U R ) = Pres(U R ). As above there is a tilting module VR such that Gen(U R ) = Gen(VR ) = V ⊥ . Also there is a, necessarily split, exact sequence 0 → K −→ U (A) −→ V → 0 with K ∈ Gen(U R ), so VR ∈ add(U R ). Therefore, U ⊥ ⊆ V ⊥ = Gen(U R ). For artinian rings, tilting torsion classes are characterized as follows. Theorem 3.3.3. Suppose T is a torsion class in Mod-R and R is right artinian. The following are equivalent: (a) T is a tilting torsion class; (b) T is closed under direct products and contains every injective module in Mod-R, and T = Gen(U R ) for some finitely generated module U R ; (c) T = Gen(U R ) for some faithful module U R , which is finitely generated both in Mod-R and in End(U R )-Mod. Proof. (a) ⇒ (b). If T = Gen(VR ) where VR is a tilting module in Mod-R then, since T = VR⊥ , T contains all injectives and is closed under direct products. (b) ⇒ (c). Assume (b). Since Gen(U R ) is closed under products, End(U R ) U is finitely generated by Proposition 1.2.3, and since U R generates E(R R ) ∈ T , U R is faithful. (c) ⇒ (a). Assuming (c), first note that, if W R is a direct summand of U R such that Gen(W R ) = Gen(U R ), then W R is faithful since U R ∈ Gen(W R ) and is finitely generated over its endomorphism ring by Proposition 1.2.3. Hence, if in addition Gen(W R ) ⊆ W R⊥ , (a) will follow from Proposition 3.3.2. Since n R is right artinian, U R = ⊕i=1 Ui where each Ui is indecomposable. From the n set {Ui }i=1 choose a subset that is minimal with respect to the property that the direct sum of its members generates Gen(U R ). Renumbering the Ui , we can k k where k ≤ n and let W R = ⊕i=1 Ui . To comassume that this set is {Ui }i=1 ⊥ plete the proof it suffices to show that Gen(W R ) ⊆ W R = U1⊥ ∩ · · · ∩ Uk⊥ .

36

Tilting Modules

Let 1 ≤ m ≤ k and suppose Ext1R (Um , M) "= 0 where M ∈ Gen(W R ). Then there is an exact sequence 0 → M → X → Um → 0 that is not split exact and X ∈ Gen(W R ) since Gen(W R ) = T is a torsion class. Hence, there is an epimorphism π : W R(J ) → X for some set J and, noting that any homomor(J ) k phism W R(J ) = Um(J ) ⊕ Y → Um where Y = ⊕i=1,i" can be written in =m Ui the form ⊕ j∈J p j ⊕ q where each p j ∈ End((Um ) R ) and q ∈ Hom R (Y, Um ), we have a commutative diagram with exact rows W (J ) = Um(J ) ⊕ Y ↓π ↓ ⊕ pj ⊕ q j∈J

0→M→

X

f

−→

Um

→ 0.

Clearly, since the given short exact sequence does not split, no p j is an isomorphism. Hence each p j ∈ Rad(End((Um ) R )) as End((Um ) R ) is a local ring by [1, Lemma 12.8]. Thus Um = f ◦ π W R(J ) = p j (Um ) + q(Y ) ⊆ Rad(End((Um ) R ))Um + q(Y ), j∈J

so q(Y ) = Um since Rad((Um ) R ))Um is superfluous in Um . But then we must k have Um ∈ Gen(⊕i=1,i" =m Ui ) contrary to the choice of W R . 3.4. Partial Tilting Modules In Theorem 3.2.1, part (iii) of each characteristic condition (b) and (c) of a tilting module is generally the most difficult to verify. Thus one is led to the following notion. Definition 3.4.1. A module U R is a partial tilting module in case U R is finitely presented and (i) proj . dim .(U R ) ≤ 1; (ii) Ext1R (U, U ) = 0. These modules are characterized in [32] as follows. Proposition 3.4.2. A finitely generated module U R is a partial tilting module if and only if Gen(U R ) ⊆ U ⊥ and U ⊥ is a torsion class. Proof. According to Proposition 1.1.1, proj . dim . U ≤ 1 if and only if U ⊥ is closed under epimorphic images. Moreover, U ⊥ is always closed under extensions.

3.4. Partial Tilting Modules

37

(⇒) Since U is finitely presented, U ⊥ is closed under direct sums by Proposition 1.1.2. Thus U ⊥ is a torsion class whenever U is a partial tilting module. Now, if 0 → K → U (A) → M → 0 is exact, we see from the exactness of Ext1R (U, U (A) ) → Ext1R (U, M) → Ext2R (U, K ) that Gen(U R ) ⊆ U ⊥ . (⇐) Since U ⊥ is a torsion class, U R is finitely presented by Proposition 1.1.3, and Ext1R (U, U ) = 0, since U R ∈ Gen(U R ) ⊆ U ⊥ . Analogous to generalized tilting modules, one has the following: Definition 3.4.3. A module U R is a generalized partial tilting module if Gen(U R ) ⊆ U ⊥ and U ⊥ is a torsion class. We want to show that every generalized partial tilting module is a direct summand of a generalized tilting module. We shall employ the following lemma to do so. Lemma 3.4.4. Given M, U ∈ Mod-R, with S = End(U R ), let {εi } I generate the right S-module Ext1R (U, M). Then there is an exact sequence 0 → M −→ X −→ U (I ) → 0 such that the connecting homomorphism ∂ : Hom R (U, U (I ) ) → Ext1R (U, M) is an epimorphism. Proof. Let fi

gi

εi : 0 → M −→ X i −→ U → 0 be extensions of M by U corresponding to the εi , and let σ : M (I ) → M with σ : (m i ) I → I m i . We shall show that the bottom row in the commutative diagram ⊕ fi

⊕gi

0 → M (I ) −→ ⊕ I X i −→ U (I ) → 0 σ ↓ h↓ f

0 → M −→

X

g

−→ U (I ) → 0,

38

Tilting Modules

in which X is the pushout of ⊕ f i and σ, satisfies the condition. Indeed, letting ιi : U → U (I ) and ιi : X i → ⊕ I X i denote the canonical injections, we see that the diagram fi

gi

0 → M −→ X i −→ U → 0 hιi ↓ ιi ↓ f

0 → M −→

X

g

−→ U (I ) → 0

commutes. Thus, according to [56], Theorem 3.4, page 74, and Lemma 1.2, page 65, ∂(ιi ) = εi for all i ∈ I. It also follows from [56], Lemma 1.2, page 65, that ∂ is an S-homomorphism, and so the lemma is proved. As we shall see in Section 3.7, the next results will allow us to circumvent part (iii) in certain cases. The first of these is from [32]. Proposition 3.4.5. Every generalized partial tilting module is a direct summand of a generalized tilting module. Proof. Let {εi } I be a set of S = End(U R )-generators for Ext1R (U, R). According to Lemma 3.4.4, there is an exact sequence i

0 → R R −→ X −→ U (I ) → 0

(3.5)

such that the connecting homomorphism ∂ : Hom R (U, U (I ) ) → Ext1R (U, R) is an epimorphism. We shall show that V =U⊕X is a generalized tilting module. From the sequence (3.5) we note that U ∈ Gen(X R ), so that Gen(VR ) = Gen(X R ), and we obtain an exact sequence ∂

Hom R (U, U (I ) ) −→ Ext1R (U, R) −→ Ext1R (U, X ) → 0 = Ext1R (U, U (I ) ). Thus, since ∂ is epic, we have X ∈ U ⊥, and since U ⊥ is a torsion class, Gen(VR ) = Gen(X R ) ⊆ U ⊥ .

(3.6)

3.4. Partial Tilting Modules

39

If M ∈ U ⊥ , then, applying Hom R ( , M) to the sequence (3.5), we obtain the exact sequence Hom R (X, M)

Hom R (i,M)

−→ Hom R (R, M) → Ext1R (U (I ) , M) → Ext1R (X, M) → 0 = Ext1R (R, M).

Thus, since Ext1R (U (I ) , M) ∼ = Ext1R (U, M) I = 0, Hom R (i, M) is epic, so M ∈ Gen(X R ) by Lemma 1.2.4, and M ∈ X ⊥ . Therefore U ⊥ ⊆ Gen(X R ) ∩ X ⊥ = Gen(VR ) ∩ X ⊥ . Now we have V ⊥ = U ⊥ ∩ X ⊥ = U ⊥ and U ⊥ ⊆ Gen(VR ) ⊆ U ⊥ . Thus we have Gen(VR ) = V ⊥ . If R is an artin algebra, partial tilting modules are direct summands of tilting modules, as K. Bongartz proved in [8]. It is worth noting here that it follows easily from part (b) of Theorem 3.2.1 that whenever VR is a tilting module over an arbitrary ring, Ext1R (V, R) is finitely generated in Mod-End(VR ). Corollary 3.4.6. If U R is a partial tilting module such that Ext1R (U, R) is finitely generated over S = End(U R ) (for example, if R is an artin algebra), then U R is a direct summand of a tilting module.

Proof. In this case we may replace I by a positive integer n in the proof of Proposition 3.4.5. Then, since Ext1R (U n , U (A) ) = 0 and, by equation (3.6), Ext1R (U n , X (A) ) = 0, we have commutative diagrams with exact rows 0 → Hom R (U n , U )(A) −→ Hom R (X, U )(A) −→ Hom R (R, U )(A) → 0 ∼ ∼ ↓ =↓ =↓ n (A) (A) 0 → Hom R (U , U ) −→ Hom R (X, U ) −→ Hom R (R, U (A) ) → 0 and 0 → Hom R (U n , X )(A) −→ Hom R (X, X )(A) −→ Hom R (R, X )(A) → 0 ∼ ∼ ↓ =↓ =↓ n (A) (A) 0 → Hom R (U , X ) −→ Hom R (X, X ) −→ Hom R (R, X (A) ) → 0 from which it follows that VR = U ⊕ X is self-small, so Proposition 3.1.7 applies. Of course, if R is an artin algebra, then VR is automatically finitely presented.

40

Tilting Modules

3.5. The Tilting Theorem The raison d’être of a tilting module VR is that it (though not as thoroughly as a progenerator) provides a connection between the categories of modules over R and S = End(VR ). This connection is described in the Tilting Theorem as presented in [18], earlier versions of which appeared in [9], [8], [47], [64]. Theorem 3.5.1 (Tilting Theorem). Suppose that VR is a tilting module in Mod-R with S = End(VR ). Let H = Hom R (V, ), H = Ext1R (V, ), T = ( ⊗ S V ), T = Tor1S ( , V ) to obtain pairs of functors H : Mod-R Mod -S : T and H : Mod-R Mod-S : T and let T = Ker H , F = Ker H, S = Ker T, E = Ker T . Then (1) (T , F ) and (S, E) are torsion theories in Mod-R and Mod-S, respectively; (2) T H = 0Mod -R = T H and H T = 0Mod-S = H T ; (3) There are natural transformations θ and ϕ that, together with the canonical natural transformations ν and η, yield exact sequences νM

θM

0 → T H M −→ M −→ T H M → 0 and ϕN

ηN

0 → H T N −→ N −→ H T N → 0 for each M ∈ Mod-R and each N ∈ Mod-S; (4) The restrictions H : T E : T and H : F S : T define category equivalences. Proof. (1) By Definition 2.4.3, Gen(VR ) = VR⊥ = T , and so it follows at once that, since VR⊥ is closed under extensions, T = Gen(VR ) is a torsion class. But then, according to Proposition 1.4.2 the corresponding torsion-free class must be Ker H = F. According to Theorem 2.4.5, VR is a ∗-module. Thus,

3.5. The Tilting Theorem

41

by Proposition 2.3.5, Cogen(VS∗ ) = Ker T = E, so E is a torsion-free class, and since, by adjointness, Hom S (N , VS∗ ) ∼ = Hom R ((N ⊗ S V ), C) with C R an injective cogenerator, the corresponding torsion class is Ker( ⊗ S V ) = S. Thus we see that (T , F) and (S, E) are torsion theories in Mod-R and Mod-S, respectively. (2) Since, by Lemma 2.1.3, H : Mod-R → Cogen(VS∗ ) = Ker T ,we have T H = 0Mod-R . Also H T = 0Mod-S , since T : Mod-S → Gen(VR ) = Ker H . Since T contains all injective modules in Mod-R, if M ∈ Mod-R there is an exact sequence 0 → M −→ E −→ D → 0 with E injective and E, D ∈ T . Thus, since VR is a ∗-module, we obtain a commutative diagram E −→ D −→ 0 νE ↑ νD ↑ T H E −→ T H D −→T H M−→0 with exact rows in which the vertical maps are isomorphisms. Thus T H = 0Mod-R . On the other hand, since SS ∈ E, so is every submodule of a projective right S-module. Thus, given N S , an exact sequence 0 → K −→ P −→ N → 0 with P projective, a similar argument shows that H T = 0Mod-S . (3) Let M ∈ Mod-R and let f

g

0 → M −→ E −→ D → 0 be exact with E injective so that E, D ∈ T . Then we have an exact sequence Hf

δ

Hg

0 → H M −→ H E −→ H D −→ H M → 0 that yields exact sequences Hf

α

0 → H M −→ H E −→ L → 0

(3.7)

42

Tilting Modules

and β

δ

0 → L −→ H D −→ H M → 0

(3.8)

in which β is the inclusion, β ◦ α = H g and L = Im H g = Ker δ ∈ E = Ker T by (1) and (2). Hence, from sequence (3.7) we obtain an exact sequence Tα

THf

0 → T H M −→ T H E −→ T L → 0,

(3.9)

and since T H D = 0 = T H M, from sequence (3.8) we obtain an exact sequence ∂

Tβ

0 → T H M −→ T L −→ T H D → 0

(3.10)

in which ν D : T H D → D and ν E : T H E → E are isomorphisms since VR is a ∗-module. Thus we obtain a commutative diagram

0 ↓

0 ↓ THf

0 → T H M −→ T H E νE ↓ νM ↓ 0→

f

M −→ πM ↓ Coker ν M

Tα

−→ TL ν D ◦Tβ ↓ g

−→

E ↓ 0

0 ↓ T HM ∂↓

D ↓ 0

→0 →0

with exact rows and columns. Thus by the Snake Lemma there is an exact sequence νM

θM

0 → T H M −→ M −→ T H M → 0 where π M is the canonical epimorphism, θ M = s −1 ◦ π M , and s = π M ◦ f −1 ◦ ν E ◦ (T α)−1 ◦ ∂ is the snake, an isomorphism. Of course, if M is another R-module and f

g

0 → M −→ E −→ D → 0

3.5. The Tilting Theorem

43

is exact with E injective, there is a similar diagram

0 ↓

0 ↓

THf

T α

0 → T H M −→ T H E νE ↓ νM ↓ f

M −→ πM ↓ Coker ν M

0→

0 ↓ T H M ∂ ↓

−→ T L ν D ◦Tβ ↓

→ 0

D ↓ 0

→ 0,

g

E ↓ 0

−→

and here let us call the snake t = π M ◦ ( f )−1 ◦ ν E ◦ (T α )−1 ◦ ∂ . Then, to establish that θ is natural, we must show that, if h : M → M , then (t −1 ◦ π M ) ◦ h = T H h ◦ (s −1 ◦ π M ), or equivalently, t ◦ T H h ◦ (s −1 ◦ π M ) = π M ◦ h. Since E is injective, there is a commutative diagram f

g

0 → M −→ E −→ D h ↓ h↓ h ↓

f

0 → M −→ E

g

−→

D

→ 0 (3.11) → 0

with exact rows. Then, letting k = H h | L : L → L , where L = Im H g and L = Im H g , and considering the commutative diagram Hf

H M −→ H E Hh ↓ H h ↓ Hf

H M −→

H E

Hg

−→ H g

δ

HD H h ↓

−→ H M H h ↓

H D

−→ H M

−→

δ

we have, corresponding to sequences (3.7), (3.9), and (3.10), the following three commutative diagrams with exact rows α

Hf

0 → H M −→ H E Hh ↓ H h ↓

Hf

0 → H M −→

HE

−→ L → k↓ α

−→ L →

THf

0 → T H M −→ T H E T Hh ↓ T H h ↓ 0

→ THM

THf

−→ T H E

0 (3.12) 0,

Tα

−→ T L → Tk ↓ T α

−→ T L

→

0 (3.13) 0

44

Tilting Modules

and 0 0

→ T HM T H h ↓

∂

Tβ

∂

Tβ

−→ T L −→ T H D Tk ↓ T H h ↓

→ T H M −→ T L −→

T H D

→

0 (3.14)

→

0.

Now, observing that s −1 ◦ π M = ∂ −1 ◦ T α ◦ ν E−1 ◦ f, we use these diagrams and the naturalness of ν to calculate t ◦ T H h ◦ (s −1 ◦ π M ) = π M ◦ ( f )−1 ◦ ν E ◦ (T α )−1 ◦ (∂ ◦ T H h) ◦ ∂ −1 ◦ T α ◦ ν E−1 ◦ f = π M ◦ ( f )−1 ◦ ν E ◦ (T α )−1 ◦ (T k ◦ ∂) ◦ ∂ −1 ◦ T α ◦ ν E−1 ◦ f = π M ◦ ( f )−1 ◦ ν E ◦ (T α )−1 ◦ (T k ◦ T α) ◦ ν E−1 ◦ f = π M ◦ ( f )−1 ◦ ν E ◦ (T α )−1 ◦ (T α ◦ T H h ) ◦ ν E−1 ◦ f = π M ◦ ( f )−1 ◦ (ν E ◦ T H h ) ◦ ν E−1 ◦ f = π M ◦ ( f )−1 ◦ (h ◦ ν E ) ◦ ν E−1 ◦ f = π M ◦ ( f )−1 ◦ (h ◦ f ) = π M ◦ ( f )−1 ◦ ( f ◦ h) = πM ◦ h as desired. A similar argument using exact sequences of the form 0 → K −→ P −→ N → 0 in Mod-S with P projective yields the natural transformation ϕ : H T → 1Mod −S with ϕN

ηN

0 → H T N −→ N −→ H T N → 0 exact for all N ∈ Mod-S. (4) This follows at once from (2) and (3). Remark 3.5.2. Assuming the notation of the Tilting Theorem, we make the following observations: 1. The torsion submodules of M ∈ Mod-R and N ∈ Mod-S are τT (M) = TrV (M) = ν M (T H M)) ∼ = T H M and τS (N ) = Ann N ( S V ) = Ker η N ∼ = H T M. 2. Since S is closed under epimorphic images and direct sums, F is closed under submodules and direct products, and T : S F : H is an equivalence, letting R VS = H ( R R R ), we see that VS is a ∗-module with End(VS ) ∼ = T H (R) ∼ = R/τT (R) such that T ∼ = Hom S (V , ) on ∼ S and H = ( ⊗ R V ) on F. (See the discussion on page 16.)

3.6. Global Dimension and Splitting

45

3.6. Global Dimension and Splitting Of course, the global dimensions of R and the endomorphism ring of an Rprogenerator are the same. For a tilting module they are close. We denote the right global dimension of a ring R by r.gl.dim. (R). Most of the results in this section were proved for artin algebras in [9], [8], [47], and [49] and extended to arbitrary rings in [18] and [64]. Proposition 3.6.1. If S VR is a tilting bimodule, then r.gl.dim. (S) ≤ r.gl.dim. (R) + 1. Proof. Let · · · → P3 → P2 −→ P1 −→ P0 → N → 0 be a projective resolution of N ∈ Mod-S. Then, since flat.dim. S V ≤ 1, by Theorem 3.2.1 and Proposition 3.2.2, the sequence T P3 → T P2 −→ T P1 is exact. Thus, if E R is injective, we see from the commutative diagram Hom R (T P1 , E) −→ Hom R (T P2 , E) −→ Hom R (T P3 , E) ∼ ∼ ∼ =↓ =↓ =↓ Hom S (P1 , H E) −→ Hom S (P2 , H E) −→ Hom S (P3 , H E) that Ext2S (N , H E) = 0 so that inj . dim . (H E) ≤ 1. Let M ∈ Mod-R with injective resolution d0

d1

dm−1

d2

0 → M −→ E 0 −→ E 1 −→ E 2 −→ · · · −→ E m → 0, and let (T , F) be the torsion theory in Mod-R induced by V. If M ∈ T , then the sequence H d0

H d1

H d2

H dm−1

0 → H M −→ H E 0 −→ H E 1 −→ H E 2 −→ · · · −→ H E m → 0 is exact by Proposition 2.3.3 since the E i and Im d1 all belong to T . Thus it follows (by induction on m) that inj . dim . H M ≤ m + 1. If M ∈ F, the exact sequence 0 → M −→ E 0 −→ Im d0 → 0 yields an exact sequence H M = 0 −→ H E 0 −→ H (Im d0 ) → H M → 0 = H E 0 .

46

Tilting Modules

Thus, since Im d0 ∈ T and has injective dimension ≤ m − 1, as we have just proved, inj . dim . (H (Im d0 )) ≤ m. Therefore, since inj . dim . (H E 0 ) ≤ 1, we see from exact sequences (N , H (Im d0 )) → Extm+1 (N , H M) → Extm+2 (N , H E 0 ), Extm+1 S S S for N ∈ Mod-S, that inj . dim .(H M) ≤ m. Now, if r.gl.dim. R = n, it follows from the exact sequences ηN

ϕN

0 → H T N −→ N −→ H T N → 0 of the Tilting Theorem that r.gl.dim. S ≤ n + 1. The most interesting case occurs when R is right hereditary, for then the torsion theory (S, E) on Mod-S induced by a tilting bimodule S VR splits in the sense that each N ∈ Mod-S is a direct sum of a module in S and a module in E. To prove this we shall employ Lemma 3.6.2. If S VR is a tilting bimodule that induces the torsion theory (T , F ) in Mod-R, then, for all L , M ∈ T and all integers i ≥ 0 ExtiS (H L , H M) ∼ = ExtiR (L , M). Proof. Since Gen(VR ) = Pres(VR ) (see Theorem 2.3.8), there is an exact sequence d3

d2

d1

d0

· · · −→ V2 −→ V1 −→ V0 −→ L → 0 with each Vi ∈ Add(VR ). Then each H Vi is projective over S, and so, being exact by Theorem 2.3.8 since each Im di ∈ T , the sequence H d3

H d2

H d1

H d0

· · · −→ H V2 −→ H V1 −→ H V0 −→ H L → 0 is a projective resolution of H L . Since H : T E : T is an equivalence, we have an isomorphism of complexes d1∗

d2∗

H d1∗

H d2∗

0 → Hom R (V0 , M) −→ Hom R (V1 , M) −→ Hom R (V2 , M) ∼ ∼ ∼ =↓ =↓ =↓

d3∗

−→ · · · H d3∗

0 →Hom S (H V0 , H M)−→Hom S (H V1 , H M)−→Hom S (H V2 , H M)−→ · · · . ∗ Thus it remains to show that Ker di+1 / Im di∗ ∼ = ExtiR (L , M) for all i ≥ 1. j To do so, first we note that, since Ext R (Vk , M) = 0 for all j ≥ 1, the exact

3.6. Global Dimension and Splitting

47

sequences βk+1

αk

0 → Im dk+1 −→ Vk −→ Im dk → 0 yield isomorphisms j−1 j Ext R (Im dk+1 , M) ∼ = Ext R (Im dk , M)

for all j > 1. Thus 1 ∼ ∼ ExtiR (L , M) ∼ = Exti−1 R (Im d1 , M) = · · · = Ext R (Im di−1 , M).

(#)

Now, from the commutative diagram with exact row and column Vi+1 di+1 ↓ Vi αi ↓ 0→ Im di ↓ 0

di

$ βi

αi−1

−→Vi−1 −→Im di−1 →0

we obtain a commutative diagram with exact row and column

di∗

% βi∗

Hom R (Vi+1 , M) ∗ di+1 ↑ Hom R (Vi , M) αi∗ ↑

Hom R (Vi−1 , M)−→Hom R (Im di , M)−→Ext1R (Im di−1 , M)→0 ↑ 0 from which, together with (#), it follows that ∗ Ker di+1 / Im di∗ = Im αi∗ /αi∗ (Im βi∗ ) ∼ = Ext1R (Im di−1 , M) ∼ = ExtiR (L , M)

as desired. Slightly more than the promised splitting is true. Proposition 3.6.3. Let S VR be a tilting bimodule. The exact sequence ϕN

ηN

0 → H T N −→ N −→ H T N → 0

48

Tilting Modules

of the Tilting Theorem splits for all N ∈ Mod-S if and only if Ext2R (X, Y ) = 0 for all X ∈ T and all Y ∈ F. In particular, (S, E) splits whenever R is right hereditary. Proof. Let X ∈ T and Y ∈ F, and let 0 → Y −→ E −→ L → 0 be exact with E injective, so that E and L ∈ T . Then since H Y = 0 we also have an exact sequence 0 → H E −→ H L −→ H Y → 0. Now, since by Lemma 3.6.2, ExtiS (H X, H E) ∼ = ExtiR (X, E) = 0 for i ≥ 1, these two exact sequences yield the diagram Ext1R (X, L) −→ Ext2R (X, Y ) → 0 ∼ =↓ 1 0 → Ext S (H X, H L) −→ Ext1S (H X, H Y ) → 0.

0→

Thus the condition is clearly sufficient, and it is necessary since, by the Tilting Theorem, any exact sequence 0 → H Y → N → H X → 0 must have H Y ∼ = H T N and H X ∼ = HT N. Definition 3.6.4. A ring S is right tilted if there is a tilting bimodule S VR with R a right hereditary ring. When S is right tilted the torsion theory (S, E) has the following properties, which also serve to determine when R is right hereditary. Proposition 3.6.5. Let S VR be a tilting bimodule. Then R is a right hereditary ring if and only if (1) the induced torsion theory (S, E) in Mod-S splits and (2) proj . dim . N ≤ 1 for all N ∈ E. Moreover, if these conditions hold, then (3) inj . dim . N ≤ 1 for all N ∈ S. Proof. (⇒). Assume that R is right hereditary. Then (1) holds by Proposition 3.6.3. Every projective PS ∈ Mod-S belongs to E; so, if N ∈ E, then, by Lemma 3.6.2 and the Tilting Theorem, Ext2S (N , P) ∼ = Ext2R (T N , T P) = 0, so, if X ∈ Mod-S, then an exact sequence 0 → K −→ P −→ X → 0,

3.7. Grothendieck Groups

49

together with Proposition 3.6.1, yields an exact sequence 0 = Ext2S (N , P) −→ Ext2S (N , X ) −→ Ext3S (N , K ) = 0 that establishes (2). Moreover, we saw in the proof of Proposition 3.6.1 that if M ∈ F = Ker H in Mod-R, then inj . dim .(H M) ≤ inj . dim . M. Thus (3) follows since S = H F. (⇐). Let X ∈ T = Ker H , let M ∈ Mod-R, and consider the exact sequence 0 → T H M −→ M −→ T H M → 0 with T H M ∈ T and T H M ∈ F. Here, by (2) and Lemma 3.6.2, Ext2R (X, T H M) ∼ = Ext2S (H X, H T H M) = 0, and by (1) and Proposition 3.6.3, Ext2R (X, T H M) = 0. Thus proj . dim . X ≤ 1 for all X ∈ T . Now suppose M ∈ Mod-R and let 0 → M −→ E −→ X → 0 be exact with E injective, so that E, X ∈ T . Then from the exact sequences Ext2R (E, L) → Ext2S (M, L) → Ext3R (X, L) with L ∈ Mod-R, we see that proj . dim . M ≤ 1. An artin algebra S of global dimension at most 2 with a torsion theory (S, E) in mod-S that satisfies conditions (1), (2), and (3) of Proposition 3.6.5 is called quasi-tilted in [45] and subsequent papers.

3.7. Grothendieck Groups We begin this section with the definition of a group that contains useful information about mod-R for a right noetherian ring R. Definition 3.7.1. If R is a right noetherian ring and | mod-R| consists of one member of each isomorphism class in mod-R, the Grothendieck group of mod-R is K 0 (mod -R) = A/R

50

Tilting Modules

with A the free abelian group with basis | mod-R|, and R is the subgroup of A generated by the elements of the form M−K−L if there is an exact sequence 0 → K −→ M −→ L → 0 in mod-R; so that letting [M] denote the coset of M in K 0 (mod −R), [M] = [K ] + [L]. An inductive argument shows that, if 0 → M1 −→ M2 −→ · · · −→ Mn−1 −→ Mn → 0 is exact, then n

(−1)i [Mi ] = 0

i=1

in K 0 (mod-R). The Tilting Theorem, together with this observation and the following lemma, allows us to prove that the Grothendieck groups of a pair of right noetherian rings R and S are isomorphic if there is a tilting bimodule S VR . Lemma 3.7.2. If R and S are right noetherian rings and S VR is a tilting bimodule, then (1) H = Hom R (V, M) and H = Ext1R (V, M) belong to mod-S whenever M ∈ mod-R; (2) T = N ⊗ S V and T =Tor1S (N , V ) belong to mod-R whenever N ∈ mod-S. Proof. (1) From the exact sequence 0 → R R −→ V0 −→ V1 → 0 with the Vi ∈ add(VR ) we obtain an exact sequence 0 → H R → H V0 → H V1 → H R → 0 in mod-S since Ext1R (V, V0 ) = 0 and the H Vi are finitely generated projective S-modules. If M ∈ mod-R, from an exact sequence 0 → K −→ R n −→ M → 0 we have an exact sequence H Rn → H M → H K → H Rn → H M → 0


51

since proj . dim .V ≤ 1. It follows that H M ∈ mod-S; hence, H K ∈ mod-S. But then, since all the other terms in the sequence belong to mod-S, so does H M. (2) One easily checks that (2) holds for any bimodule S VR such that R and S are right noetherian and VR is finitely generated. Now, as promised, we can prove Theorem 3.7.3. If R and S are right noetherian rings and S VR is a tilting bimodule, then, assuming the notation of the Tilting Theorem 3.5.1, there is an isomorphism : K 0 (mod -R) → K 0 (mod -S) such that : [M] → [H M] − [H M] and −1 : [N ] → [T N ] − [T N ]. ∧

the free abelian Proof. There is, by Lemma 3.7.2, a homomorphism from ∧ group A with basis | mod-R| to K 0 (mod-S) such that : M → [H M] − [H M] for M ∈ | mod −R|. But for each generator M − K − L of R, from the corresponding exact sequence 0 → K −→ M −→ L → 0, we obtain an exact sequence 0 → H K −→ H M −→ H L → H K −→ H M −→ H L → 0 ∧

(see Theorem 3.2.1). Thus it follows that R ⊆ Ker , and so exists. On the other hand, since S V has projective (hence flat) dimension ≤ 1 by Proposition 3.2.2, if 0 → K −→ N −→ L → 0 is exact in mod-S, then we see from Lemma 3.7.2 and the exact sequence 0 → T K −→ T N −→ T L → T K −→ T N −→ T L → 0 that there is a homomorphism : K 0 (mod-S) → K 0 (mod-R) such that : [N ] → [T N ] − [T N ] for N ∈ K 0 (mod −S). Now it follows from the Tilting Theorem that, for each generator [M] of K 0 (mod-R), ◦ ([M]) = ([H M] − [H M]) = [T H M] − [T H M] − [T H M] + [T H M] = [M]. Thus ◦ = 1 K 0 (mod −R) , and similarly ◦ = 1 K 0 (mod −S) .

52

Tilting Modules

Now suppose that R is right artinian with complete set of isomorphically distinct simple modules X 1 , . . . , X n , and for each M ∈ mod-R and each i = 1, . . . , n, let ci (M) denote the number of terms in a composition series of M that are isomorphic to X i . Then, by induction on composition length we see that if M ∈ | mod-R|, then n [M] = ci (M)[X i ] i=1

in K 0 (mod-R) = A/R. But also R is contained in the kernel the map A → Zn such that M → (c1 (M), . . . , cn (M)). Thus there are epimorphisms Zn → K 0 (mod-R) and K 0 (mod-R) → Zn from which it follows, since they must be isomorphisms (see [1, Lemma 11.6]), that Proposition 3.7.4. If R is a right artinian ring with complete set of isomorphically distinct simple modules X 1 , . . . , X n , then K 0 (mod-R) is a free abelian group with basis [X 1 ], . . . , [X n ]. Now it follows that if R and S are right artinian and S VR is a tilting bimodule, then R R , SS , VR , and S V all have the same number of isomorphism classes of indecomposable direct summands. Corollary 3.7.5. If R and S are right artinian rings and S VR is a tilting bimodule, then (1) R and S have the same number n of isomorphism classes of simple modules. (2) There are positive integers m 1 , . . . , m n and indecomposable modules m V1 , . . . , Vn such that VR ∼ = V1 1 ⊕ · · · ⊕ Vnm n . Proof. (1) This follows at once from Theorem 3.7.3 and Proposition 3.7.4. (2) According to (1), since the right artinian ring S has exactly n isomorphism classes of simple modules there is a complete orthogonal set of primitive idempotents ei j in S with i = 1, . . . , n and, for each i, j = 1, . . . , m i such that ei j S ∼ = ek S if and only if i = k. (See [1, Section 27].) Then ∼ since S = End(VR ) and Hom R (ei j V, ek V ) ∼ = ei j Sek ∼ = Hom R (ei j S, ek S), ∼ canonically, VR = ⊕i j ei j V with ei j V = ek V if and only if i = k. In practice it can be reasonably easy to determine when a given module VR is a partial tilting module, but the third condition (iii), the existence of an exact sequence 0 → R R −→ V0 −→ V1 → 0 with V0 , V1 ∈ add (VR ), is more difficult to verify. In the artinian algebra case this problem can often be circumvented by the following result that is due to K. Bongartz [8].


53

Corollary 3.7.6. Let R be a right artinian ring with n isomorphism classes of simple modules, and suppose that a partial tilting module VR = V1m 1 ⊕ · · · ⊕ Vnm n with the Vi pairwise non-isomorphic indecomposable modules. If the endomorphism ring of every finitely generated right R module is right artinian and Ext1R (V, R) is finitely generated over S = End(VR ) (for example, if R is an artin algebra), then VR is a tilting module. Proof. According to Corollary 3.4.6, VR is a direct summand of a tilting module, but then by Corollary 3.7.5 and the Krull-Schmidt theorem, VR must be a tilting module. The following example illustrates the efficacy of this corollary. Example 3.7.7. Let R be an upper triangular 4 × 4 matrix ring over a field K so that R is a hereditary serial algebra. Let J = J (R), and let ei = eii denote the i th diagonal idempotent so that the simple R-modules are Ti = ei R/ei J for i = 1, 2, 3, 4. Then we can describe the structure of R R = e1 R ⊕ e2 R ⊕ e3 R ⊕ e4 R as

RR :

1 | 2 | 3 | 4

2 3 4 | | 3 4 | 4

(see [1, Page 350]). Now, letting VR = e1 R ⊕ R/Re3 R, we see that V = V1 ⊕ V2 ⊕ V3 ⊕ V4 has the following structure:

VR :

1 1 2 4 | | 2 2 . | 3 | 4

Of course, VR is finitely presented, and (i) proj . dim .(VR ) ≤ 1 since R is hereditary. Since V1 is both injective and projective and V2 ⊕ V3 ⊕ V4 is

54

Tilting Modules

projective over R modulo, the idempotent ideal Re3 R, it easily follows that (ii) Ext1R (V, V ) = 0. Thus VR is a partial tilting module. But, although neither condition (iii) of Theorem 3.2.1 is immediately evident, since V is a direct sum of four isomorphically distinct indecomposable modules, VR is a tilting module by Corollary 3.7.6. Also, calculating S = End(VR ) as the algebra of 4 × 4 matrices [γi j ] with γi j ∈ Hom R (V j , Vi ), we find that   K 0 0 K K K K K    S∼ K e24 = 0 0 K 0 0 0 0 K so that the structure of SS is indicated by SS :

1 2 3 | ' $ 4 1 3

4 .

Thus gl.dm. S = 2. Also, since Mod-R consists of the direct sums of copies of the factors of the ei R (see [1, Theorem 32.3)]), it follows that T = Add(e1 R ⊕ e1 R/e1 J 3 ⊕ e1 R/e1 J 2 ⊕ e1 R/e1 J ⊕ e2 R/e2 J ⊕ e4 R) and F = Ker H = Add(e2 R ⊕ e3 R). So, since (S, E) splits by Proposition 3.6.3, S has exactly eight indecomposable right modules. The finitely generated projective modules over any ring yield another type of Grothendieck group that we shall use to obtain a version of Corollaries 3.7.5 and 3.7.6 for semiperfect noetherian rings of finite global dimension. Definition 3.7.8. If R is a ring and | proj-R| consists of one member of each isomorphism class of finitely generated projective modules, then the Grothendieck group of proj-R K 0 (proj -R) = A p /R p with A p the free abelian group on | proj-R| and R p is the subgroup of A p generated by the elements of the form P − P − P


55

if there is a (split) exact sequence 0 → P −→ P −→ P → 0 in proj-R; so that letting [P] p denote the equivalence class of P in K 0 (proj-R), [P] p = [P ] p + [P ] p . If R is right noetherian, then there is clearly an epimorphism ϕ : [P] p → [P] of K 0 (proj-R) onto the subgroup P of K 0 (mod-R) generated by the equivalence classes of projective modules in | mod-R|. If moreover M ∈ mod-R has a finite projective resolution 0 → Pm −→ · · · −→ P1 −→ P0 −→ M → 0, then [M] =

m

(−1)i [Pi ] ∈ P.

i=0

Thus if the right global dimension of R is finite, ϕ : K 0 (proj-R) → K 0 (mod-R) is an epimorphism. In fact, using arguments common to homological algebra, it can be shown as in [73, Theorem 4.4, page 102] that in this m (−1)i [Pi ] p . case ϕ is an isomorphism with inverse [M] → i=0 If R is semiperfect with complete irredundant set P1 , . . . , Pn of indecomposable right projective modules (see [1, Section 27]), then, since every finitely generated projective right R-module is of the form P1m 1 ⊕ · · · Pnm n , we see similarly to the discussion proceeding Proposition 3.7.4, that Proposition 3.7.9. If R is a semiperfect ring with complete set of isomorphically distinct indecomposable projective right modules P1 , . . . , Pn , then K 0 (proj-R) is a free abelian group with basis [P1 ] p , . . . , [Pn ] p . Since the simple right modules over the semiperfect ring R of Proposition 3.7.9 are just P1 /P1 J, . . . , Pn /Pn J, where J is the radical of R, that proposition and the two paragraphs preceding it now yield Corollary 3.7.10. If R and S are semiperfect right noetherian rings of finite right global dimension and S VR is a tilting bimodule, then (1) R and S have the same number n of isomorphism classes of simple modules. (2) There are positive integers m 1 , . . . , m n and pairwise non-isomorphic m indecomposable modules V1 , . . . , Vn such that VR ∼ = V1 1 ⊕ · · · ⊕ mn Vn .

56

Tilting Modules

Corollary 3.7.11. Let R be a semiperfect right noetherian ring of finite right global dimension such that the endomorphism ring of every finitely generated right R-module is a semiperfect right noetherian ring, and suppose that R has n isomorphism classes of simple modules. Let VR = V1m 1 ⊕ · · · ⊕ Vnm n be a partial tilting module with the Vi pairwise non-isomorphic indecomposable modules. If Ext1R (V, R) is finitely generated over S = End(VR ), then VR is a tilting module. Proof. According to Corollary 3.4.6, VR is a direct summand of a tilting module, say W R = V ⊕ X is tilting. But then by Corollary 3.7.10 and the Azumaya–Krull–Schmidt theorem [1, Theorem 12.6], VR must be a tilting module. Any noetherian serial ring of finite global dimension satisfies the hypotheses of Corollary 3.7.11 according to Propositions B.1.6 and B.1.7. The following is a case in point: Example 3.7.12. Let D be a local noetherian ring with maximal ideal m such that every one-sided ideal of D is a power of m (for example let D = K [[x]], the ring of power series over a field K with m = Dx), and let R be the 4 × 4 [D :m] upper triangular matrix ring   D D D D m D D D   R= m m D D  . m m m D Then, letting e1 , e2 , e3 , e4 denote the diagonal idempotents, the display

RR :

1 | 2 | 3 | 4 | 1 | 2 .. .

2 | 3 | 4 | 1 | 2 | 3 .. .

3 | 4 | 1 | 2 | 3 | 4 .. .

4 | 1 | 2 | 3 | 4 | 1 .. .

3.8. Torsion Theory Counter Equivalence

57

describes the structure of R R . Now letting VR = R/Re3 R ⊕ e4 R, we see that V = V1 ⊕ V2 ⊕ V3 ⊕ V4 has the following structure:

VR :

4 1 2 4 | | | 1 2 1 | | 2 2 | 3. | 4 | 1 .. .

Then V1 , V2 , V3 are all projective over R/Re3 R so, Ext1R (Vi , V j ) = 0 for 1 ≤ i, j ≤ 3 and Ext1R (V4 , Vi ) = 0 for 1 ≤ i ≤ 4 since V4 = e4 R. Since e4 R ∼ = e3 J, 0 → V4 −→ E(V4 ) −→ E(e3 R/e3 J ) → 0 is the minimal injective resolution of V4 (see Section B.2). Thus we have epimorphisms Hom R (Vi , E(e3 R/e3 J )) −→ Ext1R (Vi , V4 ) → 0. But for i = 1, 2, 3, no Vi has a composition factor isomorphic to e3 R/e3 J, and so Hom R (Vi , E(e3 R/e3 J )) = 0. Thus we finally see that (ii) Ext1R (V, V ) = 0. Since R is hereditary by Corollary B.1.4, we have (i) proj . dim .(VR ) = 0. Also, by Propositions B.1.2 and B.1.6, R satisfies the hypotheses of Corollary 3.7.11, and so VR is a tilting module. A complete classification of the tilting modules over noetherian serial rings and their endomorphism rings was given in [18].

3.8. Torsion Theory Counter Equivalence The Tilting Theorem (3.5.1) gives rise to torsion theories (T , F ) and (S, E) in Mod-R and Mod-S, respectively, with category equivalences T ≈ E and

58

Tilting Modules

S ≈ F. As noted in Proposition 2.3.2 and in Remark 3.5.2, these equivalences are representable by certain ∗-modules. Here we shall discuss a more general notion that was introduced in [19]. Definition 3.8.1. Let (T , F ) and (S, E) be torsion theories in Mod -R and Mod-S, respectively. A pair of category equivalences F : T E : G and F : F S : G is called a torsion theory counter equivalence. We shall show that both equivalences in a torsion theory counter equivalence are induced by representable functors and H : Mod-R Mod-S : T

H : Mod-R Mod -S : T

that satisfy orthogonality relations and admit exact sequences like those in (2) and (3) of the Tilting Theorem, and, conversely, that any two pairs of functors satisfying such orthogonality relations and admitting such exact sequences induce a torsion theory counter equivalence. Theorem 3.8.2. Let (T , F) and (S, E) be torsion theories in Mod-R and Mod-S, respectively, and suppose that F : T E : G and F : F S : G is a torsion theory counter equivalence. Let S VR

= G(S/τS (S)) and

R VS

= F (R/τT (R))

be the induced bimodules and let H = Hom R (V, ), T = (V ⊗ S ), H = ( ⊗ R V ), and T = Hom S (V , ). Then (1) F ∼ = T |E , F ∼ = H |F , and G ∼ = T |S ; = H |T , G ∼ (2) T H = 0Mod-R = T H and H T = 0Mod-S = H T ; (3) The canonical natural transformations induce exact sequences νM

θM

0 → T H M → M → T HM → 0 and φN

ηN

0 → H T N → N → H T N → 0


59

with ν M (T H M) = τT (M) and φ N (H T N ) = τS (N ) for all M ∈ Mod-R and all N ∈ Mod -S. Proof. By Proposition 1.4.3 τS (S) = ∩{r S (N )|N ∈ E} = r S (E) and τT (R) = ∩{r R (M)|M ∈ F} = r R (F). Let S¯ = S/τS (S) and R¯ = R/τT (R). By Proposition 2.3.2, (1) is true, VR is a ∗-module with S¯ = End(VR ), and VS is a ∗-module with R¯ = End(VS ). Moreover, T = Gen(VR ), E = Cogen(VS¯∗ ), S = Gen(VS ), and F = Cogen(VR¯ ∗ ) (where VS¯∗ = Hom R ( S¯ VR , C R ) and VR¯ ∗ = Hom S ( R¯ VS , C S ) for injective cogenerators C R and C S in Mod-R and Mod-S, respectively). In particular, it follows that H M ∈ E and T N ∈ F for all M ∈ Mod-R and N ∈ Mod-S. Now, VR ∈ T and so, since T N ∈ F for any N ∈ Mod-S, Hom S ((V ⊗ R V ), N ) ∼ = Hom R (V, Hom S (V , N ), = Hom R (V, T N ) = 0. Thus H T N = Hom R (V, T N ) = 0 and V ⊗ R V = 0, so H T N = (N ⊗ S V ) ⊗ R V = 0. This and a similar argument establish (2). Let ν and η be the canonical natural transformations associated with H and T and let φ and θ be the canonical natural transformations associated with T and H . As noted on page 16, we may assume that the given equivalences are given by the restrictions of these natural transformations. Then Im ν = Ker θ and Im φ = Ker η by Proposition 1.4.3. Also, ν M is monic for all M ∈ Mod-R and θ M is epic for all M ∈ Mod- R¯ by Theorem 2.3.8. To verify that θ M is epic for all M ∈ Mod-R, employing (2) and (1), we see that the exact sequence i

n

0 → τT (M) → M → M/τT (M) → 0

60

Tilting Modules

yields an isomorphism T H n

0 → T H M → T H (M/τT (M)) → 0. Thus, since T H n ◦ θ M = θ M/τT (M) ◦ n is epic, it follows that θ M is epic, so νM

θM

0 → T H M → M → T HM → 0 is exact, and similarly, ηN

φN

0 → H T N → N → H T N → 0 is too. Orthogonality relations and exact sequences like those of Theorem 3.8.2 induce torsion theories and counter equivalences between them. Theorem 3.8.3. Suppose that there are pairs of functors H : Mod-R Mod-S : T

and H : Mod -R Mod-S : T

such that T H = 0Mod-R = T H and H T = 0Mod-S = H T and that there are natural transformations ν, θ, φ, η that induce exact sequences νM

φN

θM

ηN

0 → T H M → M → T H M → 0 and 0 → H T N → N → H T N → 0 for all M ∈ Mod-R and all N ∈ Mod-S. Let T = Ker H ,

F = Ker H,

S = Ker T,

E = Ker T ,

and let R VS

= T ( S SS ) and

S VR

= H (R R R )

be the canonically induced bimodules. Then (1) (T , F ) and (S, E) are torsion theories in R-Mod and S-Mod, respectively. (2) The restrictions H : T E : T are category equivalences.

and H : F S : T


61

(3) H ∼ = Hom R (V, ), T ∼ = (V ⊗ S ), H ∼ = V ⊗ R , and T ∼ = Hom S (V , ). Proof. First note that the orthogonality relations and the exact sequences of the hypothesis establish that H M ∈ E, T N ∈ T , H M ∈ S, T N ∈ F for all M ∈ Mod-R and all N ∈ Mod-S. Also they imply ν M is an isomorphism if and only if M ∈ T , η N is an isomorphism if and only if N ∈ E, θ M is an isomorphism if and only if M ∈ F, and φ N is an isomorphism if and only if N ∈ S. Thus the equivalences of (2) are verified. For (1), suppose Hom R (M, F) = 0 for all F ∈ F. Then θ M = 0, since T H M ∈ F. Hence ν M is an isomorphism so M ∈ T . Conversely, if M ∈ T , F ∈ F, and f ∈ Hom R (M, F), then θ F is an isomorphism, θ M = 0, and θ F ◦ f = T H ( f ) ◦ θ M = 0 so f = 0. Thus we have verified condition (1) of Definition 1.4.1. Condition (2) of Definition 1.4.1 is also easily verified, and we conclude that (T , F ) and similarly (S, E) are torsion theories with torsion submodules given by τT (M) = ν M (T H M) and τS (N ) = φ N (H T N ) for all M ∈ Mod-R and N ∈ Mod-S. To establish (3) we first observe that TS ∼ = T HT S ∼ = T (S/τS (S)) and that, by Proposition 1.4.3, τS (S) = ∩{r S (N )|N ∈ E} = r S (E). Thus E is a subcategory of Mod-S/τS (S), and letting S VR

= T ( S SS ),

we have H ∼ = ⊗ S V on E as in Proposition 2.3.2. = Hom R (V, ) on T and T ∼ Now, if M ∈ Mod-R, then, since H M ∈ E, HM∼ = Hom R (V, TrV (M)) ∼ = Hom R (V, M). = HT H M ∼ = Hom R (V, T H M) ∼ Similarly, letting V = H ( R R R ) = H (R/τV (R)), ∼ Hom S (V , ) on S and H ∼ we obtain T = = ⊗ S V on F. Moreover, with the notations of Definition 2.3.1 we have F = Gen(VS ) and E = Cogen(VS∗ ), and hence Hom R (V ⊗ S VR , C R ) ∼ = Hom S (VS , VS∗ ) = 0. Thus R V ⊗ S V = 0, and hence for any N ∈ Mod-S we have H T N ⊗ S V = T N ⊗ R V ⊗ S V = 0. Thus T N ∼ = H T N ⊗S V ∼ = T HT N ∼ = N ⊗S V . ∼ Similarly, H = ⊗ R V on Mod-R.

62

Tilting Modules

In view of the last two theorems, if H, T, H , T are functors H : Mod-R Mod -S : T

and H : Mod-R Mod-S : T

such that T H = 0Mod-R = T H and H T = 0Mod-S = H T, and ν, θ, φ, η are natural transformations that induce exact sequences νM

ϕN

θM

ηN

0 → T H M → M → T H M → 0 and 0 → H T N → N → H T N → 0 for all M ∈ Mod-R and all N ∈ Mod-S, we shall let S VR

= T ( S SS ) and

R VS

= H (R R R )

be the canonically induced bimodules, identify H = Hom R (V, ), T = (V ⊗ S ), H = ( ⊗ R V ), and T = Hom S (V , ), identify ν, θ, φ, η with the canonical natural transformations, and say that the pairs of functors H, T and H , T , or the bimodules S VR and R VS , induce a torsion theory counter equivalence between Mod -R and Mod-S, or specifically, between (T , E) and (S, F ) with T = Ker H , F = Ker H, S = Ker T, and E = Ker T . As an immediate consequence of Theorems 3.8.3 and 3.8.2, we have Corollary 3.8.4. Bimodules S VR and R VS induce a torsion theory counter equivalence if and only if, for all M ∈ Mod-R and all N ∈ Mod-S, (1) ν M and φ N are monic and θ M and η N are epic; and (2) TrV (M) = Ann M (V ) and TrV (N ) = Ann N (V ). We have a seemingly more easily verified characterization in Theorem 3.8.5. Bimodules S VR and R VS induce a torsion theory counter equivalence if and only if the following four conditions hold: (1) VR and VS are finitely generated, Gen(VR ) = Pres(VR ), Gen(VS ) = Pres(VS ), End(VR ) ∼ = S/ S (V ), and End(VS ) ∼ = R/ R (V ), canonically; (2) V ⊗ R V = 0 and V ⊗ S V = 0; (3) Gen(VR ) ⊆ VR⊥ and Gen(VS ) ⊆ VS ⊥ ; (4) TrV (M) = 0 implies Ann M (V ) = 0, and TrV (N ) = 0 implies Ann N (V ) = 0 for all for all M ∈ Mod-R, N ∈ Mod-S.


63

Proof. (⇒). Note that S (V ) = r S (VS∗ ) = RejCogen(VS∗ ) (S) = TrS (S) by Proposition 1.4.3 and similarly R (V ) = TrT (R). Since VR and VS are ∗-modules, (1) follows from Theorems 2.3.6 and 2.3.8. That conditions (2) and (4) are necessary follows from the statements and proofs of Theorems 3.8.3 and 3.8.2 and Corollary 3.8.4. Since Gen(VR ) is a torsion class, (1) and Proposition 2.3.3 imply (3) since H preserves the exactness of any short exact sequence 0 → M → M → V → 0 with M, and hence also M , in Gen(VR ), yielding a splitting of the given sequence. (⇐). Note that by Theorem 2.3.8(d), conditions (1) and (3) imply that VR and VS are ∗-modules. Let T = Gen(VR ) and S = Gen(VS ). Then it follows easily from (3) (see Proposition 1.4.4) that T and S are closed under extensions. Thus T and S are torsion classes, so if H, H , T, T are induced by S V R and R VS as above and F0 = Ker H and E0 = Ker T , then (T , Fo ) and (S, Eo ) are torsion theories. Let C R be an injective cogenerator, VS∗ = H (C) and E = Cogen(VS∗ ), so that S (V ) = r S (VS∗ ). Then VR is a ∗-module with S/ S (V ) ∼ = End(VR ) and H : T E : T is an equivalence. Since Hom S (V , V ∗ ) ∼ = Hom R ((V ⊗ S V ), C) = 0, we see that E ⊆ E0 , and since Ann N ( S V ) = RejVS∗ (N ), we have E0 ⊆ E by (4). Thus E = E0 and similarly F = F0 . Thus our desired conclusion follows from Theorem 3.8.2. In view of Theorem 3.8.5(3), Proposition 3.3.2 and Theorem 2.3.8(c) yield the following corollary. Corollary 3.8.6. Suppose S VR and R VS induce a torsion theory counter equivalence. If VR is faithful and S V is finitely generated, then VR is a tilting module. If S is a ring with subring A and ideal I such that S = A ⊕ I, we say that S is a split extension of A by I, and we write S = A ∝ I. If, in addition, I 2 = 0, then S = A ∝ I is called a trivial extension of A by I. One can use these notions and the following corollary to find non-tilting examples of torsion theory counter equivalences Corollary 3.8.7. Let A VR and R V A induce a torsion theory counter equivalence. If S = A ∝ I with I ⊗ A V = 0 and V ⊗ A I = 0, then S VR and R VS induce a torsion theory counter equivalence.

64

Tilting Modules

Proof. Clearly we need only verify the “S” part of conditions (1)–(4) of Theorem 3.8.5. Condition (1) holds because VS and every module it generates or cogenerates is annihilated by I, and (2) follows because V ⊗ S V = V ⊗ A V. If N ∈ Gen(VS ), and 0 → N −→ X −→ V → 0 in Mod-S, then N ⊗ A I = 0 = V ⊗ A I implies X ⊗ A I = 0, so X I = 0. Thus the sequence splits in both Mod-A and Mod-S, and therefore (3) is verified. Finally, observing that ∼ ∼ S V = S ⊗ A V and VS = V ⊗ A S, we have Hom S (V , N ) ∼ = Hom S ((V ⊗ A S), N ) ∼ = Hom A (V , Hom S (S, N )) ∼ = Hom A (V , N )) and N ⊗S V ∼ = N ⊗ S (S ⊗ A V ) ∼ = (N ⊗ S S) ⊗ A V ∼ = N ⊗A V so that, for all N ∈ Mod-S, TrV (N ) = 0 implies Ann N (V ) = 0. Example 3.8.8. Suppose that R is a hereditary artin algebra with duality D = Hom K ( , E(K /J (K ))), and let R U R = D(R). Then it follows easily from Theorem 3.2.1 that R U R is a tilting bimodule. Suppose further that R has no non-zero injective projective modules, and let S denote the trivial extension r u S= R∝U = | r ∈ R, u ∈ U . 0 r Then we may view U as an ideal of S and let S VR be the bimodule with underlying set U and U V = 0. Let T = Gen(VR ) and F = Ker Hom R (V, ), so that (T , F) is a tilting torsion theory, and note that T = {M R | M is injective} and F = {M R | M has no injective direct summands}. Let R VS = Ext1R (V, R). Then by the Tilting Theorem V ⊗ R U = T H (R) = 0. Moreover D(U ⊗ R V ) ∼ = Hom R (U, DV ) ∼ = Hom R (U, R) = 0, since R has no non-zero injective projective modules. Thus by Corollary 3.8.7, S VR and R VS induce a torsion theory counter equivalence. Now VS∗ = Hom R ( S VR , D(R)) = R S ∼ = S/U S


65

so that, since VR is a ∗-module with End(VR ) ∼ = S/U, E = Cogen(VS∗ ) = {N S | N R is projective} is a torsion-free class in Mod-S. Moreover, noting that if N ∈ Mod-S and N R = N ⊕ P with PR projective, then NU ⊆ N , one easily checks that the corresponding torsion class in Mod-S is S = {N S | N R has no projective direct summands}.

4 Representable Dualities

Suppose that A R and S B are subcategories of Mod-R and S-Mod, respectively. A duality between A R and S B is a pair of contravariant functors DA : A R S B : DB such that DB ◦ DA ∼ = 1A R and DA ◦ DB ∼ = 1S B . As pointed out in [1, Lemma 23.4], DA and DB are right adjoints of one another in the sense that there are isomorphisms µ M N : Hom R (M, DA (N )) → Hom S (N , DB (M)) that are natural in M ∈ A R and N ∈ S B. If S U R is a bimodule we let U R = Hom R ( , U ) and S U = Hom S ( , U ) to obtain a pair of contravariant functors U R : Mod -R S- Mod : S U (both of which will, when convenient, simply be denoted by U or ).We say that a duality DA : A R S B : DB is representable by S U R if there are natural isomorphisms DA ∼ = U R |A R and DB ∼ = S U | S B. When this is the case, we make the identifications DA = U R = and DB = S U = and write the duality as : A R S B : . 66

4.1. The U -dual

67

4.1. The U -dual Throughout this section we assume that U = S U R is a given bimodule and that all modules are either right R-modules or left S-modules. The functors U R : Mod -R S- Mod : S U are adjoint on the right via µ M N : Hom R (M, S U (N )) → Hom S (N , U R (M)) with

µ M N ( f )(n) : m → f (m)(n).

(In fact, it is not difficult to show that any pair of functors D : ModR S-Mod : D that are adjoint on the right are representable by the S − R-bimodule D( R R R ) ∼ = D ( S SS ).) Associated with this adjunction are the evaluation maps δ X : X → 2 X with δ X (x) : f −→ f (x) for X in Mod-R or S-Mod, x ∈ X , and f ∈ X. They satisfy (δ X ) ◦ δX = 1X

(4.1)

and yield natural transformations δ : 1Mod -R → 2 and δ : 1 S- Mod → 2 . A module X is (U -)reflexive if δ X is an isomorphism, and (U -)torsionless if δ X is a monomorphism. Note that Ker δ X = RejU (X ), so that X is U -torsionless if and only if X is cogenerated by U . On the other hand, we have Proposition 4.1.1. If X is a module, then δ X is an epimorphism if and only if X/ RejU (X ) is U -reflexive. p

Proof. Let X −→ X/ RejU (X ) be the canonical epimorphism. Then by definition of the reject, ( p) is an isomorphism. Thus 2 ( p) is also an isomorphism and the commutativity of the diagram X δX ↓ 2 X

p

−→ 2 p

X/ RejU (X ) δ X/ RejU (X ) ↓

−→ 2 (X/ RejU (X ))

completes the proof. Dual to Theorem A.2.2 one obtains

→0

68

Representable Dualities

Proposition 4.1.2. If S U R induces a representable duality : A R S B : between subcategories of Mod-R and S-Mod, then the modules in A R and S B are U -reflexive. The equation (4.1) yields Proposition 4.1.3. For all X in Mod-R or S-Mod, (1) (δ X ) is a split epimorphism and δX is a split monomorphism; (2) X is U -torsionless; (3) If X is U -reflexive, then so is X. Thus, in particular, the U ’s induce a duality between the categories of U S R -reflexive modules in Mod-R and S-Mod . If M ∈ Mod-R, then, applying to an exact sequence R (B) −→ R (A) −→ M → 0, we obtain an exact sequence 0 → M −→ U A −→ U B in S-Mod so that M ∈ Copres( S U ). If M R is finitely presented, we may assume that A and B are finite, and then we say that M is finitely copresented by S U. If M ∈ mod-R, so that A may be finite, we say that M is semi-finitely copresented by S U. Thus we have Proposition 4.1.4. For all X in Mod-R or S-Mod, and a bimodule S U R , (1) X ∈ Copres(U ); (2) If X is finitely presented, then X is finitely copresented by U ; (3) If X is finitely generated, then X is semi-finitely copresented by U. Note that, since Ker δ R = RejU (R) = r R (U ) and 2 ( S S) ∼ = End(U R ), we have Proposition 4.1.5. Relative to a bimodule S U R , (1) U R is faithful if and only if R R is U -torsionless; (2) S ∼ = End(U R ), canonically, if and only if S S (and hence U R ) is U -reflexive.

4.2. Costar Modules

69

4.2. Costar Modules As we just observed in Proposition 4.1.5, if S = End(U R ), then both U R and S S are U -reflexive. A module U R with S = End(U R ) is called a costar module if every U -torsionless right R-module whose U -dual is finitely generated is U -reflexive and every finitely generated torsionless left S-module is U reflexive. Thus a module U R with S = End(U R ) is a costar module if S U R induces a duality U : D R S C : U where D R ={M R ∈ Cogen(U R ) | U M ∈ S- mod} and S C = S- mod ∩ Cogen( S U ). Here, and in later sections, we shall investigate the extent to which the role of costar modules is dual to that of ∗-modules. Throughout the section we shall assume that S = End(U R ) and D R and S C are the categories of modules whose objects are those described in this last display. Our goal is to determine when S U R induces a duality between them. We begin with the following lemma, dual to Lemma 2.3.7, in order to obtain a characterization of D R . Lemma 4.2.1. Given a bimodule S U R , suppose M ∈ Mod-R and the left S-module M is generated by elements {γa | a ∈ A}. Then there is a f monomorphism M/ RejU (M) −→ U A such that f is an epimorphism. Thus if C = Coker f, there are exact sequences f

0 → C −→ (U A ) −→ (M/ RejU (M)) → 0 and Ext1R ( f,U )

0 → Ext1R (C, U ) −→ Ext1R (U A , U ) −→ Ext1R (M/ RejU (M), U ). Proof. Letting πa : U A → U denote the canonical projections for the direct product U A , define ϕ : M → U A via πa (ϕ(m)) = γa (m). Then Ker ϕ = RejU (M) and ϕ induces the desired monomorphism f since f (πa ) = πa ◦ f = γa for all a ∈ A. Now we have Proposition 4.2.2. A module M ∈ D R if and only if there is a positive intef

ger n and monomorphism M −→ U n such that f is an epimorphism. In particular, D R ⊆ cogen(U R ).

70


Proof. The condition is necessary by Lemma 4.2.1. Conversely, the condition f yields an epimorphism S n ∼ = U n −→ M. As suggested earlier, we intend to prove dual versions of results in Chapter 2. To do so we need dual versions of Lemmas 2.2.1 and 2.2.2. f

Lemma 4.2.3. Suppose 0 → M −→ X → L → 0 is exact where X is U -reflexive and f is an epimorphism. Then M is U -reflexive if and only if L is U -torsionless. f

Proof. Applying we obtain the exact sequence 0 → L → X −→ M → 0 and then the commutative diagram with exact rows 0→

M δM ↓

f

−→

→

X δX ↓

2 f

0 → 2 M −→ 2 X

L δL ↓

→0

→ 2 L .

Since δ X is an isomorphism, the lemma follows from the Snake Lemma. f

Lemma 4.2.4. Let 0 → M −→ X −→ L → 0 be exact in Mod-R. If X is U -reflexive and L is U -torsionless, then f is epic if and only if Im f is U -reflexive. Proof. From the exact sequence π

0 → L −→ X −→ Im f → 0 and the inclusion j

0 → Im f −→ M we obtain the commutative diagram with exact rows 0→

M α↓

f

−→ π

X δX ↓

0 → Im f −→ 2 X

−→

L δL ↓

→

→0

2 L

where α = j ◦ δ M . Since X is reflexive and L is torsionless, α is an isomorphism by the Five Lemma, so α is also an isomorphism. By the adjointness of the functors we have (δ M ) ◦ δM = 1M ,

4.2. Costar Modules

71

and since δ is a natural transformation, δM ◦ j = 2 j ◦ δIm f . Also α = (j ◦ δ M ) = (δ M ) ◦ 2 ( j), so j = 1M ◦ j = (δ M ) ◦ δM ◦ j = (δ M ) ◦ 2 j ◦ δIm f = α ◦ δIm f . Hence f is epic if and only if j is an isomorphism if and only if δIm f is an isomorphism. For convenience we shall denote the category of modules semi-finitely copresented by U R as scopres(U R ). Thus, according to Proposition 4.1.4, U : S C → scopres(U R ) for an arbitrary bimodule S U R . One may view the modules satisfying the conditions of the next proposition as dual to weak ∗-modules. Proposition 4.2.5. Assume S = End(U R ). The following are equivalent: (a) U : scopres(U R ) S C : U is a duality; (b) Every finitely generated U -torsionless left S-module is U -reflexive; f (c) If 0 → M −→ U n → L → 0 is exact with L ∈ Cogen(U R ), then ( f ) is an epimorphism. Proof. (a) ⇒ (b). This is obvious, in view of Proposition 4.1.2. (b) ⇒ (c). Since Im f is an epimorph of S S n and is contained in M f whenever 0 → M −→ U n is exact, this implication follows from Lemma 4.2.4. (c) ⇒ (a). By (c), it follows from Proposition 4.2.2 that scopres(U R ) ⊆ D R , so : scopres(U R ) → S C. Thus we have : scopres(U R ) S C : . If M ∈ scopres(U R ), then, applying (c) and Lemma 4.2.3, we conclude that M is reflexive. Suppose S N ∈ S C. Then there is an exact sequence g 0 → K → S S m −→ N → 0 and, applying , we obtain an exact sequence g 0 → (N ) −→ (S m ) → L → 0 where (S m ) ∼ = U m and L ≤ K ∈ 2 Cogen(U R ). Again by (c), g is epic, and we obtain that N is reflexive from the commutative diagram Sm δS ↓

g

−→ 2 g

N δN ↓

2 S m −→ 2 N with exact rows.

→0 →0

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Any injective module U R satisfies condition (c) of Proposition 4.2.5. More generally Example 4.2.6. A module U R with S = End(U R ) induces a duality U : scopres(U R ) S C : U whenever Cogen(U R ) ⊆ Ker Ext1R ( , U ). The next theorem serves to characterize costar modules with conditions that are dual to those of Theorem 2.3.8. Theorem 4.2.7. Assume that S = End(U R ), D R ={M R ∈ Cogen(U R ) | U M ∈ S- mod} and S C = S- mod ∩ Cogen( S U ). Then the following are equivalent: (a) : D R S C : is a duality. That is, U R is a costar module; (b) : scopres(U R ) S C : is a duality and D R = scopres(U R ); (c) δ M is an epimorphism if M ∈ S-mod, and δ N is an epimorphism if N ∈ S-mod; f (d) D R ⊆ scopres(U R ) and, if 0 → M −→ U n → L → 0 is exact with L ∈ Cogen(U R ), then ( f ) is an epimorphism; f

(e) If 0 → M −→ U n → L → 0 is exact in Mod-R, then L ∈ Cogen (U R ) if and only if ( f ) is an epimorphism. Proof. (a) ⇔ (b). One implication is obvious. Assuming (a), D R = ( S C) ⊆ scopres(U R ) by Proposition 4.1.4. But If M ∈ scopres(U R ) via an exact sef quence 0 → M −→ U n → L → 0 with L ∈ Cogen(U R ), then im f ∈ S C is U -reflexive by (a), so M ∈ D R by Lemma 4.2.4 and Proposition 4.2.2. (b) ⇔ (d). This equivalence is by Propositions 4.2.5 and 4.2.2. (a) ⇒ (c). Lemma 4.1.1 applies. (c) ⇒ (a). This will follow from Lemma 4.1.1 if we show that : S C → D R . As in the proof that (a) implies (b), we see that scopres(U R ) ⊆ D R . Now in view of Proposition 4.1.4, we do have : S C → D R . f (d) ⇒ (e). Suppose 0 → M −→ U n → L → 0 is exact with f epic. Then M ∈ D R by Proposition 4.2.2, so M is reflexive since (d) implies (a). Since U n is reflexive by hypothesis, L ∈ Cogen(U R ) by Lemma 4.2.3. (e) ⇒ (b). By Proposition 4.2.5 it suffices to show that D R = scopres(U R ). f

If M ∈ D R , then by Proposition 4.2.2 there is an exact sequence 0 → M −→ U n → L → 0 with f epic. Hence (e) implies that M ∈ scopres(U R ). If M ∈ scopres(U R ), then (e) also implies M ∈ D R by Proposition 4.2.2.

4.2. Costar Modules

73

A costar module actually satisfies a condition that is a little stronger than condition (c) of Theorem 4.2.7. Proposition 4.2.8. Let U R be a costar module inducing the duality U : f D R S C : U . If 0 → K −→ M −→ L → 0 is exact with M ∈ D R , then f is epic if and only if L is U -torsionless. In this case, K ∈ D R . f

Proof. Assume M ∈ D R and 0 → K −→ M −→ L → 0 is exact. First suppose that f is epic. Then M ∈ S C, so K is finitely generated. Thus by Theorem 4.2.7(c), δ K is epic and hence is an isomorphism (and also K ∈ D R ). Thus L ∈ Cogen(U R ) by Lemma 4.2.3. Conversely, assume that L ∈ Cogen(U R ). By hypothesis M is finitely generated, and so is Im f ⊆ K . But then Im f is U -reflexive because it belongs to S C. Thus f is epic by Lemma 4.2.4. Corollary 4.2.9. If U R is a costar module inducing the duality U : D R S C : U , then the ’s preserve exactness of short exact sequences of modules in both D R and S C. Proof. By Proposition 4.2.8, U preserves exactness of sequences of modules in D R . f g If 0 → K → N → L → 0 is exact with K , N , L in S C, then in the exact sequence f

g

0 → L → N → K N ∈ D R is U -reflexive and Im g is U -torsionless. Thus the bottom row is exact in the commutative diagram 0→

K ↓

g

→

N δN ↓

f

→ 2 f

→ 0

→ 0, ∼ so, since N and L are U -reflexive, (Im g) = K is finitely generated and Im g ∈ D R is U -reflexive. Thus g is epic by Lemma 4.2.4. 0 → (Im g) → 2 N

→

L δL ↓ 2 L

Now, modulo the next lemma, we are in position to show that over an artin algebra, a finitely generated costar module is just the dual of a ∗-module. Lemma 4.2.10. If R is an artin algebra and U R ∈ mod-R, then D R = cogen(U R ) and S C = cogen( S U ).

74


Proof. Since R is an artin algebra, so is S = End(U R ), and a module is finitely generated if and only if it is finitely cogenerated. Thus C S = cogen(U S ). Also cogen(U R ) ⊆ D R since Hom R (M, U ) ⊆ Hom K (M, U ) where K is the center of R, and D R ⊆ cogen(U R ) by Proposition 4.2.2. Proposition 4.2.11. Let R be an artin algebra and let D : mod-R R-mod : D denote the artin algebra dual. Then a finitely generated module U R is a costar module if and only if R V = D(U ) is a ∗-module. Proof. (⇒). From Lemma 4.2.10, assuming that U R is a costar module with S = End(U R ), it follows that U R : cogen(U R ) cogen( S U ) : S U is a duality. According to Corollary 2.4.13, we may assume that R V is faithful and prove that it is a tilting module. Now U R is also faithful, so that R R ∈ cogen(U R ). Thus D(R R ) ∈ gen( R V ) = D(cogen(U R )) and so is every finitely generated injective left R-module. Let H = U R ◦ D : R-mod → S-mod, and observe that H (M) = Hom R (D M, U ) ∼ = Hom R (V, M), f

naturally. Let M ∈ R-mod to obtain an exact sequence 0 → M −→ g E(M) −→ L → 0 from which we obtain an exact sequence Hg

H (E(M)) −→ H (L) → Ext1R (V, M) → 0. But Dg

Df

0 → D(L) −→ D(E(M)) −→ D(M) → 0 is exact, and if M ∈ gen( R V ), all three modules in this sequence belong to cogen(U R ) so that, by Corollary 4.2.9, H g = U R (Dg) is epic. Thus we see that gen( R V ) ⊆ Ker Ext1R (V, ). On the other hand, suppose that Ext1R (V, M) = 0. Then H g = U R (Dg) is epic, so by Proposition 4.2.8, D(M) ∈ cogen(U R ) and hence M ∈ gen( R V ). Thus the implication follows from Proposition 3.2.3. (⇐). Suppose that R V is a ∗-module with S = End( R V ) and let S U R = D( R VS ). Then ∗ S VR

= Hom R (V, D(R)) ∼ = Hom R (R, D(V )) ∼ = SUR .

Thus letting H = Hom R (V, ) and T = ( ⊗ S V ), we see from Definition 2.3.1 that H ◦ D : cogen(U R ) cogen( S U ) : D ◦ T ∼ U and D ◦ T ∼ is a duality. But H ◦ D = = S U . R

4.3. Quasi-Duality Modules

75

Now applying Lemma 4.2.10, Proposition 4.2.11, and the artin algebra dual again, one can easily obtain Corollary 4.2.12. A finitely generated module R V with S = End( R V ) over an artin algebra R is a ∗-module if and only if Hom R (V, ) : gen( R V ) cogen( S D(V )) : V ⊗ S is an equivalence of categories.

4.3. Quasi-Duality Modules A subcategory C of Mod-R or S-Mod is finitely closed if all submodules, epimorphic images, and finite direct sums of modules in C belong to C. Definition 4.3.1. A module U R with S = End(U R ) is a quasi-duality module if the S U R -duals induce a duality : gen(U R ) gen( S S) : between the smallest finitely closed subcategories of Mod-R and S-Mod that contain U R and S S, respectively. Just as costar modules are “dual to” ∗-modules, so are their historical predecessors, quasi-duality modules, dual to quasi-progenerators. Here we shall present results on quasi-duality modules that arose from the work of several authors. This section is based on material contained in the papers [39], [40], [41], [42], [59], [61], [70], [83]. We shall soon obtain characterizations of quasi-duality modules that include the fact that they are costar modules. To do so we first need the following notions and results. A module U is injective relative to a module M if Hom R ( f, U ) is an epif morphism whenever 0 → K → M is exact. If U is injective relative to itself, U is called a quasi-injective module. According to [1, Propositions 16.13 and 16.10], the class of modules that U is injective relative to is closed under sub modules, epimorphic images, and direct sums, and A Uα is injective relative to M if and only if so is each Uα . Dual to Lemma 2.4.4 we have Lemma 4.3.2. If U is injective relative to each Mi for i ∈ I, and cogenerates all factors of each Mi , then U cogenerates every factor of ⊕ I Mi .

76


Proof. We first show that if U R is M1 ⊕ M2 -injective and cogenerates all factors of M1 and of M2 , then U R cogenerates all factors of M1 ⊕ M2 . Suppose f : M1 ⊕ M2 → L is an epimorphism and that 0 "= x ∈ RejU (L). Then x = f (m 1 ) + f (m 2 ) where m i ∈ Mi . If f (m 1 ) ∈ f (M2 ), then, since f (M2 ) is U -torsionless, there is an α ∈ Hom R ( f (M2 ), U ) with α(x) "= 0. But then, since U R is L-injective, α extends to L , contradicting the choice of x. Hence f (m 1 ) "∈ f (M2 ). Thus, since L/ f (M2 ) is an epimorphic image of M1 and is hence U -torsionless, there is a β ∈ Hom R (L/ f (M2 ), U ) with β( f (m 1 ) + f (M2 )) "= 0, and this induces a map in Hom R (L , U ), which contradicts the choice of x. It follows that the lemma is true if I is finite. And also, since U is L-injective if ⊕i∈I Mi → L is an epimorphism, it then follows for arbitrary I. En route to our characterizations of quasi-duality modules, we determine which quasi-injective modules are costar modules in Proposition 4.3.3. A quasi-injective module is a costar module if and only if it cogenerates all its factors. Proof. This is immediate from Lemma 4.3.2 and Theorem 4.2.7(e). As well as being a key to our characterizations of quasi-duality modules, the following lemma will be employed later to characterize (Morita) duality modules. Lemma 4.3.4. Suppose that S = End(U R ) and S U is a cogenerator. If E R satisfies Cogen(E R ) = Cogen(U R ), then, for any M ∈ Mod-R, U R is M-injective whenever E R is M-injective. In particular, if E(U R ) is U -torsionless, then U R is injective. i

Proof. Let denote the S U R -dual. Fix a monomorphism U R → E RA , and for any γ ∈ (E RA ) denote γ ◦ i ∈ End(U R ) by sγ . Note that ∩{Ker sγ |γ ∈ f (E RA )} = 0. Now suppose 0 → K R → M R is exact where E R is M-injective, and consider the induced exact sequence ( f )

(M) −→ (K ) −→ C −→ 0 with C = Coker ( f ). Suppose α ∈ (K ). Then, since E RA is also M-injective, there is a β : M R → E RA such that f

K −→ M α↓ β↓ U

i

−→ E A


77

commutes. Hence for any γ ∈ (E RA ), sγ α = γ ◦ i ◦ α = γ ◦ β ◦ f = f (γ ◦ β). Thus sγ (K ) ⊆ Im f and hence sγ C = 0 for all γ ∈ (E RA ). If ν ∈ ( S C), we now have sγ (ν(C)) = ν(sγ (C)) = 0 for all γ ∈ (E RA ), so ν(C) ⊆ ∩{Ker sγ |γ ∈ (E RA )} = 0. Thus we have shown that ( S C) = 0; therefore, S C = 0 since S U is a cogenerator. Now we are ready for Theorem 4.3.5. Assume S = End(U R ). The following are equivalent: (a) U R is a quasi-duality module; (b) U R is a costar module such that every factor of S S is U -torsionless and every factor of U R is U -reflexive; (c) U R is quasi-injective and cogenerates all its factors and S U is an injective cogenerator; (d) Gen(U R ) ⊆ Cogen(U R ) and S U is an injective cogenerator. Proof. (a) ⇒ (b). It suffices to prove that any quasi-duality module is a costar module since the defining duality shows that it satisfies the remaining two conditions. So assume that U R is a quasi-duality module and let D R and S C be as in Section 4.2. It is immediate that : D R → S C and that all modules in S C are U -reflexive. And from this latter fact it follows that : S C → D R . If M ∈ D R , then (M) ∈ gen( S S); thus, by hypothesis 2 (M) ∈ gen(U R ), and since M embeds in 2 (M) it is U -reflexive. (b) ⇒ (c). Assume that U R is a costar module satisfying the stated conditions. By Theorem 4.2.7(e) U R is quasi-injective. Also, S U is injective g because, if 0 → K → S → N → 0 is exact, then, since Im g is a factor of U R = (S) and is hence U -reflexive, and since S N is U -torsionless, we can conclude that g is epic by Lemma 4.2.4. Since, also by hypothesis, S U cogenerates all simple left S-modules, S U is an injective cogenerator. (c) ⇒ (a). Assuming (c), U R is costar by Proposition 4.3.3. If 0 → K R → U Rn → X R → 0 is exact, then so is S S n → S K → 0, and therefore K ∈ D R and U n are U -reflexive. Thus the exactness properties of U imply that X is U -reflexive and that X ∈ gen( S S). Now if M ∈ gen(U R ), we may assume that there is also an exact sequence 0 → M −→ X −→ L → 0

78


with X as above. But then X and L , and hence M, are U -reflexive. Also M ∈ gen( S S), being an epimorphic image of X because U is injective relative to X. If N is a submodule of a finitely generated (U -torsionless, by hypothesis) left S-module Y , then Y and Y /N belong to S C. Thus the exactness properties of U yield that N is U -reflexive and that N ∈ gen(U R ). (d) ⇒ (c). Let E R = TrU R (E(U R )). Then E R is quasi-injective ([1, Exercise 18.17]), so E R is U R -injective and clearly Cogen(U R ) ⊆ Cogen(E R ). By hypothesis Cogen(E R ) ⊆ Cogen(U R ). Hence U R is quasi-injective by Lemma 4.3.4. (c) ⇒ (d). This follows from Lemma 4.3.2. The next two results give insight into the structure of the modules involved in a duality between finitely closed subcategories of Mod-R and S-Mod . Lemma 4.3.6. Let A R and S B be full subcategories of Mod-R and S-Mod, respectively, that are closed under finite direct sums and isomorphisms, and suppose that D : AR S B : D is a duality. f

(1) If S B is closed under epimorphic images and 0 → K → M is exact Df in A R , then D M → D K → 0 is exact in S B. f (2) If A R is closed under submodules and N → L → 0 is exact in S B, Df then 0 → DL → D N is exact in A R . Df

g

Proof. (1) If D M → D K → N → 0 is exact, then g ∈ S B and 0 = D(g ◦ D f ) = D D f ◦ Dg. So, since D D f is monic, Dg = 0. But D is faithful, so g = 0 and D f is epic. (2) This is dual. We recall that a module M is linearly compact if the inverse limit of any pλ inverse system of epimorphisms M −→ L λ is also an epimorphism. This is equivalent to the condition that any collection of cosets of submodules of M whose finite intersections are all non-empty has a non-empty intersection itself (see [80, 29.7], for example).


79

Proposition 4.3.7. If U : A R S B : U is a representable duality between finitely closed subcategories of Mod-R and S-Mod, then (1) the modules in A R and S B are linearly compact; (2) if X ∈ A R or S B, then the lattices of submodules of X and U X are iK anti-isomorphic via K → Ker U (i K ) where K −→ X is the inclusion map. Proof. (1) Suppose that M ∈ A R and iλ

pλ

0 → K λ −→ M −→ L λ → 0 is an inverse system of exact sequences. Then by Lemma 4.3.6 i λ

pλ

0 → L λ −→ M −→ K λ → 0 is a direct system of exact sequences in S B. Thus lim − →i λ

lim − →pλ

0 → lim − →L λ −→ M −→ lim − →K λ → 0 is an exact sequence of modules in S B. Now, applying and Lemma 4.3.6 to this sequence, we obtain an exact sequence lim − →i λ

lim − →pλ

2 0 → lim − →K λ −→ M −→ lim − →L λ → 0.

But since converts direct limits to inverse limits (see [69, Theorem 2.27] ∼ or [80, 29.5]) and M and the L λ are U -reflexive, it follows that lim ← − pλ = lim − →pλ is an epimorphism. (2) We omit this proof, which also employs Lemma 4.3.6. Now we can characterize quasi-duality modules without reference to their endomorphism rings. Theorem 4.3.8. A module U R is a quasi-duality module if and only if U R is quasi-injective, finitely cogenerated, linearly compact, and cogenerates all its factor modules. Proof. (⇒). If U R is a quasi-duality module it is finitely cogenerated by Proposition 4.3.7(2). It is quasi-injective and cogenerates all its factor modules by Theorem 4.3.5. And according to Lemma 4.3.7(1) it is linearly compact. (⇐). Assume that U R satisfies the conditions. By Proposition 4.3.3, U R is a costar module inducing a duality : D R S C : as in Section 4.2. According to Theorem 4.3.5, we need only prove that S U is an injective

80


cogenerator. So let L be a left ideal of S with inclusion map i : L → S, and let {L α | α ∈ A} be the family of finitely generated left ideals contained in L with canonical exact sequences nα

iα

0 → L α −→ S −→ S/L α → 0. Consider the canonical exact sequences n α

π

0 → (S/L α ) −→ S −→ Im i α → 0 and j

p

0 → Im i α −→ L α −→ Coker j → 0. In the latter sequence, L α ∈ D R ⊆ cogen(U R ) and by hypothesis and Lemma 4.3.2 Coker j is U -torsionless; thus, by Proposition 4.2.8, j is epic. Now from the commutative diagram 0→

Lα ϕ↓

iα

−→

S δS ↓

nα

−→ 2 n α

π

S/L α δ S/L α ↓

→0

0 → Im i α −→ 2 S −→ 2 S/L α where ϕ = j ◦ δ L α is epic and δ S is an isomorphism, we see that δ S/L α is monic, that is, S/L α is U -torsionless and hence belongs to S C. Thus by Corollary 4.2.9 the sequences n α

i α

0 → (S/L α ) −→ S −→ L α → 0 are exact, and since S ∼ = U R is linearly compact, so are the rows in lim ← −i α

lim ← −L α → 0 ← −S −→ lim ∼ ∼ =↓ =↓ S

lim − →i α

−→ lim − → L α → 0.

Thus, since lim − →i α = i : L → S, we see that S U is injective. To show that S U is a cogenerator it suffices to show that for each maximal left ideal L of S, S/L is U -torsionless. But L = A L α where each S/L α is U -torsionless. Thus {rU (L α ) | α ∈ A} is a family of non-zero submodules of U R , and since U R is finitely cogenerated, there is a finite subset F ⊆ A such that U (S/L) ∼ = rU (L)) = ∩ A rU (L α ) = ∩ F rU (L α ) "= 0.

The following propositions, whose proofs are dual to those of Propositions 2.4.9 and 2.4.10, yield examples of quasi-duality modules.


81

Proposition 4.3.9. If U R is a quasi-duality module with S = End(U R ) and B = BiEnd(U R ), then U B is a quasi-duality module and R is U -dense in B. Proposition 4.3.10. If U B is a quasi-duality module and R is U-dense in B, then U R is a quasi-duality module. Example 4.3.11. If p is any prime integer, then Z p∞ is faithfully balanced and is the minimal injective cogenerator over the linearly compact ring A( p) of p-adic integers. One can show that any subring of A( p) is Z p∞ -dense in A( p). Thus, in particular the Z p∞ is a quasi-duality module over Z. Examples of different flavors can be found in [39] and [40]. One of the problems that arises naturally when considering a duality represented by a bimodule S U R is the characterization of the U-reflexive modules. We conclude this section with a series of results that lead to a characterization of the S U R -reflexive modules in Mod-R relative to a quasi-duality module U R with End(U R ) = S. To do so we need a notion more general than linearly compact. (Unfortunately, in this case we know of no characterization of the U -reflexive modules in S-Mod .) Definition 4.3.12. If M is a module with elements m λ and submodules K λ , the family {m λ , K λ } I is solvable if there is an m ∈ M such that m − m λ ∈ K λ for each λ ∈ I. The family {m λ , K λ } I is finitely solvable if for each finite subset F ⊆ I there is an m F ∈ M such that m F − m λ ∈ K λ for each λ ∈ F. Note that m is a solution for {m λ , K λ } I if and only if m ∈ ∩ I (m λ + K λ ). The connection between solvability and inverse limits is given in Lemma 4.3.13. Let {(L λ ), ( f λµ )}(I,≤) be an inverse system of modules with inverse system of epimorphisms pλ : M → L λ (so that f λµ ◦ pµ = pλ whenever λ ≤ µ in I ). If K λ = Ker pλ (λ ∈ I ), then lim ← − pλ : M → lim ← − L λ is an epimorphism if and only if every finitely solvable family {m λ , K λ } I in M is solvable. Proof. We may consider ← lim − L λ = {(λ ) I | f λµ (µ ) = λ } whenever λ ≤ µ (m) = ( p (m)) in I and lim p λ λ I ([69, Theorem 2.22 and page 55]). ← − (⇒). Consider ( pλ (m λ )) I ∈ I L λ . If λ ≤ µ in I, then, by assumption there is an m 0 ∈ (m λ + K λ ) ∩ (m µ + K µ ). But then f λµ ( pµ (m µ )) = f λµ ( pµ (m 0 )) = pλ (m 0 ) = pλ (m λ ).

82


Thus ( pλ (m λ )) I ∈ lim lim ← − L λ , so since we are assuming that ← − pλ : M → lim ← −L λ is an epimorphism, there is an m ∈ M such that pλ (m) = pλ (m λ ) for all λ ∈ I, that is, m ∈ ∩ I (m λ + K λ ). (⇐). Suppose that (λ ) I = ( pλ (m λ )) I ∈ lim ← − L λ . If F is a finite subset of I , let λ ≤ k ∈ F for all λ ∈ F. Then pλ (m λ ) = λ = f λk (k ) = f λk ◦ pk (m k ) = pλ (m k ), so m k ∈ ∩ F (m λ + K λ ). Thus by hypothesis there is an m ∈ M such that pλ (m) = λ for all λ ∈ I. Taken together, the notions in the following definition suffice to ensure reflexivity. Definition 4.3.14. Let S U R be a bimodule. If M ∈ Mod-R, then (1) M is U -dense (in 2 M) if for each h ∈ 2 M and each finite set F ⊆ M there is an m ∈ M such that h( f ) = f (m) for all f ∈ F; (2) M is U -linearly compact if M is U -torsionless, and lim ← − pλ : M → lim ← − L λ is an epimorphism for every inverse system of epimorphisms pλ : M → L λ with each L λ U -torsionless. Note that if S = End(U R ), then BiEnd(U R ) ∼ = 2 (R R ), via the canonical ∼ isomorphism U R = Hom R (R, U ), and so R R is U -dense if and only if R is U -dense in B = BiEnd(U R ). It follows from Lemma 4.3.13 that, as observed in [41], M is U -linearly compact if and only if every finitely solvable family {m λ , K λ } I with each m λ ∈ M and M/K λ ∈ Cogen(U R ) is solvable. If {m λ , K λ } I is a finitely solvable family in a module M with m F − m λ ∈ K λ for each finite subset F ⊆ I, then, letting K F = ∩λ∈F K λ , one checks that, if I denotes the set of finite subsets of I, then {m F , K F }I is finitely solvable. Indeed, if F1 , . . . , Fn are finite subsets of I, and m ∪nj=1 F j − m λ ∈ K λ , for λ ∈ ∪nj=1 F j then m ∪nj=1 F j − m F = m ∪nj=1 F j − m λ + m λ − m F ∈ K F for = 1, . . . , n. Lemma 4.3.15. If S U R is a bimodule and M R is U-dense and U-linearly compact, then M is U -reflexive. Proof. Let h ∈ 2 M. Since M is U-dense there is, in particular, for each f ∈ M an m f ∈ M with h( f ) = f (m f ). Consider {m f + Ker f | f ∈ M}. If F is a finite subset of M, then there is an m F such that h( f ) = f (m F )


83

for all f ∈ F and so f (m F ) = h( f ) = f (m f ). Thus {m f , Ker f }M is finitely solvable. But then for each finite subset F ⊆ M we let m F − m f ∈ Ker f for f ∈ F and K F = ∩ F Ker f so that the system {m F , K F }I is finitely solvable (where I denotes the set of finite subsets of M). Now M/K F ∈ Cogen(U R ) and the canonical maps p F : M → M/K F form an inverse system of epimorphisms. Thus, since M is U-linearly compact, there is by Lemma 4.3.13 an m ∈ M such that m − m F ∈ K F for all F ⊆ M, and so if f ∈ F, δ M (m)( f ) = f (m) = f (m F ) = f (m f ) = h( f ); therefore, δ M is an isomorphism. Lemma 4.3.16. If gen(U R ) ⊆ Cogen(U R ), then every right R-module is U-dense. Proof. This is proved in [80, 47.6(4)] and is a part of the proof of [81, Theorem 4.1]. Also connecting U-reflexivity and U-linearly compactness, we have Lemma 4.3.17. Let S U R be a bimodule. If M R is a U-reflexive module such that S U is M-injective, then M is U-linearly compact. pλ

Proof. Let M −→ L λ → 0 be an inverse system of epimorphisms with the L λ ∈ Cogen(U R ), and apply to obtain a direct system of monomorphisms pλ

0 → L λ −→ M. Then lim − →pλ

0 → lim − →L λ −→ M is exact, and so since S U is M-injective, we obtain a commutative diagram with exact top row (lim − →pλ )

2 M −→

2 lim ← −( pλ )

(lim − →L λ ) →0 ∼ =↑

2 lim 2 M −→ ← −( L λ ) ∼ lim =↑ ← −(δ L λ ) ↑ lim ← − pλ lim M −→ ← −L λ

84


from which, since ← lim lim −(δ L λ ) is a monomorphism, it follows that ← − pλ is an epimorphism. Finally, we have the promised characterization. Proposition 4.3.18. If U R is a quasi-duality module, then M R is U-reflexive if and only if M is U-linearly compact. Proof. Since U is injective over S = End(U R ), the condition is necessary by Lemma 4.3.17. It is sufficient by Lemmas 4.3.16 and 4.3.15.

4.4. Morita Duality The various forms of duality theories for modules sprang from what has come to be known as Morita duality, which originated in the papers of G. Azumaya [6] and K. Morita [65]. Most of the results in this brief section, and several further details about Morita duality, can be found in [1] and [81]. Definition 4.4.1. A module U R with S = End(U R ) is a (Morita) duality module if U R is faithful and balanced (so that R ∼ = End( S U ), canonically), and the S U R -reflexive modules are closed under submodules and epimorphic images. In this case the faithfully balanced bimodule S U R is said to induce a Morita duality. Note that a bimodule S U R is faithfully balanced if and only if R R and S S are U-reflexive. Theorem 4.4.2. The following statements about a bimodule S U R are equivalent. (a) S U R induces a Morita duality; (b) Every epimorphic image of R R , S S, U R , and S U is U-reflexive; (c) S U R is faithfully balanced and U R and S U are injective cogenerators; (d) S U R is faithfully balanced and U R and S U are cogenerators; Proof. (a) ⇒ (b). This is clear. (b) ⇒ (c). Since every cyclic module and every epimorphic image of U is U-reflexive, and hence U-torsionless, we can apply Lemma 4.2.4 to an exact f sequence 0 → I −→ R R −→ R/I → 0 to see that f is epic. Thus U R is injective and a cogenerator.

4.4. Morita Duality

85

(c) ⇒ (a). This follows easily from the implied exactness of 2 . (c) ⇒ (d) is obvious and (d) ⇒ (c) follows from Lemma 4.3.4. From Theorem 4.3.5 and Theorem 4.4.2 we obtain Corollary 4.4.3. A module U R is a duality module if and only if it is a quasiduality module that is faithfully balanced and a cogenerator. If S U R is a duality module, then both U R and S U are quasi-duality modules. Thus from Proposition 4.3.18 we have the following result of B.J. Müller [66]. Corollary 4.4.4. If S U R induces a Morita duality, then the U-reflexive modules are precisely the linearly compact modules in Mod-R and S-Mod . Corollary 4.4.5. A module U R is a duality module if and only if U R is a balanced linearly compact finitely cogenerated injective cogenerator. Proof. By Theorems 4.3.8 and 4.3.5, if S = End(U R ), then S U is an injective cogenerator Corollary 4.4.6. A module U R is a duality module if and only if R R and U R are linearly compact and U R is a finitely cogenerated injective cogenerator. Proof. If U R is a duality module, then by Corollary 4.4.5 we need only note that R R is linearly compact, and this follows from Theorem 4.4.2 and Corollary 4.4.4. Conversely, U R is a quasi-duality module by Theorem 4.3.8. Thus, since R R is linearly compact, R R is U-reflexive by Proposition 4.3.18, so U R is faithfully balanced and Corollary 4.4.3 applies.

5 Cotilting

We consider dual notions of tilting modules and the Tilting Theorem. We begin by examining a type of theorem that is dual to the Tilting Theorem in a manner similar to the way Morita duality is related to Morita equivalence. We continue our practice of using the term subcategory of Mod-R or S-Mod to indicate a full subcategory that is closed under isomorphisms. And we say that an abelian subcategory of Mod-R or S-Mod is a full subcategory that is closed under finite direct sums and contains the kernels and cokernels of all of its homomorphisms. In addition to the contravariant Hom functors U R : Mod -R S-Mod : S U (both of which we usually denote by ) derived from a bimodule S U R , henceforth, we shall let !U R = Ext1R ( , U ) and !S U = Ext1S ( , U ) to obtain another pair of contravariant functors !U R : Mod -R S-Mod : !S U , both of which will usually be denoted by !.

5.1. Cotilting Theorem As we shall see, there are several versions of “cotilting” modules that lead to a “cotilting theorem” as defined here. Definition 5.1.1. Let A R and S A be abelian subcategories of Mod-R and of S-Mod, respectively, such that R R ∈ A R and S S ∈ S A. Let S U R be a bimodule, and let T R = Ker ∩ A R , F R = Ker ! ∩ A R , S T = Ker ∩ S A, S F = Ker ! ∩ S A. 86

5.1. Cotilting Theorem

87

Then S U R induces a cotilting theorem between A R and S A if the following four conditions are satisfied: (1) (T R , F R ) and ( S T , S F) are torsion theories in A R and S A, respectively; (2) : A R → S F, ! : A R → S T , : S A → F R , ! : S A → T R ; (3) There are natural transformations γ : ! 2 → 1A R and γ : ! 2 → 1S A that, together with the evaluation maps δ : 1A R → 2 and δ : 1S A → 2 , yield exact sequences γM

δM

0→! 2 M −→ M −→ 2 M → 0 and γN δN 0 → ! 2 N −→ N −→ 2 N → 0 for each M ∈ A R and each N ∈ S A; (4) The restrictions : F R S F : and ! : T R S T : ! define category equivalences. Suppose that S U R induces a cotilting theorem between A R and S A. Then, since the abelian subcategories A R and S A contain R R and S S, respectively, they also contain all finitely presented modules. Also S U ∼ = (R R ) ∈ S F and UR ∼ = ( S S) ∈ F R so that Ext1S (U, U ) = 0 and Ext1R (U, U ) = 0, and S U R is faithfully balanced since R R ∈ F R and S S ∈ S F are U -reflexive. Also dual to Theorem 3.2.1 (c), Ker ∩ Ker ! ∩ A R = 0 and Ker ∩ Ker ! ∩ S A = 0. Clearly, condition (4) of Definition 5.1.1 follows from the other conditions. In this section we employ a series of lemmas to show that, given some further restrictions, condition (3) follows from (1), (2), and the first part of (4). For the remainder of this section we assume that A R and S A are abelian subcategories of Mod-R and S-Mod . Moreover, we shall use A to denote both A R and S A, etc. Lemma 5.1.2. Suppose S U R satisfies conditions (1) and the first half of (4) of Definition 5.1.1. If X ∈ A, then δ X is epic and X has torsion subobject (torsion submodule) τ (X ) = Ker δ X . Proof. Let X ∈ A and let T ∈ T be a submodule of X such that X/T ∈ F. If p : X → X/T is the natural epimorphism, then ( p) is an isomorphism by hypothesis, so 2 ( p) is an isomorphism. Also δ X/T is an isomorphism by hypothesis (see Proposition 4.1.2). Since p is an epimorphism, the conclusions

88

Cotilting

follow from the commutative diagram X δX ↓

p

−→

→0

X/T δ X/T ↓

2 ( p)

X −→ (X/T ). 2

2

Lemma 5.1.3. Suppose S U R satisfies conditions (1), (2), and the first half of (4) of Definition 5.1.1. If Y ∈ T ∩ gen(F), then there is an X in F and a commutative diagram 0→

L δL ↓

i

→

p

→

X δX ↓

2 i

Y →0 νY ↓

∂

0 → 2 L −→ 2 X → ! 2 Y → 0 with exact rows in which all vertical maps are isomorphisms. Proof. Let X ∈ F and suppose p

i

0 → L −→ X −→ Y → 0 is exact with Y ∈ T . Then by definition of T and F, we have an exact sequence ∂1

i

0 → X −→ L −→ !Y → 0. But by (2), !Y = 0 = !L , so this sequence yields the desired bottom row. Commutativity of the left square and exactness of the rows allow us to define νY = ∂ ◦ δ X ◦ p −1 . By hypothesis and Lemma 5.1.2, both δ X and δ L are isomorphisms. Thus, so is νY . Lemma 5.1.4. Suppose F is closed under extensions of modules. If 0 → f L −→ X −→ Y → 0 is exact with L ∈ F and Y ∈ gen(F), then X ∈ gen(F). Proof. There is an H ∈ F, an epimorphism g and a pullback of f and g that yield a commutative diagram 0→L→Q→ H →0 ↓ g↓ f

0→L→ X → Y →0 ↓ ↓ 0 0

5.1. Cotilting Theorem

89

with exact rows and columns. Thus, since F is closed under extensions, X ∈ gen(F). Lemma 5.1.5. Suppose F is closed under extensions of modules and S U R satisfies conditions (1), (2), and the first half of condition (4) of Definition 5.1.1. The collection of mappings ν = (νY ) is a natural isomorphism on the full category T ∩ gen(F). Proof. Given a map f : Y → Y between modules in τ ∩ gen(F), there is a pull-back diagram j

n1

0 → L → P → Y → 0 f1 ↓ f ↓ i

0→L→

n

→ Y

X

→0 g

with X ∈ F. Now by Lemma 5.1.4, there is an epimorphism X → P → 0 with X ∈ F. Then, letting L = Ker n 1 ◦ g ≤ X , we obtain a commutative diagram with exact rows i

n

0 → L → X → Y → 0 f ↓ ↓ f ↓ i

0→ L →

n

→ Y

X

→0

in which n = n 1 ◦ g and

f = f 1 ◦ g.

Thus, since Y = 0 = ! X and Y = 0 = ! X , this yields a commutative diagram with exact rows 0→

i

X −→ L −→ !Y → 0 , ↓ !f ↓ f ↓ i

0 → X −→ L −→ !Y → 0 which, in particular, yields a commutative diagram ∂

2 X → ! 2 Y !2 f ↑ 2 f ↑ 2 X

∂

→ !2Y

(see [69, Theorem 6.4, p. 173]). Now, defining νY via the exact sequence 0 → L −→ X −→ Y → 0,

90

Cotilting

we obtain a cube n

−→

X f

% X

%

νY |

−→ Y −→ 0 ↓ ↓ 2 νY ↓ → !2Y X−

↓ 2 X

|

n

−→ 0

Y f

%

!2 f

%

!2Y

−→

in which all squares, except possibly the right one, commute. Then, since n is epic, so does the desired right square commute. Finally, since the sequence defining νY was derived from the sequence defining νY , we need to show that, if νY is derived from 0 → L −→ f f X −→ Y → 0 and νY is derived from 0 → L −→ X −→ Y → 0, then νY = νY . So, following [58, Proposition 9], we employ the commutative pull-back diagram with exact rows, columns, and diagonal 0 ↑

0 0 ↑ % f

0→ L → X → Y 0→ L → % K % 0

g

h

→0

↑ % f ↑ g P → X → 0 ↑ ↑ L = L ↑ ↑ 0 0 h

with P ∈ F since L , X ∈ F. Now let νY0 be derived from 0 → K −→ P −→ Y → 0 so the commutative square f

X → Y 1Y ↑ g ↑ h

P →

Y

yields (see the cube) νY = νY ◦ 1Y = 1!2 Y ◦ νY0 = νY0 . Similarly, νY0 = νY . Finally we can show how condition (3) follows from the other conditions and ones that we shall see are satisfied by various versions of cotilting modules.

5.2. Cotilting Modules

91

Proposition 5.1.6. Suppose S U R satisfies conditions (1), (2), and the first half of condition (4) of Definition 5.1.1. If, in addition, both F R and S F are closed under extensions in Mod-R and S-Mod, T R ⊆ gen(F R ) and S T ⊆ gen( S F), and S U R satisfies Ext2 ( , U ) = 0 on F R and S F, then condition (3) is also satisfied, that is, S U R induces a cotilting theorem between A R and S A. Proof. According to Lemma 5.1.5, the additional conditions imply that all of condition (4) is satisfied. Let X ∈ A. By Lemma 5.1.2, if T = Ker δ X , then T is a torsion module in A. Also, if i T : T → X is the inclusion map, then, since !2 X = 0, we see from δX

iT

0 → T −→ X −→ 2 X → 0 that !i T

0 → ! X −→ !T → 0 is exact, and so ! 2 i T is an isomorphism. Now employing (4) and noting that if f : X → X , then f : Ker δ X → Ker δ X , we see that γ X = i T ◦ νT−1 ◦ ! 2 (i T )−1 is the desired natural transformation. Indeed, if f : X → X, then all the squares commute in i

νT

i

νT

X ← T → !2 T f ↓ f ↓ !2 f ↓ X ← T →

!2 T

!2 i T

−→

!2 X !2 f ↓

!2 i T

−→

!2 X .

5.2. Cotilting Modules In this section we discuss the dual version of generalized tilting modules, which we shall simply refer to as cotilting modules. In the next section we shall establish a cotilting theorem for so-called cotilting bimodules and employ a principal result of this section to characterize their reflexive modules. We begin with a result dual to one that was valuable in our study of generalized tilting modules. For any R module U we denote the kernel of Ext1R ( , U ) by ⊥ U. Proposition 5.2.1. ≤ 1.

⊥

U is closed under submodules if and only if inj . dim .U

Proof. This is dual to the proof of Proposition 1.1.1.

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Cotilting

Now, as promised, we have the definition of a cotilting module. It first appeared in the work of R. Colpi, G. D’Este, and A. Tonolo [28], where they proved the characterization in Proposition 5.2.6 to follow. Definition 5.2.2. A module U R is a cotilting module if Cogen(U R ) = ⊥ U. For a bimodule S U R , we shall, when convenient, continue to denote the contravariant functors Hom R ( , U ) and Hom S (U, ) by and Ext1R ( , U ) and Ext1S ( , U ) by !, adding subscripts when necessary. f

If U R is a cotilting module with S = End(U R ), and 0 → M −→ X −→ L → 0 is exact with X R a S U R -reflexive module (for example X = U n ), then f we have an exact sequence X −→ M → !L → ! X = 0, from which we see that f is epic if and only if L ∈ ⊥ U = Cogen(U R ). Thus, employing Theorem 4.2.7(e), we obtain Proposition 5.2.3. Every cotilting module is a costar module. Dual to Proposition 3.1.3 we have Proposition 5.2.4. If U R is a cotilting module, then Cogen(U R ) = Copres(U R ). Proof. Let M ∈ Cogen(U R ) (so that M = M/ RejU (M)) and apply Lemma 4.2.1. Now we can verify duals to Theorem 3.1.4 and Proposition 3.1.5 to obtain the following two characterizations of cotilting modules. The first of these was obtained by L. Angeleri Hügel, A. Tonolo, and J. Trlifaj in [3]. Theorem 5.2.5. A module U R is a cotilting module if and only if (i) inj . dim .U ≤ 1, (ii) Ext1R (U A , U ) = 0 for all sets A, (iii) an injective cogenerator C R admits an exact sequence 0 → U1 −→ U0 −→ C → 0 with U0 , U1 ∈ Prod(U R ). Proof. Assume that U R is a cotilting module. (i) follows from Proposition 5.2.1. (ii) is obvious from the definition.


93

(iii) Since R R ∈ ⊥ U = Cogen(U R ), it follows that every injective right R-module is an epimorphic image of a direct product of copies of U. Thus there is an exact sequence f

0 → K −→ U A −→ C → 0, and by Proposition 5.2.4 another exact sequence g

0 → K −→ U B −→ L → 0 with L ∈ Cogen(U R ). An argument dual to that of Theorem 3.1.4 using the pushout diagram of f and g completes the proof of this implication. Conversely, assume (i), (ii), and (iii). If M ∈ Cogen(U R ), then an exact sequence 0 → M −→ U A −→ L → 0 yields an exact sequence 0 = Ext1R (U A , U ) → Ext1R (M, U ) → Ext2R (L , U ) = 0. On the other hand, assume that M ∈ ⊥ U and let p : M → M/RejU0 (M) be the canonical epimorphism to obtain, from (iii), a commutative diagram α

Hom R (M, U0 ) −→ Hom R (M, C) Hom R ( p, C) ↑ Hom R ( p, U 0 ) ↑ Hom R (M/RejU0 (M), U0 ) −→ Hom R (M/RejU0 (M), C) in which Hom R ( p, U 0 ) is an isomorphism by definition of RejU0 (M), and α is an epimorphism since U1 ∈ Prod(U R ). But now it follows that Hom R ( p, C) is an isomorphism, and since C is a cogenerator, so is p. Thus M ∈ Cogen(U0 ) ⊆ Cogen(U ). Proposition 5.2.6. A module U R is a cotilting module if and only if (i) inj . dim .U ≤ 1; (ii) Ext1R (U A , U ) = 0 for all sets A; (iii) Ker Hom R ( , U ) ∩ ⊥ U = 0. Proof. If U is cotilting, then (i) and (ii) follow from Theorem 5.2.5, and if Hom R (M, U ) = 0 = Ext1R (M, U ), then M ∈ ⊥ U = Cogen(U ), so M = 0 and (iii) is verified. Conversely, ⊥ U is closed under submodules by (i) and Proposition 5.2.1, so by (ii) Cogen(U R ) ⊆ ⊥ U. If M ∈ ⊥ U, then from 0 → RejU (M) −→ M −→ M/ RejU (M) → 0

94

Cotilting

we obtain the exact sequences ∼ =

(M/ RejU (M)) −→ M → (RejU (M)) → !(M/ RejU (M)) = 0 and ! M → !(RejU (M)) → Ext2R (M/ RejU (M), U ) = 0, from which we see that (RejU (M)) = 0 = !(RejU (M)). Thus by condition (iii), ⊥ U ⊆ Cogen(U R ). As one might hope, if R is an artin algebra, finitely generated cotilting modules are just the artin algebra duals of tilting modules. Proposition 5.2.7. Let R be an artin algebra with D the artin algebra dual. Let U R be finitely generated. Then U R is a cotilting module if and only if R V = DU is a tilting module. Proof. (⇒) If M ∈ gen( R V ), then D M ∈ cogen(U R ) ⊆ ⊥ U. But then Ext1R (V, M) ∼ = Ext1R (D M, U ) = 0, so M ∈ V ⊥ ∩ R-mod . On the other hand, if M ∈ V ⊥ ∩ R-mod, then D M ∈ ⊥ U = Cogen(U R ), and so, being finitely cogenerated, D M ∈ cogen(U R ), that is, M ∈ gen( R V ). Thus R V = DU is a tilting module by Proposition 3.2.3. (⇐) If R V = DU is a tilting module, then U R ∼ = DV clearly satisfies conditions (i) and (iii) of Theorem 5.2.5, and by Corollary 1.3.3 U A ∈ Add(U R ), so, since Ext1R (U (B) , U ) ∼ = Ext1R (U, U ) B for any set B, we see that condition (ii) of Theorem 5.2.5 also holds. Any finitely generated module over an artin algebra is pure-injective in the sense of the following definition. Here we shall see that, according to a recent result of S. Bazzoni [7] (Theorem 5.2.12 below), so are all cotilting modules. Definition 5.2.8. A short exact sequence in Mod-R is said to be pure-exact if Hom R (M, ) preserves its exactness for every finitely presented module M. A module U R is pure-injective if Hom R ( , U ) preserves the exactness of every pure-exact sequence. (For details, see [52].) To reach our present objective of proving that all cotilting modules are pure-injective, we need the following characterization of pure-injectivity.


95

Proposition 5.2.9. A module U R is pure-injective if and only if, for every set A, Hom R ( , U ) preserves the exactness of the canonical short exact sequence 0 → U (A) → U A → U A /U (A) → 0. Proof. See [52, Theorem 7.1]. Thus it is straightforward that, if U R is a cotilting module, then U R is pure-injective if and only if Ext1R (U A /U (A) , U ) = 0 (equivalently, U A /U (A) is U -torsionless) for every set A. The following simple lemma reduces the task of proving that this condition holds for all cotilting modules to the countable case. Lemma 5.2.10. If U N /U (N) ∈ Cogen(U R ), then U A /U (A) ∈ Cogen(U R ) for all sets A. Proof. Under the hypothesis it clearly suffices to show that the U A /U (A) ∈ Cogen(U N /U (N) ). So assume that x = (yα )α∈A + U (A) is a non-zero element of U A /U (A) . Then there is a countably infinite subset B ⊆ A such that yα "= 0 for all α ∈ B, and so the the kernel of the canonical mapping U A /U (A) → U B /U (B) ∼ = U N /U (N) does not contain x. We need two results from set theory. A family of sets is almost disjoint if the intersection of any two distinct elements of the family is finite. Lemma 5.2.11. (1) For any infinite set A there is a family of card(AN ) countable almost disjoint subsets of A. (2) For any cardinal ν there is a cardinal λ ≥ ν such that λℵ0 = 2λ . Proof. (1) Let T = {t : {1, . . . , n} → A | n ∈ N}. Then we have card(T ) = card({F ⊆ A | F is finite}) · ℵ0 = card(A). For every function f ∈ AN , let T f = { f |{1,...,n} | n ∈ N}. If f and g are two different functions in AN , then T f ∩ Tg is finite. Indeed, if m is the least element of N such that f (m) "= g(m), then T f ∩ Tg = { f |{1,...,n} | n < m} has just m − 1 elements. Thus {T f } f ∈AN is a family of card(AN ) countable almost disjoint subsets of T . Considering a bijection of A to T , we have the asserted conclusion. (2) We reproduce the proof given in [43, Lemma 3.2]. Let α1 = ν and for n ≥ 1 define αn+1 = 2αn . Let αn ≥ ν. λ= n∈N

96

Cotilting

Then λ ℵ0 ≥

αn =

n∈N

2αn+1 = 2

n∈N

αn+1

=2

n∈N

αn

= 2λ .

n∈N

Since clearly λℵ0 ≤ 2λ , the conclusion follows. Now we obtain Bazzoni’s theorem. Theorem 5.2.12. If U R is a cotilting module, then U R is pure-injective. Proof. According to Lemma 5.2.10 and the remarks preceding it, it suffices to prove that Ext1R (U N /U (N) , U R ) = 0. Let X = U N /U (N) . Consider an infinite set A and, applying Lemma 5.2.11(1), let {Aβ }β∈AN be an almost disjoint family of card(AN ) countable subsets of A. For each β ∈ AN , let ηβ : U Aβ → U A /U (A) be the restriction to U Aβ of the canonical mapping of U A onto U A /U (A) . Then Ker ηβ = U (Aβ ) and Im ηβ = X β ∼ = X. We now claim that, since the family {Aβ } is almost disjoint, the sum n ηβi (z βi ) = 0 for some AN X β ≤ U A /U (A) is a direct sum. For suppose i=1 n Aβi (A) z βi ∈ U . Then y = i=1 z βi ∈ U . Let F be the support of y and G = ∪1≤i"= j≤n (Aβi ∩ Aβ j ). Then F ∪ G is finite. Fix an index i ∈ {1, . . . , n}. For every α ∈ Aβi \(F ∪ G), the α-component y(α) of y is 0, and hence z βi (α) = 0 too since α "∈ Aβ j whenever j "= i. Thus z βi ∈ U (A) , and hence ηβi (z βi ) = 0 for every i ∈ {1, . . . , n}, proving our claim. Let V be the submodule of U A with V /U (A) = ⊕ AN X β . By Lemma 5.2.11(2), we can choose A such that λ = card(A) ≥ card(U ) and λℵ0 = 2λ . Since Ker(!) = Cogen(U R ) is closed under submodules and direct products, we have !(V ) = 0, so the exact sequence N

0 → U (A) −→ V −→ X (A ) → 0 induces an exact sequence N

(U (A) ) −→ !(X (A ) ) → 0. The first term has card((U ) A ) ≤ λλ = 2λ . The second term is isomorphic N ℵ to !(X ) A . Thus, if !(X ) "= 0, the second term has cardinality at least 2λ 0 = λ 22 , contradicting the existence of this epimorphism. Now we have the following result, which appeared in [59] and will prove useful in the following section.

5.3. Cotilting Bimodules

97

Corollary 5.2.13. If U R is a pure-injective module, then Ker ! is closed under direct limits. In particular, if U R is a cotilting module, then direct limits of U -torsionless modules are U -torsionless. Proof. Consider a direct system {Mi }i∈I in Ker(!). According to [80, 33.9(2)], the canonical exact sequence f

0 → K → ⊕i∈I Mi → lim − → Mi → 0 is pure-exact, so since U R is pure-injective, f is surjective in the induced exact sequence f

(⊕i∈I Mi ) → (K ) → !(lim − → Mi ) → !(⊕i∈I Mi ) = 0 and the first statement follows. The final statement follows from Theorem 5.2.12.

5.3. Cotilting Bimodules According to Propositions 3.2.2 and 5.2.7, if U R is a finitely generated cotilting module over an artin algebra R and S = End(U R ), then S U R is faithfully balanced and S U is a finitely generated cotilting module. Thus we are led to the following Definition 5.3.1. A cotilting bimodule is a faithfully balanced bimodule S U R such that U R and S U are cotilting modules. Remark 5.3.2. Since, according to Proposition 5.2.3, cotilting modules are costar modules, if S U R is a cotilting bimodule, then all finitely generated U -torsionless right R-modules and all finitely generated U -torsionless left S-modules are U -reflexive. We shall employ Proposition 5.1.6 to obtain the cotilting theorem of R. Colpi [26] induced by a cotilting bimodule. To do so we need the following lemmas. Lemma 5.3.3. Let R U S be a cotilting bimodule and let 0 → K −→ M −→ L → 0 be exact. Then (1) if L ∈ Ker ! and any two of K , M, L are reflexive, then so is the third;

98

Cotilting

(2) if K , L ∈ Ker !, then M is U -reflexive if and only if K , L are U reflexive. Proof. Since, in either case L ∈ Ker !, and ! = 0 when S U R is a cotilting bimodule, we have a commutative diagram K → M → L →0 δM ↓ δL ↓ δK ↓ 0→ 2 K → 2 M → 2 L →0 0→

with exact rows. In each case the Five Lemma applies. The next lemma is derived from the work of F. Mantese in [58]. Lemma 5.3.4. Let R U S be a cotilting bimodule, let F denote the class of U -reflexive modules in Mod-R, and let A = {M1 /M2 | Mi ∈ F for i = 1, 2}. Suppose that 0 → K −→ M −→ L → 0 is exact in Mod-R. Then (1) (2) (3) (4)

if if if if

M ∈ F and L ∈ A, then K ∈ F; K ∈ F, then M ∈ A if and only if L ∈ A; M ∈ A, then K ∈ A if and only if L ∈ A; M ∈ A and L ∈ Cogen(U ), then L ∈ F.

Proof. Let 0 → L 2 −→ L 1 −→ L → 0 be exact with L i ∈ F. If M ∈ F, then in the pullback diagram 0 0 ↑ ↑ 0→K −→ M −→ L →0 ↑ ↑ 0→K −→ P −→L 1 →0 , ↑ ↑ L2 = L2 ↑ ↑ 0 0 P ∈ F by the first column and Lemma 5.3.3(2), so K ∈ F by Lemma 5.3.3(2) and the second row. Thus (1) follows. Also if K ∈ F, then P ∈ F by Lemma 5.3.3(2), and so M ∈ A by the first column, proving one implication of (2).


99

For the remainder of the proof we consider the commutative diagram with exact rows and columns and the Mi ∈ F

0 ↓

α

β

0 ↓ K f ↓

0→M2 −→M1 −→ M →0 ↓ g↓ g◦β

0→ X → M1 −→ L →0 ↓ ↓ 0 K ↓ 0 where K ∼ = K by the Snake Lemma. If K ∈ F, then X ∈ F by Lemma 5.3.3(2) applied to the first column. Thus the second row shows that L ∈ A. Thus (2) holds. To verify (3), from the first column, if K ∈ A it follows from (2) that X ∈ A, so since X ∈ Ker !, X ∈ F by Lemma 5.3.3(2). Thus by (2) and the second row, L ∈ A. Also from the second row, if L ∈ A, then X ∈ F by (1), so K ∼ = K ∈ A. Finally, to verify (4), if L ∈ Cogen(U R ), then L ∈ F by Lemma 5.3.3(2) applied to the second row. These last two lemmas provide the tools to prove that the category of R-maps with objects A, as in Lemma 5.3.4, is an abelian subcategory of Mod-R. We also need Lemma 5.3.5. Let S U R be a bimodule and let denote the U -dual. (1) If !2 M = 0, then Coker (δ M ) ∈ Ker !. (2) If M is U -reflexive, then Coker (δ M ) ∈ Ker . Proof. Since M ∼ = (M/ RejU (M)), the exact sequence 0 → M/ RejU (M) −→ 2 M −→ Coker δ M → 0 yields, under an application of , an exact sequence (δ M )

0 → (Coker δ M ) −→ 3 M −→ M → !(Coker δ M ) → 0

100

Cotilting

if 2 M ∈ Ker !. But (δ M ) is epic since (δ M ) ◦ δM = 1M , so (1) holds. If δM is an isomorphism, then so is (δ M ) and (2) follows. Note that the abelian subcategories A R and S A of Mod-R and S-Mod in a cotilting theorem, contain the finitely presented modules in Mod-R and S-Mod, respectively. Theorem 5.3.6. Let S U R be a cotilting bimodule. Let F R and S F denote the classes of U -reflexive modules in Mod-R and S-Mod, respectively, and let A R = {M1 /M2 | M1 , M2 ∈ F R } and

SA

= {N1 /N2 | N1 , N2 ∈ S F}.

Then S U R induces a cotilting theorem (with classes of torsion free objects F R and S F) between A R and S A. Proof. We begin by showing that A R is an abelian subcategory of Mod-R. (This, too, employs the work of Mantese [58].) Clearly A R is closed under finite direct sums. So suppose that f : M → M in A R and consider the pushout diagram where M ∼ = M1 /M2 with the Mi ∈ F R . 0 ↑

0 ↑ f

0→Ker f → M −→ M ↑ n↑ f

0→Ker f → P −→ ↑ M2 = ↑ 0

M1 ↑ M2 ↑ 0.

Applying Lemma 5.3.4(2) to the left column, we see that P ∈ A. Then Im f ∈ F R by Lemma 5.3.4(4), and so Ker f ∼ = Ker f ∈ A R by Lemma 5.3.4(3). Finally, applying Lemma 5.3.4(3) to the exact sequences 0 → Ker f → M → Im f → 0 and 0 → Im f → M → Coker f → 0, we see that A R and, similarly, S A are abelian subcategories of Mod-R and S-Mod, respectively. Clearly F R ⊆ Ker ! ∩ A R . But if M ∈ Ker ! = Cogen(U R ) and 0 → M2 −→ M1 −→ M → 0


101

is exact with the Mi ∈ F R , then by Lemma 5.3.3(1), M ∈ F R . Thus F R = Ker ! ∩ A R . By definition T R = Ker ∩ A R . Next we verify condition (2) of Definition 5.1.1. From an exact sequence f

g

0 → M2 −→ M1 −→ M → 0 with the Mi ∈ F R , we obtain an exact sequence f

g

∂

0 → M −→ M1 −→ M2 −→ ! M → 0

(ex-1)

from which, letting I = Im f, we obtain two short exact sequences g

α

0 → M −→ M1 −→ I → 0

(ex-a)

and β

∂

0 → I −→ M2 −→ ! M → 0

(ex-b)

where f = β ◦ α. In ex-a, I and M are torsionless and M1 is U -reflexive. Thus, by Lemma 5.3.3, M and I are reflexive. Therefore : A R → S F, and by ex-b, ! M ∈ S A. Applying to ex-1, we obtain a commutative diagram 0

→ ∂

M2 δ M2 ↓

f

−→ M1 δ M1 ↓ 2 f

0→! M −→ 2 M2 −→ 2 M1 from which, since the Mi are U -reflexive, we see that ! : A R → S A ∩ Ker = S T . Now we turn to condition (1) of Definition 5.1.1. Since U R ∈ F R ⊆ Cogen(U R ), T R = {T ∈ A R | Hom R (T, F) = 0 for all F ∈ F}. Since !2 = 0 and M is U -reflexive whenever M ∈ A R , we see from Lemma 5.3.5 and Proposition 5.2.6 that δ M is epic for all M ∈ A R . But then the exact sequence δM

0 → Ker δ M −→ M −→ 2 M → 0 shows that Ker δ M ∈ A R , and applying , we obtain an exact sequence δ M

0 −→ 3 M −→ M → (Ker δ M ) → 0 = !2 M,

102

Cotilting

which, since δ M is an epimorphism, shows that (Ker δ M ) = 0, that is, Ker δ M ∈ T R . Thus if M ∈ A R \F R there is a T = Ker δ M ∈ T R such that Hom R (T, M) "= 0. So we see that (T R , F R ) is a torsion theory in A R with τ (M) = Ker δ M . Clearly the first part of condition (4) of Definition 5.1.1 and the remaining conditions of Proposition 5.1.6 are satisfied, and so the theorem is proved. Since, according to Corollary 5.2.13, if S U R is a cotilting bimodule, Cogen(U R ) = Ker !U R and Cogen( S U ) = Ker !S U are closed under direct limits, using a proof similar to Proposition 4.3.18, we have the following characterization of the U -reflexive modules that is a consequence of results in [7] and [59]. Proposition 5.3.7. If S U R is a cotilting bimodule, then M R is U -reflexive if and only if M is U -dense and U -linearly compact. Proof. In view of Lemma 4.3.15, we only need to prove necessity. So assume that M is U -reflexive. Then M is trivially U -dense. So suppose that ιλ

pλ

0 → K λ −→ M −→ L λ → 0 is an inverse system of exact sequence with M reflexive and L λ ∈ Cogen(U R ) = Ker !. Then, according to Lemma 5.3.3, each K λ and each L λ is reflexive. An application of yields a direct system of exact sequences pλ

ιλ

0 → L λ −→ M −→ K λ → 0 so that lim − →ιλ

lim − →pλ

0 → lim − →L λ −→ M −→ lim − →K λ → 0 is exact. Now, since by Corollary 5.2.13 lim − →K λ ∈ Cogen( S U ) = Ker !, we have a commutative diagram with exact rows (lim − →ιλ )

(lim − →pλ )

2 lim ← − ιλ

2 lim ← − pλ

2 0 → (lim − →L λ )→0 − →K λ ) −→ M −→ (lim ∼ ∼ ↑ = =↑ 2 0 → lim ← − K λ ∼ =↑

0→

lim ← −Kλ

−→ 2 M −→ ∼ =↑ lim ← −ιλ

−→

so that lim ← − pλ is an epimorphism.

M

lim ← − pλ

−→

2 lim ← − L λ ∼ =↑

lim ← −L λ

5.4. Cotilting via Tilting and Morita Duality

103

5.4. Cotilting via Tilting and Morita Duality Here, following [29], we present examples of cotilting bimodules over noetherian serial rings that are not finitely generated. To do so we first consider when the Morita dual of a tilting bimodule is a cotilting bimodule. Suppose that A VS is a tilting bimodule, and let H = Hom A (V, ), H = Ext1A (V, ), T = (V ⊗ S ), T = Tor1S (V, ) so that H : Ker H Ker T : T

and H : Ker H Ker T : T

are pairs of category equivalences, as in the Tilting Theorem 3.5.1. Suppose further that a bimodule A W R induces a Morita duality W = W R : L R A L : A W = W between the categories of linearly compact modules in Mod-R and A-Mod (Corollary 4.4.4), and let S U R = Hom A (V, W ), so that SUR

= A W ( A VS ) = H ( A W R ).

Then S U is a cotilting module according to Proposition 2.3.5, and noting that ∼ ∼ A V ∈ A L and A W ∈ Ker H , we see that End(U R ) = End( A V ) = S, canon∼ ically, ([1, Proposition 23.3]) and End( S U ) ∼ End( W ) R, canonically = = A ([1, Proposition 21.2]). Thus S U R is a faithfully balanced bimodule such that S U is a cotilting module. We shall show that in some cases S U R is actually a cotilting bimodule, that is, that U R is a cotilting module. But first we investigate the properties of U R in this general case. Using the usual covariant and contravariant adjointness conditions [1, Propositions 20.7 and 20.6], one easily checks that there are natural isomorphisms U R ∼ = H ◦ W R : Mod -R −→ S- Mod

(delta-1)

S U ∼ = A W ◦ T : S- Mod −→ Mod -R.

(delta-2)

and

To obtain analogous descriptions of the !’s we need the case n = 1 of the following lemma in which the first isomorphism is [11, Proposition 5.1, page 120], and the second can easily be obtained by applying Hom A ( , Hom R (M, W )) ∼ = Hom R (M, Hom A ( , W )) to a projective resolution of A V.

104

Cotilting

Lemma 5.4.1. Let S N and M R be modules and A VS and A W R be bimodules. (1) If A W is injective, then there are natural isomorphisms Hom A (TornS (V, N ), W ) ∼ = ExtnS (N , Hom A (V, W )) for n = 1, 2, . . . . (2) If A W and W R are both injective, then there are natural isomorphisms ExtnA (V, Hom R (M, W )) ∼ = ExtnR (M, Hom A (V, W )) for n = 1, 2, . . . . Now we see that there are also natural isomorphisms !U R ∼ = H ◦ W R : Mod -R −→ S- Mod

(gamma-1)

!S U ∼ = A W ◦ T : S- Mod −→ Mod -R.

(gamma-2)

and

As we noted above, S U is a cotilting module when S U R is the Morita dual of a tilting bimodule. Next, we observe that S U R is nearly a cotilting bimodule. Proposition 5.4.2. Suppose that A W R induces a Morita duality and A VS is a tilting bimodule. If S U R = Hom A (V, W ), then S U R is a faithfully balanced bimodule such that S U is a cotilting module, and U R satisfies (1) There is an exact sequence 0 → U R −→ W0 −→ W1 → 0 with W0 , W1 ∈ add(W R ). In particular, inj . dim .U R ≤ 1 and U R is finitely cogenerated and linearly compact; (2) !U R (M) = 0 whenever M R → U n , for n = 1, 2, . . . . In particular, Ext1R (U, U ) = 0; (3) There is an exact sequence 0 → U1 −→ U0 −→ W R → 0 with U0 , U1 ∈ add(U R ). Proof. (1) and (3) follow from applications of A W to the exact sequences 0 → P1 −→ P0 −→ A V → 0 with P0, P1 ∈ add( A A) and 0 → A A −→ V0 −→ V1 → 0 with V0 , V1 ∈ add( A V ) that exist because A V is a tilting module. According to (1), inj . dim .U R ≤ 1, so !U R is right exact. Thus if M R → U n , there is an exact sequence !U R (U )n → !U R (M) → 0. But !U R (U ) ∼ = H ◦ W R ( A W (V )) ∼ = H (V ) = 0.


105

As we have just seen, the module U R that we are considering satisfies conditions (i) and (iii) of the characterization of cotilting modules in Theorem 5.2.5. Thus U R is a cotilting module when condition (ii) of that theorem holds. Corollary 5.4.3. If S U R is as in Proposition 5.4.2, then S U R is a cotilting bimodule if and only if Ext1R (U A , U ) = 0 for all sets A. Suppose R is a noetherian serial ring. Then either R is artinian, R has no artinian direct summands, or R is a ring direct sum R = R1 + R2 such that R1 has no artinian ring direct summands and R2 is artinian. Since a module U R is a cotilting module if and only if the component of U belonging to each each indecomposable ring direct summand of R is a cotilting module, to show that U R is a cotilting module, we may assume that R is indecomposable and either artinian or not artinian. We shall show that if R has self-duality induced by a bimodule R W R and R VS is a tilting bimodule, then S U R = W (V ) is a cotilting bimodule. In view of the preceding discussion, to do so we need only show that Ext1R (U A , U ) = 0 for all sets A, which is a fact that will become apparent from Lemma 5.4.4. If U R is a finitely cogenerated module over a noetherian serial ring such that Ext1R (U, U ) = 0, then Ext1R (U A , U ) = 0 for any set A. Proof. Assume that R is indecomposable and Ext1R (U, U ) = 0. If R is artinian, then U R is finitely generated and, according to Propositions B.1.6 and B.1.7, if S = End(U R ), then S is artinian and S U ∼ = Hom R (R, U ) is finitely generated. Thus by Proposition 1.3.2 U A # Add(U). But then, Ext1R (U A , U ) = 0. If R is not artinian, then we may assume that U R = E i 1 ⊕ · · · ⊕ E i k ⊕ M1 ⊕ · · · ⊕ M where the E i j are injective envelopes of their socles and the Mi are of finite length. Then we see from Proposition B.2.2 and the argument in the first paragraph of this proof that (B ) (B ) (C) (D ) (D ) UA ∼ = E i 1 1 ⊕ · · · ⊕ E i k k ⊕ E 0 ⊕ M1 1 ⊕ · · · ⊕ M

for some sets B j , C, Di . Now to see that Ext1R (U A , U ) = 0 we need only check that each Ext1R (E 0 , Mi ) = 0. To do so, observe that, by Proposition

106

Cotilting

B.2.1, there is an injective resolution f

0 → Mi −→ E j −→ E k → 0 with E j and E k indecomposable and artinian. Then we need to show that Hom R (E 0 , E j )

Hom R (E 0 , f )

−→

Hom R (E 0 , E k ) → 0

is exact. So, let β ∈ Hom R (E 0 , E k ) with K = Ker β. Then by Proposition B.2.1 there is an m ∈ N such that E 0 /K J m ∼ = E i1 , and since Ext1R (E i1 , Mi ) = 0 Hom R (E 0 /K J m , E j )

Hom R (E 0 /K J m , f )

−→

Hom R (E 0 /K J m , E k ) → 0

is exact. Thus, letting η : E 0 → E 0 /K J m be the natural map, we have a map γ ∈ Hom R (E 0 /K J m , E j ) and a commutative diagram E0 η↓

β

−→E k β

E 0 /K J m −→E k γ ↓ Ej

f

−→E k

so that Hom R (E 0 , f )(γ ◦ η) = f ◦ γ ◦ η = β ◦ η = β. Now, as promised, we have Proposition 5.4.5. Let R be a noetherian serial ring with self-duality induced by a duality bimodule R W R . If R VS is a tilting bimodule, then S U R = W (V ) is a cotilting bimodule. Proof. Since U R is finitely cogenerated and Ext1R (U, U ) = 0 by Proposition 5.4.2, Lemma 5.4.4 and Corollary 5.4.3 complete the proof. Non-artinian noetherian serial rings with self-duality are rather abundant (see Appendix B), and so we can obtain examples of non-finitely generated cotilting modules. Indeed, if in Example 3.7.12 we insist that D = K [[x]] (or any other linearly compact noetherian uniserial ring), let R VS be the given tilting module and let R W be the minimal cogenerator, then R W R induces a self-duality (Proposition B.2.3), and so S U R = W (V ) is a cotilting bimodule


107

that is artinian but not finitely generated. Perhaps a simpler example will be edifying. Example 5.4.6. Let D be a linearly compact local noetherian serial ring with maximal ideal m, let J be the radical of

D D 1 0 0 0 R= with e1 = and e2 = , m D 0 0 0 1 and let Si = Rei /J ei and Ti = ei R/ei J, for i = 1, 2. Here, as in Example 3.7.12, if P2 = Re2 , we see that V = P2 ⊕ S2 is a tilting module with, as one may calculate,

D C S = End( R V ) ∼ = 0 C with C = D/m. Now R has self-duality induced by a bimodule R W R , and by Proposition 5.4.5 S U R = W (V ) is a cotilting bimodule. We can write RW

= Q 1 ⊕ Q 2 and W R = E 1 ⊕ E 2

where Q i = E(Si ) and E i = E(Ti ). Let us assume that W (P2 ) = E 2 and, consequently, W (S2 ) = T2 . Then UR ∼ = E 2 ⊕ T2 . On the other hand, letting maps operate on the opposite side of modules, SU

∼ = Hom R (V, W )

Hom R (P2 , Q 1 ) Hom R (P2 , Q 2 ) ∼ = Hom R (S2 , Q 1 ) Hom R (S2 , Q 2 )

I I ∼ = 0 C

where, after identifying D = e2 Re2 , we see that D I = E( D C) ∼ = = e2 Q 1 ∼ e2 Q 2 , the unique artinian uniserial D-module of infinite length. Moreover it is interesting to check that the following diagrams indicate the structure of S and U.

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Cotilting

Structure of S : 1 ↓ 1 S f1 : ↓ 1 ↓ .. .

2 S f2 : ↓ 1

1 ↓$ 1 2 f1 S : ↓ 1 ↓ .. .

f2 S : 2

Structure of U : .. . ↓ 1 ↓ UR : 2 ⊕ 2 ↓ 1 ↓ 2

.. .. . . ↓ ↓ 1 1 SU : ↓ ⊕ ↓ 1 2. 1 ↓' ↓ 1 1

5.5. Weak Morita Duality As a consequence of Corollary 4.4.4, a Morita duality is a duality between the categories L R and S L of linearly compact modules in Mod-R and S-Mod with R R ∈ L R and S S ∈ S L. In any case L R and S L are closed under epimorphic images, submodules, and extensions [81, Proposition 3.3]. Since, when there is a Morita duality between them, L R and S L are precisely the U -reflexive modules relative to a faithfully balanced module S U R , we now focus on just the closure properties of L R and S L. We are led to the following more general notion. Definition 5.5.1. A weak Morita duality (WMD) is a duality D R : F R S F : DS between full subcategories F R of Mod-R and S F of S-Mod such that (1) R R ∈ F R and S S ∈ S F; (2) F R and S F are closed under submodules and extensions.

5.5. Weak Morita Duality

109

According to a theorem of Morita [1, Theorem 23.5], we may assume that D R = Hom R ( , U ) = U R (= ) and D S = Hom S ( , U ) = S U (= ) for a faithfully balanced bimodule S U R with U R ∈ F R and S U ∈ S F and that all modules in F R and S F are U -reflexive; then we say that S U R induces a weak Morita duality (WMD) between F R and S F. A WMD between the subcategories of all U -reflexive modules in Mod-R and S-Mod is called a generalized Morita duality (GMD). Proposition 5.5.2. A bimodule S U R is a duality module if and only if it induces a weak Morita duality between categories C R and S C, which are closed under epimorphic images and contain R R and S S, respectively. Proof. See [1, Theorem 24.1]. Note that when a subcategory F of Mod-R is closed under extensions (and so finite direct sums), gen(F ) is the collection of epimorphic images of modules in F. Suppose now, for the next two results, that the S U R -dual induces a WMD between F R and S F, and that f

g

0 → K −→ M −→ X → 0 is exact with M (and hence K ) ∈ F R (or S F). Then, according to (1) of the following proposition, the sequence g

f

0 → X −→ M −→ K → ! X → 0

(#) is also exact.

Proposition 5.5.3. If S U R induces a WMD between F R and S F, then (1) Ext1R (M, U ) = 0 for all M ∈ F R ; (2) Id(U R ) ≤ 1; (3) Ker ∩ Ker ! ∩ gen(F R ) = 0. Proof. (1) Since S S is U -reflexive, so is U R ∼ = S S. Thus = ( S S) and (U R ) ∼ if M ∈ F R and 0 → U −→ X −→ M → 0 is exact, then by Lemma 4.2.4, so is 0 → M −→ X −→ U → 0. But this sequence is split exact, so, since U, X, and M are all U -reflexive, 0 → U −→ X −→ M → 0 splits.

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Cotilting

(2) If I ≤ R R , then according to (1) Ext2R (R/I, U ) = Ext1R (I, U ) = 0. Thus, if 0 → U −→ E −→ C → 0 is exact in Mod-R with E injective, then Ext1R (R/I, C) = Ext2R (R/I, U ) = 0, so C is injective by Baer’s Criterion. For (3), in the exact sequence (#), if X = 0 = ! X, f and hence 2 f are isomorphisms. But then, since δ M and δ K are isomorphisms, so is f, that is, X = 0. Throughout the remainder of this section, when a result or proof refers symmetrically to both F R and S F, we shall simply denote both categories by F. Proposition 5.5.4. Let S U R induce a WMD between F R and S F, and suppose that X ∈ gen(F). Then (1) (2) (3) (4) (5)

!X = 0; ! X = 0; δ X is epic; (Ker δ X ) = 0; ! X = 0 if and only if X ∈ F.

Proof. Since is left exact we see, as in (#), that : gen(F R ) → S F. Thus (1) follows from Proposition 5.5.3. For (2), applying to the sequence (#) we obtain an exact sequence 2 f

0 → ! X −→ 2 K −→ 2 M in which 2 f = δ M ◦ f ◦ δ −1 K is monic, that is, ! X = 0. Let I = Im f in (#). Since I ⊂ K ∈ F, by (1) of Proposition 5.5.3 ! I = 0. Thus from the commutative diagram with exact rows g

M −→ X → 0 ∼ δX ↓ =↓ 2 g

2 M −→ 2 X → 0 = ! I we see that δ X is epic and (3) holds. Now, by (3) we have an exact sequence δX

0 → Ker δ X −→ X −→ 2 X → 0,


111

which, since by (1) !2 X = 0, yields an exact sequence δ X

0 → 3 X −→ X −→ (Ker δ X ) → 0, so (Ker δ X ) = 0 because δ X ◦ δX = 1X . Finally, for (5), if X ∈ F, then ! X = !2 X = 0 by (1). Conversely, suppose ! X = 0. Then, applying ! to δX

0 → Ker δ X −→ X −→ 2 X → 0, we obtain, using (1) and Proposition 5.5.3, an exact sequence 0 = !2 X → ! X −→ !(Ker δ X ) −→ Ext2 (2 X, U ) = 0, so !(Ker δ X ) ∼ = ! X = 0. But Ker δ X ∈ gen F, and by (4) (Ker δ X ) = ∼ 2 0 also. Thus by Proposition 5.5.3, Ker δ X = 0, and so by (3) X = X ∈ F. Since they are U -reflexive, the modules in F R and S F are U -torsionless when S U R induces a WMD between them. In fact F R and S F are precisely the U -torsionless modules in gen(F R ) and gen( S F ). Corollary 5.5.5. If S U R induces a WMD between F R and S F, then F R and S F consist of the U -torsionless modules in gen(F R ) and gen( S F ). In particular, the finitely generated U -torsionless left R-modules and right S-modules all belong to F R and S F, respectively. Proof. Proposition 5.5.4(3) shows that, if X ∈ gen F is U -torsionless, then X∼ = 2 X ∈ F. Now we are in position to prove that a bimodule that induces a WMD between F R and S F induces a cotilting theorem between gen(F R ) and gen( S F ). Theorem 5.5.6. Let S U R induce a WMD between F R and S F. Then S U R induces a cotilting theorem (with classes of torsion free objects F R and S F ) between gen(F R ) and gen( S F ). Proof. Clearly gen(F R ) is an abelian subcategory of Mod-R. According to Proposition 5.5.4 (5), F R = Ker ! ∩ gen(F R ), and, by definition, T R = Ker ∩ gen(F R ). Now Hom R (T, C) = 0 for all T ∈ T R , C ∈ F R , since F R consists of U -torsionless modules, and if Hom R (X, C) = 0, for all C ∈ F R , then X = 0. If X ∈ gen(F R ) and Hom R (T, X ) = 0 for all T ∈ T R , then

112

Cotilting

Ker δ X = 0 by Proposition 5.5.4 (4), and so X ∈ F R by Corollary 5.5.5. Also by Proposition 5.5.4 (4), Ker δ M = RejU (M) is the torsion submodule of M whenever M ∈ gen(F R ). Since analogous results hold in gen( S F ), (1) of Definition 5.1.1 is verified. Condition (2) follows from Proposition 5.5.4 (1) and (2). The first part of condition (4) of Definition 5.1.1 holds since S U R induces a WMD between F R and S F. According to Proposition 5.5.3, Ext2R ( , U ) = 0, and the remaining conditions of Proposition 5.1.6 are obvious. The preceding results also yield Proposition 5.5.7. If S U R induces a WMD, then U R and S U are costar modules. In fact, if M ∈ F and f

g

0 → K −→ M −→ X → 0 is exact, then f is epic if and only if X ∈ F. Proof. From the exact sequence (#) we see that f is epic if and only if ! X = 0, so the last statement follows from Proposition 5.5.4(5). But by Corollary 5.5.5, this is equivalent to X ∈ Cogen(U ), and thus the first statement follows from Theorem 4.2.7(e). In view of Proposition 5.5.7, the next corollary follows from Proposition 4.2.11 and Corollary 2.4.13, according to which a faithful costar module over an artin algebra is a cotilting module. Corollary 5.5.8. Let U R be finitely generated over an artin algebra R. If S U R induces a WMD, then U R is a cotilting module. Conversely, if U R is a cotilting module with S = End(U R ), then S U R induces a WMD between the categories F R and S F of finitely generated U -torsionless left R- and right S-modules. We shall show that, when R is an artin algebra and S U R is a finitely generated cotilting bimodule, then the abelian categories of Theorem 5.3.6 are just the finitely generated modules. Immediately we have Corollary 5.5.9. If S U R is a finitely generated cotilting bimodule over an artin algebra, then S U R induces a cotilting theorem between mod-R and S-mod.


113

Our next goal is to show that this is the “largest possible” cotilting theorem in the artin algebra case. Suppose R is an artin algebra with center K and dual D, and let R W R = D( R R R ). Then R W R is a balanced two-sided injective cogenerator, and by adjointness there are natural isomorphisms W ∼ =D on both Mod-R and R-Mod . Let U R be a finitely generated module with S = End(U R ). Then, as we saw in Proposition 5.2.7, U R is a cotilting module if and only if there is a tilting bimodule R VS with S U R = D( S VR ), and then S U R is a cotilting bimodule. When a bimodule S U R induces and ! dualities U R : F R S F : S U

and !U R : T R S T : !S U ,

as in a cotilting theorem (see Definition 5.1.1), for convenience, let us say that the modules in F R and S F are U -reflexive and the modules in T R and S T are !U -reflexive. Also, if there are no maps other than 0 from M to U, we say that M is U -torsion. Theorem 5.5.10. Let R be an artin algebra and let U R be a finitely generated cotilting module with S = End(U R ). If M is either a right R-module or a left S-module,then (1) M is U -reflexive if and only if M is finitely generated and U -torsionless; (2) M is !U -reflexive if and only if M is finitely generated and U -torsion. Proof. Let R VS be a tilting module with S U R = D(V ) and recall from the Tilting Theorem 3.5.1 that the bimodule S VR induces pairs of equivalences H : Ker H Ker T : T

and H : Ker H Ker T : T .

If M R is finitely generated and U -torsionless, then D M ∈ gen( R V ) = Ker H ∩ R-mod, and so from (delta-1) and (delta-2) on page 103 we have natural isomorphisms S U ◦ U R (M) ∼ = D 2 (M) ∼ = DT H D(M) ∼ = M. If M R is U -torsion, then D M ∈ Ker H , and so from (gamma-1) and (gamma-2) on page 104 we have natural isomorphisms !S U ◦ !U R ∼ = D 2 (M) ∼ = M. = DT H D(M) ∼

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Cotilting

Thus, since VS is also a tilting module we see that the conditions in each part of the theorem are sufficient. To see that the conditions are each necessary recall that, since R is an artin algebra, the D-reflexive R-modules are just the finitely generated R-modules [1, Theorem 24.8]. (1) According to Corollary 1.3.3, if M R is U -torsionless, then there is an embedding 0 → M → U (A) of M into a direct sum of copies of U. But then we have an epimorphism RV

A

∼ = D(U R ) A → D(M) → 0,

and since R V A ∈ Gen( R V ) = Ker H (Proposition 1.2.3) so is D(M) whenever M R is U -torsionless. Thus for any such M R we finally have, employing the isomorphisms (delta-1) and (delta-2) on page 103, S U ◦ U R (M) ∼ = D 2 (M). = DT H D(M) ∼ Thus if M R is U -reflexive, then M R is D-reflexive and U -torsionless, or equivalently, M R is finitely generated and U -torsionless. (2) From (delta-1) on page 103 we see that D(M) ∈ Ker H whenever U R (M) = 0. So from the isomorphisms (gamma-1) and (gamma-2) on page 104 we have, for any M ∈ Mod-R, !S U ◦ !U R (M) ∼ = DT H D(M) ∼ = D 2 (M). Thus if M ∈ Mod-R is !U -reflexive, then M is D-reflexive, and hence finitely generated. But then !U (M) is finitely generated, and M is U -torsion since U R M ∼ = U R !S U !U R (M) ∼ = H D DT !U R (M) = 0.

Now we are in position to establish the connection between GMD’s and cotilting modules over artin algebras. Theorem 5.5.11. Let R be an artin algebra and let U R be a finitely generated module with S = End(U R ). Then U R is a cotilting module if and only if S U R induces a generalized Morita duality. Proof. If U R is a finitely generated cotilting module, then according to Theorem 5.5.10(1), the U -reflexive modules are just the finitely generated U torsionless modules. Thus s U R induces a GMD. The converse follows from Corollary 5.5.8, since a GMD is a WMD.

5.6. Finitistic Cotilting Modules and Bimodules

115

Actually, Morita duality and the cotilting dualities of Theorem 5.5.11 are the only examples of generalized Morita dualities that we are aware of. 5.6. Finitistic Cotilting Modules and Bimodules We now consider WMD’s in the noetherian case. We begin with a specialized definition of a cotilting module. Definition 5.6.1. A finitistic cotilting module is a module U R in mod-R such that (1) inj . dim .U R ≤ 1; (3) Ext1R (U, U ) = 0; (3) Ker(U R ) ∩ Ker(!U R ) ∩ mod-R = 0. We pointed out in Example 3.7.12 and Proposition 5.4.5 that noetherian serial rings provide examples of tilting modules and cotilting bimodules over non-artinian rings. They also supply examples of finitistic cotilting modules. Proposition 5.6.2. If R is right noetherian and right hereditary, and VR is a tilting module, then VR is a finitistic cotilting module. Proof. Suppose VR is a tilting module that is not a finitistic cotilting module; then by Definition 5.6.1 there is a non-zero, finitely generated module M R such that V (M) = 0 = !V (M). Also by Theorem 3.2.1 there is an exact sequence 0 → R → V0 → V1 → 0 with Vi ∈ add(VR ). Hence Hom R (M, R) = 0, and the exactness of Hom R (M, V1 ) → Ext1R (M, R) → Ext1R (M, V0 ) shows that Ext1R (M, R) = 0. But since R is right hereditary, considering a projective presentation of M R , we see that this is not possible. If R is an artin algebra and U R is a finitely generated cotilting module, then in the exact sequence 0 → U1 −→ U0 −→ C → 0 of Theorem 5.2.5 one can have the Ui ∈ add(U R ). Thus a dual argument serves to verify the converse of Proposition 5.6.2 when R is a hereditary artin algebra. Lemma 5.6.3. If S U R is a bimodule such that U R and S U are noetherian, then Cogen(U R ) ∩ mod-R = cogen(U R ) and Cogen(U R ) ∩ S-mod = cogen( S U ).

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Cotilting

Proof. The proof is straightforward. Regarding the finitely generated reflexive modules relative to a finitistic cotilting module we have Proposition 5.6.4. Suppose R is right noetherian and U R is a finitistic cotilting module with End(U R ) = S such that S U is noetherian. If M is a module in mod-R, then the following are equivalent. (a) M R is U -reflexive; (b) M R is U -torsionless; (c) !(M R ) = 0. Proof. (a) ⇒ (b) is clear. Assuming (b) and referring to Lemma 5.6.3, the hypothesis that U R is a finitistic cotilting module clearly implies (c). Similarly, assuming (c), we see that !(RejU (M R )) = 0, and since (b) implies (c) we have !(M/ RejU (M R )) = 0. Hence the canonical sequence 0 → (M/RejU (M R )) → (M) → (RejU (M R )) → 0 is exact, so we can conclude that (RejU (M R )) = 0. Since U R is a finitistic cotilting module and R is right noetherian, we conclude that RejU (M R ) = 0, so M R is U -torsionless (and we have proved (c) ⇒ (b)). We now apply Lemma 4.2.1, noting that we can take A finite since (M R ) embeds in a finite direct sum of copies of S U, to obtain an exact sequence f

0 → MR → U n → L R → 0 such that ( f ) is an epimorphism and !L → !(U n ) = 0. But then since (c) ⇒ (b), L R is U -torsionless, so M R is U -reflexive by Lemma 4.2.3. Now we can establish the connection between weak Morita duality and finitistic cotilting modules. Theorem 5.6.5. The following are equivalent for any bimodule S U R . (a) S U R induces a WMD between the classes of U -torsionless modules in mod-R and S-mod . (b) (i) S U R is faithfully balanced and R R , S S, U R , and S U are noetherian, (ii) Ext1R (U, U ) = 0 = Ext1S (U, U ), (iii) inj . dim .U R ≤ 1 and inj . dim . S U ≤ 1.

5.6. Finitistic Cotilting Modules and Bimodules

117

(c) (i) S U R is faithfully balanced and R R , S S, U R , and S U are noetherian, (ii) S U R induces a duality between the classes of U -torsionless modules in mod-R and S-mod . (d) (i) R is right noetherian, S is left noetherian, and U R and S U are finitely generated and faithful, (ii) Every finitely generated U -torsionless module is U -reflexive. (e) (i) S U R is faithfully balanced, R is right noetherian, and S is left noetherian, (ii) S U and U R are finitistic cotilting modules. (f) (i) S = End(U R ) is left noetherian, R is right noetherian, and S U ∈ S-mod, (ii) U R is a finitistic cotilting module. Proof. (a) ⇒ (b). (i) is clear since the classes of finitely generated U -reflexive modules are closed under submodules and include R R and S S. (ii) and (iii) follow from Proposition 5.5.3. (b) ⇒ (c). By Lemma 5.6.3 it suffices to show that all modules in cogen(U ) are U -reflexive. Suppose M ∈ cogen(U R ). By (ii) and (iii) it follows that !(M) = 0. Since (K ) ∈ cogen( S U ), we also have !K = 0. Hence 2 preserves the exactness of an exact sequence of the form 0 → K → R n → M → 0, and since M is U -torsionless and R n is U -reflexive, we obtain that M is U -reflexive. (c) ⇒ (a). Referring to Lemma 5.6.3, since cogen(U ) is closed under submodules, it suffices to show that cogen(U ) is closed under extensions, and, for this, it suffices to show that cogen(U ) = Ker !∩ mod-R (or S-mod). Let f M ∈ mod-R and suppose 0 → K → R n → M → 0 is exact. If !(M) = 0, then, since K is U -reflexive, M is U -torsionless by Lemma 4.2.3. On the other hand, if M is U -torsionless, then f is epic by Lemma 4.2.4; thus, !(M) = 0. (c) ⇒ (d) is immediate and, since the hypotheses imply that R R , U R , S S, and S U are finitely generated and U -torsionless, (d) ⇒ (c) is straightforward. Now, (e) follows immediately from (b) and (a) via Theorem 5.5.6, and it is clear from the definition that (e) implies (b). Since (e) implies (f) is clear, it suffices to show that (f) implies (d). Assume (f). Since !(R R ) = 0, R R is U -reflexive by Proposition 5.6.4, so (d)(i) is satisfied. Also by Proposition 5.6.4, all U -torsionless modules in mod-R are U -reflexive. Suppose S N is a U -torsionless module in S-mod and let j

0→K →Q→N →0

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be an exact sequence in S-mod with Q projective. Then S Q is U -reflexive and Im(( j)) is U -reflexive since it is a submodule of ( S K ) ∈ cogen(U R ). Thus ( j) is epic by Lemma 4.2.4, so K is U -reflexive by Lemma 4.2.3. Finally, since !(K ) = 0 by Proposition 5.6.4, we see that 2 preserves the exactness of the preceding sequence and obtain that N is U -reflexive. As a corollary to Theorem 5.6.5 we have the following result, which is due to J. P. Jans [51], whose work has provided inspiration for many of the results of this section. Corollary 5.6.6. Let R be a noetherian ring. Then the bimodule R R R induces a duality between the R-torsionless modules in mod-R and R-mod if and only if inj . dim . R R ≤ 1 and inj . dim .R R ≤ 1. According to Corollary 5.5.8, if U R is a finitely generated module with S = End(U R ) over an artin algebra R, then S U R induces a weak Morita duality if and only if U R is a cotilting module. Thus Theorem 5.6.5 also yields Corollary 5.6.7. A finitely generated module U R is a finitistic cotilting module over an artin algebra R if and only if U R is a cotilting module. Definition 5.6.8. A bimodule S U R satisfying the conditions of Theorem 5.6.5 is called a finitistic cotilting bimodule. From Theorem 5.5.6 and Theorem 5.6.5(a) we obtain Corollary 5.6.9. If S U R is a finitistic cotilting bimodule, then S U R induces a cotilting theorem between mod-R and S-mod . Of course Z ZZ is a finitistic cotilting bimodule. The following corollary yields further examples. (See Examples 3.7.7 and 3.7.12.) Corollary 5.6.10. If R is a hereditary noetherian serial ring with tilting bimodule S VR , then S VR is a finitistic cotilting bimodule. Proof. According to Proposition 5.6.2, VR is a finitistic cotilting module, and by Propositions B.1.6 and B.1.7, S is noetherian and S V is finitely generated. Thus Theorem 5.6.5(f) applies.

5.7. U -torsionless Linear Compactness

119

5.7. U -torsionless Linear Compactness Recall that a module M is linearly compact if, for every inverse system of pλ epimorphisms M −→ Mλ , the inverse limit lim ← − pλ is an epimorphism. Here we shall consider a more general notion (a version of which appeared in [42]) and its connection to U -reflexivity in regard to weak and generalized Morita duality and cotilting bimodules. Many of the proofs in this section have their roots in the papers [26] and [29]. Definition 5.7.1. Let S U R be a bimodule and let denote the U -dual. A U -torsionless module M is said to be U -torsionless linearly compact if, for pλ every inverse system of maps {M −→ Mλ } such that Mλ is U -torsionless and lim (Coker( pλ )) = 0 for all λ, the inverse limit ← − pλ has (Coker(lim ← − pλ )) = 0. If S U R is a bimodule, we continue to let denote the U -dual, and we begin with a lemma that applies to costar modules. Lemma 5.7.2. Let S U R be a bimodule. If M is U -torsionless linearly compact iλ and M is a direct limit of a directed system {Nλ −→ M} of U -reflexive submodules Nλ , then M is U -reflexive. iλ

Proof. Given M and {Nλ −→ M} as in the hypothesis, let pλ = i λ ◦ δ M : pλ M → Nλ to obtain an inverse system {M −→ Nλ } in which each Nλ cλ is U -torsionless, and let Nλ −→ Cλ = Coker pλ . From the commutative diagram 0→

Nλ δ Nλ ↓

iλ

−→ M = M δM ↓ 2 i λ

(δ M )

2 Nλ −→ 3 M −→ M with exact rows, we see, since δ Nλ is an isomorphism and (δ M ) ◦ δM = 1M , that ( pλ ) = (δ M ) ◦ 2 i λ is a monomorphism. Thus it follows from the exact sequence ( pλ )

0 → Cλ −→ 2 Nλ −→ M that (Coker pλ ) = 0, so by hypothesis, (Coker(lim ← − pλ )) = 0. Now since lim i is an isomorphism and − →λ ∼ lim ← − pλ = lim ← −((i λ ) ◦ δ M ) = (lim − →i λ ) ◦ δ M ,

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Cotilting

we have ∼ Coker(lim ← − pλ ) = Coker δ M , and so applying to δM

M −→ 2 M → Coker δ M → 0, we see that the epimorphism (δ M ) is an isomorphism and so then is δM . As an immediate consequence of Lemma 5.7.2 we have Proposition 5.7.3. If U R is a costar module with S = End(U R ) and M R is U -torsionless linearly compact, then M is S U R -reflexive. Proof. Since U R is a costar module, by definition every finitely generated U -torsionless left S-module is U -reflexive. Thus Lemma 5.7.2 applies. The class of linear compact modules in Mod-R is closed under submodules, epimorphic images, and extensions (see [81]). Very little is known about the closure properties of the class of U -torsionless linearly compact modules; however, employing the next lemma we shall see that it is closed under U -torsionless epimorphic images. Lemma 5.7.4. Let S U R be a bimodule. If M, N , and L are in Mod-R, f ∈ Hom R (M, N ), and g ∈ Hom R (N , L), then there is an exact sequence Coker f → Coker(g ◦ f ) → Coker g → 0. Hence (1) Coker g ∈ Ker if Coker(g ◦ f ) ∈ Ker , (2) Coker g ◦ f ∈ Ker if Coker f, Coker g ∈ Ker . Proof. Applying the Snake Lemma to the commutative diagram with exact rows M g◦ f ↓ 0 → g ◦ f (M)

f

−→ N → Coker f →0 g↓ g↓ → L → Coker(g ◦ f ) → 0

where g is induced by g, we obtain that Coker g is isomorphic to Coker g, and hence we have the exact sequence as asserted.


121

Now we have the following closure property of the class of U -torsionless linearly compact modules in Mod-R. It seems not even to be known whether the class is closed under finite direct sums. Proposition 5.7.5. Let S U R be a bimodule. If M is U -torsionless linearly compact, L is U -torsionless, and p : M → L with Coker p ∈ Ker , then L is U -torsionless linearly compact. Proof. Suppose that {qλ : L → L λ } is an inverse system with Coker qλ ∈ Ker for all λ. By Lemma 5.7.4(2), Coker(qλ ◦ p) ∈ Ker for all λ, and so since M is U -torsionless linearly compact, Coker(lim ← −(qλ ◦ p)) ∈ ) ◦ p) ∈ Ker . Hence Coker(lim q Ker , that is, Coker((lim q λ λ ← − ) ∈ Ker by ← − Lemma 5.7.4(1). In order to examine the connection between U -torsionless linearly compactness and weak Morita duality, we need the next two lemmas. Lemma 5.7.6. If M R is S U R -reflexive, every U -torsionless M admitting an exact sequence M −→ M −→ C → 0 with C = 0 is U -reflexive, and Coker(i) = 0 for every embedding i S L → M, then M is U -torsionless linearly compact. pλ

Proof. Suppose that {M −→ Mλ } is an inverse system of maps with Mλ ∈ Cogen(U R ), and Cλ = Coker pλ such that each Cλ = 0. From the latter it pλ follows that each map in the direct system {Mλ −→ M} is monic. Thus we have an exact sequence lim − →pλ

0 → lim − →Mλ −→ M. So by hypothesis (Coker((lim − →pλ )) = 0. In the commutative diagram 2 M δM ↑ M

(lim − →pλ )

−→ lim ← − pλ

−→

(lim − →Mλ ) −→ Coker((lim − →pλ )) → 0 ↑ lim ← − Mλ

→

Coker(lim ← − pλ )

→0

with exact rows, the vertical maps are both isomorphisms since M and the Mλ 2 ∼ are U -reflexive and (lim ← −( Mλ ). But then we can complete the − →Mλ ) = lim

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Cotilting

diagram to obtain an isomorphism to which, applying , we have ∼ (Coker(lim − →pλ ))) = 0. ← − pλ )) = (Coker((lim

Theorem 5.7.7. If S U R induces a WMD between F R and S F, then every module in F R and S F is U -torsionless linearly compact. p

Proof. Suppose that M ∈ F R , and 0 → M −→ M −→ C → 0 with M U p torsionless and C = 0. From the latter it follows that M −→ M is monic, so M is U -reflexive. Thus by Lemma 5.3.5, Proposition 5.5.4, and Proposition 5.5.3, M is U -reflexive. If we have an exact sequence i

0 → L −→ M → X → 0, then, since by Proposition 5.5.4 !M = 0, we see that Coker(i) ∼ = ! X. But also by Proposition 5.5.4, ! X = 0. Thus, Lemma 5.7.6 applies. Now we have the tools to characterize WMD’s. Theorem 5.7.8. Let F R and S F be subcategories of Mod-R and S-Mod that are closed under submodules and extensions and contain R R and S S, respectively. Then the S U R -dual induces a WMD between F R and S F if and only if the following five conditions hold: (1) (2) (3) (4) (5)

SUR

is faithfully balanced; : F R S F : ; ! M = 0, if M ∈ F R , and ! N = 0, if N ∈ S F; Ker ∩ Ker ! ∩ gen(F R ) = 0 and Ker ∩ Ker ! ∩ gen( S F) = 0; All modules in F R and S F are U -torsionless linearly compact.

Proof. We know that all the conditions are necessary by Propositions 4.1.5, 5.5.3, and 5.5.4 and Theorem 5.7.7. For sufficiency, let us denote both F R and S F by F. Suppose that X ∈ gen(F ) so that there is an exact sequence f

0 → K −→ M → X → 0 with M and K in F. Then, from the exact sequence 0 → X → M we have X ∈ F, 2 M ∈ F and, since it embeds in 2 X, X/ RejU (X ) ∈ F. In particular, if X ∈ Cogen(U ), then X ∈ F ⊆ Ker !. Now from the exact sequence


123

0 → RejU (X ) −→ X −→ X/ RejU (X ) → 0 we obtain an exact sequence X → (RejU (X )) → !(X/ RejU (X )) = 0 from which it follows that (RejU (X )) = 0. Since right ideals of R (or left ideals of S) belong to F, it follows, as in the proof of Proposition 5.5.3(2) that Id(U ) ≤ 1. Thus if ! X = 0, we have an exact sequence 0 = ! X → !(RejU (X )) → Ext2 (X/ RejU (X )) = 0, and so we have shown that X ∈ Cogen(U ) if and only if ! X = 0. Now from the exact sequence f

M −→ K −→ ! X → 0 we see that f is epic if and only X ∈ Cogen(U ). In particular, letting M = U n , it follows from Theorem 4.2.7(e) that U is a costar module, and thus every finitely generated U -torsionless module is U -reflexive. In particular, if M ∈ F, then every finitely generated submodule of M is U -reflexive. Thus by Lemmas 5.7.2 and 5.3.5 and condition (4) every module in F is U -reflexive, and S U R induces a WMD between F R and S F. Lemma 5.7.9. Let S U R be a bimodule. Assume that U -torsionless modules in mod-R are U -reflexive and suppose M ∈ Mod-R is U -torsionless and M i is U -torsionless linearly compact. If L → M, then (Coker i) = 0, and so L is U -torsionless linearly compact. iλ

Proof. Let{L λ → M} be the upward directed family of finitely generated submodules of L so that L = lim − → L λ . Then we obtain exact sequences i λ

M −→ L λ → Coker (i λ ) → 0, and since M is U -torsionless and the L λ are U -reflexive by hypothesis, it follows that (Coker (i λ )) = 0. Since M is U -torsionless linearly compact, we obtain (Coker ← lim −(i λ )) = 0. But ∼ ∼ (Coker ← lim − →i λ )) = (Coker(i)). −(i λ )) = (Coker (lim And so, by Proposition 5.7.5, L is U -torsionless linearly compact. If S U R induces a Morita duality, then U R and S U are injective cogenerators (Theorem 4.4.2). Thus, in this case, U -torsionless linearly compact is simply linearly compact, ! M = 0 for all M, and Ker = 0.

124

Cotilting

Our concluding characterization of generalized Morita duality complements the characterization, Corollary 4.4.4, of the reflexive modules under a Morita duality as the linearly compact modules. Theorem 5.7.10. Let S U R be a bimodule. Let F R and S F denote the classes of all U -reflexive modules in Mod-R and S-Mod. Then U induces a GMD if and only if (1) (2) (3) (4)

S U R is faithfully balanced, !(M) = 0 for all M in F R and !(N ) = 0 for all N in S F, Ker ∩ Ker ! ∩ gen(F R ) = 0 and Ker ∩ Ker ! ∩ gen( S F) = 0, F R and S F consist precisely of the U -torsionless linearly compact modules in Mod-R and S-Mod .

Proof. First assume that U induces a GMD. Since R R and S S are U -reflexive, S U R is faithfully balanced. Then (2) and (3) follow from Theorem 5.7.8. Also by Theorem 5.7.8, for (4), we need only show that any U -torsionless linearly compact module is U -reflexive. According to Proposition 5.5.7, U R and S U are costar modules. Hence if M in Mod-R is U -torsionless linearly compact, then M is U -reflexive by Proposition 5.7.3. But then M is U -reflexive by Lemma 5.3.5 and Proposition 5.5.3. Conversely, assume (1)–(4). By (1), R R and S S, are U -reflexive. By the symmetry of the hypotheses it suffices to prove that F R is closed under submodules and extensions. For closure under extensions suppose that 0 → M1 → M → M2 → 0 is exact in Mod–R where M1 and M2 are U -reflexive. Since ! M2 = 0 and !M1 = 0 by (2), we can apply twice to obtain the straightforward proof that M is U -reflexive. Finally, suppose that M in Mod-R is U -reflexive and i that L is a submodule of M. Applying to L → M we have the exact sequence i

M → L → Coker i → 0. Note that, since finitely generated free modules are U -reflexive and hence U -torsionless linearly compact by (4), we can conclude from Lemma 5.7.5 that finitely generated U -torsionless modules are U -torsionless linearly compact and therefore U -reflexive, again by (4). Since M is U -reflexive, it is U -torsionless linearly compact by (4). Hence (Coker i) = 0 by Lemma 5.7.9, and L is U -torsionless linearly compact by Lemma 5.7.5. Thus


125

L is U -reflexive by (4), and so L is U -reflexive by Lemma 5.3.5 and condition (3). Several of the preceding Lemmas also prove useful in characterizing the U -torsionless linearly compact modules when U R is a cotilting module. Theorem 5.7.11. Let U R be a cotilting module with S = End(U R ). Then M R is U -torsionless linearly compact if and only if M R is S U R -reflexive and i Coker i ∈ Ker whenever L → M in S-Mod . Proof. (⇒) Suppose that M R is U -torsionless linearly compact. Since the cotilting module U R is a costar module by Proposition 5.2.3, satisfies !2 = 0 by definition, and has Ker ∩ Ker ! = 0 by Proposition 5.2.6, the fact that M is reflexive follows from Proposition 5.7.3 and Lemma 5.3.5. That i Coker i ∈ Ker whenever L → M follows from Lemma 5.7.9 (with R and S interchanged) since M is U -reflexive. (⇐) Assuming the conditions, according to Lemma 5.7.6 we need only show that a U -torsionless module M R is S U R -reflexive whenever there is an exact sequence p

g

M −→ M −→ C → 0 in which M is S U R -reflexive and C = 0. To this end, let K = Ker p and I = Im p and consider the exact sequences f

α

0 → K −→ M −→ I → 0 and β

g

0 → I −→ M −→ C → 0 with β ◦ α = p. Since I ∈ Cogen(U R ) = Ker !, we see that f is epic. Thus K is U -reflexive by Lemma 4.2.3. From the commutative diagram M δM ↓

α

−→ I −→ δI ↓

0

2 α

2 M −→ 2 I −→ Coker(2 α)→0 we see that Coker(2 α) ∼ = Coker δ I . By hypothesis (Coker(2 α)) = 0 α since I → M, and !(Coker δ I ) = 0 by Lemma 5.3.5. Thus by Proposition 5.2.6 Coker δ I = 0, so I is U -reflexive. Since C = 0 and ! M = 0,

126

Cotilting

we obtain a commutative diagram with exact rows β

0→ I −→ M δM ↓ δI ↓ ∼ =

→C → 0

2 β

2 I −→ 2 M →C → 0 p

in which C = Coker(2 β). Since C = 0 we have M → M, so by hypothesis Coker(2 p) = 0. Thus, since 2 p = 2 β ◦ 2 α we see from Lemma 5.7.4 that C = 0. Now we can apply to obtain the commutative diagram with exact rows 3 β

0→ 3 M −→ 3 I δ I ↓ ∼ δ M ↓ = 0→

M

2 β

−→

I

from which it follows that the epimorphism δ M is actually an isomorphism, and then so is δM , since δ M ◦ δM = 1M . Finally, since M is S U R reflexive and Ker ∩ Ker ! = 0, so is M by Lemma 5.3.5. Since a cotilting bimodule satisfies the first three conditions of Theorem 5.7.10, these last two theorems easily yield Corollary 5.7.12. A cotilting bimodule S U R induces a GMD if and only every U -reflexive module is U -torsionless linearly compact.

5.8. Examples and Questions In this section we shall present examples showing that, although, as we have seen, they all induce cotilting theorems, cotilting bimodules, bimodules inducing weak Morita dualities, and those inducing generalized Morita dualities, are all distinct notions. It follows from Lemma 5.6.3 and Theorem 5.6.5 that if R is a hereditary noetherian ring, then R R R induces a weak Morita duality between the projective modules in mod-R and R-mod. The ring Z of integers is such a ring, and Z ZZ is neither a cotilting bimodule nor does it induce a generalized Morita duality. Example 5.8.1. According to Theorem 5.6.5, Z ZZ induces a (self) WMD between the finitely generated Z-torsionless (i.e., the finitely generated free modules). However !(ZN ) = Ext1Z (ZN , Z) "= 0 (i.e., Z N is not a Whitehead

5.8. Examples and Questions

127

group [36, Theorem 95.3, page 165 and Proposition 99.2, page 179]), so Z ZZ is not a cotilting bimodule. On the other hand, Z ZZ does not induce a GMD by Theorem 5.7.10, since ZN = (Z(N) ) is Z-reflexive ([35, Corollary 2.5, page 61]), but !(ZN ) "= 0. There are non-noetherian examples of domains that induce weak Morita dualities. Indeed, according to a theorem of E. Matlis, (see [37, Proposition 5.6, page 145]), if D is an integral domain with quotient field Q, then every module in cogen(D) is D reflexive if and only if Q/D is an injective cogenerator in Mod-D. In view of the following proposition we see that in this case D D D induces a WMD on cogen(D). (According to [37, Proposition 5.7, page 146], there are non-noetherian domains satisfying this condition.) Proposition 5.8.2. Let W be a module over a ring R. If Ext1R (W, W ) = 0 and inj . dim .(W ) ≤ 1, then cogen(W ) is contained in Ker Ext1R ( , W ) and is closed under extensions. Proof. Suppose first that there is an exact sequence 0 → M −→ W n −→ C → 0. This yields an exact sequence Ext1R (W n , W ) → Ext1R (M, W ) → Ext2R (C, W ), which, by hypothesis, shows that Ext1R (M, W ) = 0. Let f

g

0 → M1 −→ X −→ M2 → 0 be exact and suppose that also M1 → W m and M2 → W n so that we have maps θ1 , . . . , θn ∈ Hom R (M2 , W ) and ϕ1 , . . . , ϕm ∈ Hom R (M1 , W ) 1 with ∩n1 Ker θi = 0 and ∩m 1 Ker ϕi = 0. Since Ext R (M2 , W ) = 0, we see that ϕ1 , . . . , ϕm extend to ϕ 1 , . . . , ϕ m ∈ Hom R (X, W ). Thus, if x ∈ (∩n1 Ker θi ◦ m+n . g) ∩ (∩m 1 Ker ϕ i ) we see that x = 0 so that X → W We began this section by recalling that hereditary noetherian rings induce weak Morita dualities. Next we shall show that, if R is a hereditary perfect ring, then R R R is a cotilting bimodule for which the reflexive modules are the finitely generated projective modules, and thus over such a ring the reflexive modules need not be closed under submodules. We begin with

128

Cotilting

Proposition 5.8.3. If R is a right perfect right hereditary left coherent ring, then R R is a cotilting module and Ext1R (M, R R ) = 0 if and only if M is projective. Proof. Since R is right hereditary, every right R-module has injective dimension ≤ 1. Since direct products of projective right R-modules are projective (see [1, Theorems 19.20 and 28.4]), Ext1R (R RA , R R ) = 0 for any set A. If M ∈ Mod-R, then the terms P1 and P0 in a projective resolution 0 → P1 −→ P0 −→ M → 0 of M belong to Prod(R R ) by Lemma 1.3.1. Thus, letting M be a cogenerator, we see that the conditions of Theorem 5.2.5 are verified and R R is a cotilting module. On the other hand, if Ext1R (M, R R ) = 0, then since Ext1R (M, ) commutes with direct products ([69, Theorem 7.14]), the sequence splits. Thus Ext1R (M, R R ) = 0 if and only if M is projective. Our next proposition may be of independent interest, but it is nearly all the more we need to obtain the desired example. Proposition 5.8.4. If R is a basic two-sided perfect coherent ring, then a projective R-module P is R R R -reflexive if and only if P is finitely generated. Proof. Finitely generated projective R modules are reflexive over any ring R ([1, Proposition 20.17]). So assume that R is a two-sided perfect coherent ring with radical J and that PR is projective but not finitely generated. According to [1, Propositions 20.13 and 28.13] we may assume that P = e R (A) for some primitive idempotent e in a basic set of primitive idempotents {e1 , . . . en } for R and some infinite set A. Then P ∼ = Re A . Since J (Re A ) ⊆ J e A , we see A that (Re/J e) is semisimple, and since e is the only member of {e1 , . . . , en } that does not annihilate it, (Re/J e) A is a vector space over the division ring R/ R (Re/J e), which is isomorphic to Re/J e as left R-modules. But then by [50, Theorem 2, Page 247], (Re/J e) A ∼ = (Re/J e)(B) with card(B) > card(A). A Thus there is an epimorphism Re → (Re/J e)(B) , so, since Re A is projective, the projective cover Re(B) of (Re/J e)(B) ([1, Proposition 28.13]) is a direct summand of P ∼ = Re A ([1, Proposition 17.15]). But then, a similar argument shows that the projective module 2 P has a direct summand isomorphic to e R (C) with card(C) > card(B) > card(A), and thus it cannot be isomorphic to P = e R (A) according to [1, Theorem 28.14.]. Immediately now, in view of Theorem 5.3.6, we have

5.8. Examples and Questions

129

Corollary 5.8.5. If R is a two-sided perfect hereditary ring, then R R R is a cotilting bimodule that induces a cotilting theorem between the categories of finitely presented modules A R and R A with torsion theories (T R , F R ) and ( R T , R F) in which classes of the R-reflexive modules F R and R F are just the projective modules in mod-R and R-mod . Now we can easily verify an example, due to G. D’Este [33], of a finitely generated cotilting bimodule that does not induce a weak Morita duality. Example 5.8.6. Let L be an infinite dimensional vector space over a field K , and let

K 0 , R= L K the ring of lower triangular matrices with entries as indicated. Then R R R is a cotilting bimodule whose reflexive modules are precisely the R-torsionless modules in mod-R and R-mod, but R R R does not induce a WMD. Of course, we already know from Corollary 5.7.12 that the algebra R of Example 5.8.6 must have an R-reflexive module that is not R-torsionless linearly compact. In [33, Corollary 2.6] D’Este characterized the R-torsionless linearly compact R-reflexive modules as those finitely generated projective modules that contain no finite dimensional indecomposable direct summands. Thus, though they are both R-reflexive, if e1 , e2 denote the primitive diagonal idempotents in R, then e2 R is R-torsionless linearly compact, while the simple projective right R-module e1 R is not. Although we have made use of the notion of U -torsionless linearly compactness in the preceding section, we view it as a rather mysterious one. For example, we would be interested to see a direct proof that Z is Z-linearly compact. The fact that the U -reflexive modules relative to a Morita duality bimodule U S R are just the linearly compact modules suggests the problem of determining the classes of U -reflexive modules when S U R merely induces a weak Morita duality. Are they the U -torsionless linearly compact modules? In particular, since Z(N) and ZN are Z-reflexive, we wonder if they are Z-torsionless linearly compact. As we have just suggested, given a bimodule S U R , the class of U torsionless linearly compact R-modules is not well understood. That it is closed under U -torsionless factor modules is a consequence of Lemma 5.7.5. The known properties of the class of linearly compact modules (see [81, Section 3]) lead us to ask:

130

Cotilting

Is the class of U -torsionless linearly compact R-modules closed under (a) (b) (c) (d)

submodules? U -torsionless extensions? finite direct sums? any infinite direct sums?

Proposition 5.8.7. Let S U R be a cotilting bimodule. If S is left noetherian, then U R is U -torsionless linearly compact. Proof. Since U R is reflexive and S S ∼ = (U R ), according to Theorem 5.7.11 it will suffice to observe that Coker i ∈ Ker for every left ideal I of S and inclusion map i : I → S S. But, since !( S S) = 0, there is an exact sequence i ( S S) −→ I → !(S/I ) → 0, and since I, being finitely generated and U -torsionless, and S S are reflexive, S/I ∈ S A in Theorem 5.3.6. Thus (see Definition 5.1.1) Coker i ∼ = !(S/I ) ∈ Ker . In view of this proposition and of Corollary 5.7.12 and Theorem 5.7.8, we wonder (as was suggested to us by R. Colpi) if the cotilting bimodules of Proposition 5.4.5 (or more specially the one of Example 5.4.6) might induce a GMD or a WMD. In particular, are the modules R R and S S U -torsionless linearly compact in these cases? Results in Sections 4.3 and 4.4 suggest the presently open problems of determining one-sided characterizations of weak and generalized Morita duality and cotilting bimodules.

Appendix A Adjoints and Category Equivalence

In this appendix we will give an exposition of some results from category theory that suffice to justify our assumption in Chapter 2 that the functors of interest can be taken to be Hom and tensor functors. Let C be a category. If X is an object of C we write X ∈ Ob C or just X ∈ C if no confusion results. If X, Y ∈ C we denote the set of morphisms with domain X and codomain Y by HomC (X, Y ). We write elements in HomC (X, Y ) in f the form X −→ Y or f : X → Y and we denote the identity morphism in HomC (X, X ) by 1 X . If D is another category and F is a functor from C to D we write F : C → D; and F : C D : G is shorthand for F : C → D and G : D → C. We denote the identity functor on C by 1C . If X is an object of C, then HomC (X, ) and HomC ( , X ) denote the canonically induced (covariant and contravariant) functors from C into the category Set of all sets.

A.1. The Yoneda–Grothendieck Lemma Lemma A.1.1 (Yoneda Lemma). Suppose X is an object in the category C and F : C → Set is a covariant functor. If u ∈ F X, letting $(u) Z : HomC (X, Z ) → F Z for each Z ∈ C via $(u) Z : f −→ F f (u) for f ∈ HomC (X, Z ), one obtains a natural transformation $(u) from HomC (X, ) to F. The mapping $ defines a bijection between F X and the set of all natural transformations from HomC (X, ) to F that is natural in both X and F. Moreover, if τ is such a natural transformation, then τ X (1 X ) → τ under this correspondence. Proof. It is straightforward that $(u), as defined, is a natural transformation. Let (τ ) = τ X (1 X ) if τ is a natural transformation. Then $(u) = ($(u)) X (1 X ) = F(1 X )(u) = 1 F X (u) = u, 131

132

Adjoints and Category Equivalence

so $ = 1 F X . Moreover, $(τ ) = $(τ X (1 X ) = τ where the last equality follows from the commutativity of the diagram HomC (X, X ) HomC (X, f ) ↓ HomC (X, Z )

τX

−→ τZ

−→

FX ↓ Ff F Z,

that is, ($(τ X (1 X ))) Z ( f ) = F f (τ X (1 X )) = τ Z ( f ), completing the proof that $ and are inverse functions. The proof that $ is natural in X and F is straightforward. Corollary A.1.2. If X, Y ∈ C, then the correspondence u → HomC (u, ), where HomC (u, Z )( f ) = f u for u ∈ HomC (Y, X ) and f ∈ HomC (X, Z ), defines a bijection between HomC (Y, X ) and the set of natural transformations from HomC (X, ) to HomC (Y, ) that is natural in X and Y . Moreover, under this correspondence, isomorphisms correspond to natural isomorphisms. Proof. It suffices to note that u is an isomorphism if and only if HomC (u, ) is a natural isomorphism. If HomC (u, ) is an isomorphism, then HomC (u, Y ) is an isomorphism, so there exists v ∈ HomC (X, Y ) such that vu = 1Y . Since HomC (u, X ) is an isomorphism, HomC (u, X )(1 X ) = u, and Hom(u, X ) (uv) = u, we have uv = 1 X . Hence u is an isomorphism. A.2. Adjoint Covariant Functors Adjoints and Arrows of Adjunction Given categories C and D and a pair of covariant functors H : C D : T , T is a left adjoint of H and H is a right adjoint of T if, for each pair of objects N of D and M of C, there is a bijection α N ,M : HomD (N , H M) → HomC (T N , M) such that the diagrams HomD (N , H M ) HomD ( f, H M ) ↓

α N ,M

−→ α N ,M

HomD (N , H M )

−→

HomD (N , H M) HomD (N , H g) ↓

−→

HomD (N , H M )

−→

HomC (T N , M ) ↓ HomC (T f, M ) HomC (T N , M )

and α N ,M

α N ,M

HomC (T N , M) ↓ HomC (T N , g) HomC (T N , M )

A.2. Adjoint Covariant Functors

133

f

g

are commutative for all morphisms N −→ N in D and M −→ M in C. That u v is, if N −→ H M and N −→ H M are morphisms we have the equations α N ,M (v) ◦ T f = α N ,M (v ◦ f )

(A.1)

g ◦ α N ,M (u) = α N ,M (H g ◦ u).

(A.2)

and In particular, for N = H M and u = 1 H M , we have from (A.2) g ◦ α H M,M (1 H M ) = α H M,M (H g)

and for N = H M , f = H g, and v = 1 H M , (A.1) becomes α H M,M (H g) = α H M ,M (1 H M ) ◦ T H g. Hence the diagram THM T Hg ↓ T H M

α H M,M (1 H M )

−→

α H M ,M (1 H M )

−→

M ↓g M

v

u

is commutative. Similarly, if T N −→ M and T N −→ M are morphisms, we have the equations −1 α −1 N ,M (v) ◦ f = α N ,M (v ◦ T f )

(A.3)

−1 H g ◦ α −1 N ,M (u) = α N ,M (g ◦ u).

(A.4)

and Hence if M = T N , M = T N , and g = T f, we have −1 α −1 N ,T N (1T N ) ◦ f = α N ,T N (T f ) = H T f ◦ α −1 N ,T N (1T N ),

so the diagram N f ↓ N

α −1 N ,T N (1T N )

−→

(1 ) α −1 N ,T N T N

−→

HT N ↓ HT f HT N

is also commutative. In this situation we denote the morphism α −1 N ,T N (1T N ) : N → H T N by θ N and the morphism α H M,M (1 H M ) : T H M → M by µ M . Then, because of the observations above, θ : 1D → H T and µ : T H → 1C are natural transformations, and then θ is called the arrow of adjunction associated with α and µ is its quasi-inverse.

134


Equation (A.1) with f = θ N yields µT N ◦ T θ N = α H T N ,T N (1 H T N ) ◦ T (θ N ) = α N ,T N (θ N ) = α N ,T N (α −1 N ,T N (1T N )) = 1T N . Briefly, we write µT ◦ T θ = 1T .

(A.5)

Similarly, we can apply equation (A.4) to conclude that H µ M ◦ θ H M = 1 H M ; that is, H µ ◦ θ H = 1H .

(A.6)

Theorem A.2.1. Given categories D and C, a pair of covariant functors H : C D : T, and natural transformations θ : 1D → H T and µ : T H → 1C , suppose f equations (A.5) and (A.6) are satisfied. Then α( f ) = µ M ◦ T f for N −→ H M defines bijections that give H as a right adjoint of T . Proof. The proof is straightforward, if one notes that the inverse β of α is g given by β(g) = H g ◦ θ N if T N −→ M.

Faithfully Full Adjoints Theorem A.2.2. Suppose H : C D : T is a pair of covariant functors and H is right adjoint to T with arrow of adjunction θ with quasi–inverse µ. Then (1) µ is a natural isomorphism if and only if H is full and faithful; (2) θ is a natural isomorphism if and only if T is full and faithful. fi

Proof. (1) Assume that µ is a natural isomorphism. If M −→ M , i = 1, 2, are morphisms in C with H f 1 = H f 2 , then −1 f 1 = µ M ◦ T H ( f 1 ) ◦ µ−1 M = µM ◦ T H ( f2) ◦ µM = f2. g

Suppose H M −→ H M . Letting f = µ M ◦ T (g) ◦ µ−1 M , the commutativity

A.3. Equivalence of Categories

135

of the diagram HM ↓ θH M

−→

g

H M ↓ θH M

HTg

H T H M ↓ H (µ M )

HT H M −→ ↓ H (µ M ) Hf

−→

HM

H M

(using (A.6)) shows that H ( f ) = g. Hence H is full. Conversely, fix M ∈ C. Since H is full and faithful, the composite α H M,M

H

HomC (M, M ) −→ HomD (H M, H M ) −→ HomC (T H M, M ) for M ∈ C defines a natural isomorphism τ : HomC (M, ) −→ HomC (T H M, ), which, according to Corollary A.1.2, is induced by the isomorphism τ M (1 M ) = α H M,M (1 H M ) = µ M : T H M → M. Thus µ : T H → 1C is a natural isomorphism. The proof of (2) is similar.

A.3. Equivalence of Categories Definition A.3.1. If C and D are categories, a covariant functor H : C−→D is an equivalence if there is a covariant functor T : D−→C such that T ◦ H and H ◦ T are naturally isomorphic to the identity functors 1C and 1D , respectively. In this case, T is also an equivalence, and we refer to the equivalence H : C D : T. Theorem A.3.2. Let C and D be categories and let T : D−→C be a covariant functor. Then T is an equivalence if an only if T is full and faithful and every object M in C is isomorphic to an object T N for some object N ∈ D. Proof. Suppose T is an equivalence and let η : 1D → H T be a natural isofi morphism. If N −→ N , i = 1, 2 are morphisms in D and we assume that T f 1 = T f 2 , then the commutativity of the diagrams N fi ↓ N

ηN

−→

HT N H T fi ↓ ηN −→ HT N

136


and the assumption show that f 1 = f 2 , so T is faithful. Similarly, H is faithful. g If T N −→ T N is a morphism, then letting f = η−1 N H gη N , the commutativity of the diagram ηN

ηN

H T N ←− N Hg ↓ f ↓ ηN H T N ←− N

−→

HT N HT f ↓ ηN −→ HT N

with isomorphisms η N , η N shows that H T f = H g and hence that T f = g since H is faithful. Hence T is full. Clearly, any object M in C is isomorphic to the object T H M. Conversely, if M is an object in C we choose an object H M in D such that T H M is isomorphic to M and we choose an isomorphism µ M : T H M → M. −1 µM f µM

f

If M −→ M is a morphism in C, then T H M −→ T H M is a morphism in Hf C, and since T is full and faithful, there is a unique morphism H M −→ H M in D such that T H f = µ−1 M f µ M , that is, we have the commutative diagram THM THf ↓ T H M

µM

−→

M f ↓ µM −→ M

for exactly one choice of H f . One verifies that H : C−→D is a functor and that µ : T H → 1C is a natural isomorphism. Now, in view of Theorems A.2.2 and A.3.2, we have Corollary A.3.3. Suppose H : C D : T is a pair of covariant functors and H is right adjoint to T with arrow of adjunction θ : 1D −→ H T and its quasi-inverse µ : T H −→ 1C . Then H : C D : T is an equivalence if and only if θ and µ are natural isomorphisms. Theorem A.3.4. Let C and D be categories. If H : C D : T is an equivalence, there exist natural isomorphisms θ : 1D → H T and µ : T H → 1C such that equations (A.5) and (A.6) are satisfied. Proof. Suppose H : C D : T is an equivalence and µ : T H → 1C is a natural isomorphism. Since, for any object N in D, µ−1 TN

T N −→ T H T N

A.3. Equivalence of Categories

137

is a morphism, and since T is full and faithful, there is a unique morphism θN N −→ H T N such that T θ N = µ−1 TN,

(A.7)

and a simple argument, again since T is full and faithful, shows that θ N is an f isomorphism. If N −→ N is a morphism in D, the diagram TN Tf ↓ T N

µT N

←−

T HT N T HT f ↓ µT N ←− T HT N

is commutative. Hence, using equation (A.7), we have T f = µT N ◦ T H T f ◦ µ−1 TN = T θ N−1 ◦ T H T f ◦ T θ N = T (θ N−1 H T f θ N ), so f = θ N−1 H T f θ N , that is, the diagram θN

N f ↓

−→

N

−→

θN

HT N HT f ↓ HT N

is commutative. Thus θ : 1D → H T is a natural isomorphism. Equation (A.7) shows that equation (A.5) is satisfied. Since µ is a natural isomorphism, the diagram THM T H µ−1 M ↓ T HT H M

µM

−→

M µ−1 ↓ M µT H M −→ T H M

is commutative for any object M ∈ C. Hence we have T H µ M = µT H M = T θ H−1M where we have used equation (A.7). Since T is full and faithful, we obtain equation (A.6). Theorem A.3.5. Let D and C be categories and let T : D−→C be a covariant functor. The following are equivalent.

138


(a) T is an equivalence; (b) T is full and faithful and has a full and faithful right adjoint; (c) T is full and faithful and has a full and faithful left adjoint. Proof. (b) ⇒ (a) follows from Theorem A.2.2. (a) ⇒ (b) is clear from Theorems A.2.1, A.3.2 and A.3.4. (b) ⇒ (c). Having proved the equivalence of (a) and (b), we note that the symmetry afforded by (a) makes plain the equivalence of (b) and (c).

Appendix B Noetherian Serial Rings

A serial ring is a semiperfect ring R such that the submodule lattice of each right and each left indecomposable projective R-module is linearly ordered. In this appendix we present results concerning noetherian serial rings that are assumed in some examples given in the text. We let J denote the radical of R.

B.1. Finitely Generated Modules Lemma B.1.1. Let R be a noetherian serial ring. If PR is finitely generated and projective and K ≤ P, then P has a decomposition into local direct summands P = P1 ⊕ · · · ⊕ Pn such that K = K ∩ P1 ⊕ · · · ⊕ K ∩ P for some ≤ n. Proof. The proof is by induction on the composition length c(K /K J ). First, suppose that e is a primitive idempotent in R and that K = xe R. We may assume that K ⊆ P J, since otherwise (see [1, Section 27]) K would be a direct summand of P. Let N = P/xe R so that η

⊆

0 → xe R −→ P −→ N → 0 is a projective cover of N with η the natural map. Now N = N ⊕ Q where q Q is projective and N has no projective direct summands. Letting Q −→ N be a projective cover, we obtain another projective cover (q ⊕ 1 Q ) : Q ⊕ Q −→ N ⊕ Q = N of N . Thus, by uniqueness of projective covers [1, Lemma 17.17] there is an

139

140

Noetherian Serial Rings

isomorphism f : P → Q ⊕ Q such that η = (q ⊕ 1 Q ) ◦ f. Letting P = f −1 Q and P = f −1 Q , we see that ⊆

η|P

0 → xe R −→ P −→ N → 0 η|P

is exact with P −→ N a projective cover of N . But then, since N has no projective direct summands, the transpose t N has minimal projective resolution Hom R (P , R) → Hom R (e R, R) → t N → 0 ∼ Re is uniserial, we must (see [1, Theorem 32.13]). Since Hom R (e R, R) = ∼ have Hom R (P , R) = R f for some primitive idempotent f ∈ R. Thus P ∼ = f R is uniserial and K = xe R ⊆ P , so the case = Hom R (R f, R) ∼ c(K /K J ) = 1 is established, since by the Azumaya-Krull-Schmidt theorem [1, Theorem 12.6] we may renumber the Pi so that P = P ⊕ P2 ⊕ · · · ⊕ Pn . Suppose that c(K /K J ) = m, and that the result is true for submodules L of finitely generated projective R-modules with c(L/L J ) ≤ m − 1. Then clearly m ≤ n. Let K = L + xe R ≤ P for some primitive idempotent e ∈ R, and suppose that P = P1 ⊕ · · · ⊕ Pn with L = L ∩ P1 ⊕ · · · ⊕ L ∩ Pm−1 . By the first step in the proof, there is a indecomposable direct summand P of P with xe R ≤ P . Thus, again by the Azumaya-Krull-Schmidt theorem, the given decomposition of P complements P , so that one of the projections π1 , . . . , πn restricts to an isomorphism (πi |P ) : P → Pi (see [1, Proposition 5.5]). If i ≥ m, we are done. Otherwise we may assume that (π1 |P ) : P → P1 is an isomorphism. Since P1 is uniserial, either π1 (xe R) ⊆ L ∩ P1 or L ∩ P1 ⊆ π1 (xe R). In the first case we see that π1 (K ) ⊆ K ∩ P1 ⊆ π1 (K ) so that K = K ∩ (π1 (K ) ⊕ π2 (P) ⊕ · · · ⊕ πn (Pn )) = (K ∩ P1 ) ⊕ (K ∩ (π2 (P) ⊕ · · · ⊕ πn (Pn )),

B.1. Finitely Generated Modules

141

where, modulo its radical, the latter term has length ≤ m − 1. In the second case, letting π , π2 , . . . , πn be the projections for the decomposition P = P ⊕ P2 ⊕ · · · ⊕ Pn , the isomorphism (π |P1 ) : P1 → P has π (K ) = π (xe R + L ∩ P1 ) = xe R ⊆ K ∩ P ⊆ π (K ) so that K = (K ∩ P ) ⊕ (K ∩ (π2 (P) ⊕ · · · ⊕ πn (Pn )). In either case the inductive hypothesis applies. Immediately, now, we have Theorem B.1.2. If R is a noetherian serial ring, then every finitely generated R-module is a direct sum of local modules. Corollary B.1.3. If R is a noetherian serial ring, then every uniform R module is uniserial. Proof. If U R is uniform and x, y ∈ U, then x R + y R is uniserial by Theorem B.1.2. Thus x R and y R are comparable. Let R be an indecomposable noetherian serial ring, and let e1 , . . . , en be a basic set of primitive idempotents for R. Since R is indecomposable and each ei J/ei J 2 is simple, the ei can be renumbered so that ei R is a projective cover of ei−1 J for i = 2, . . . , n and e1 R is a projective cover of en J, if en J "= 0. (See [1, Section 32].) When this is the case, e1 R, . . . , en R is called a Kupisch series for R. We note here that if e1 R, . . . , en R is a Kupisch series for an indecomposable noetherian serial ring R that is not artinian, then ei J "= 0 for all i = 1, . . . , n. Corollary B.1.4. If R is a noetherian serial ring and e is a primitive idempotent in R, then every non-zero submodule of e R is of the form e J k for some k k ∈ N. In particular, ∩∞ k=1 J = 0, and every finitely generated R-module is the direct sum of a projective module and a module of finite length. k Proof. We may assume that R is indecomposable. Suppose that ∩∞ k=1 e J = N "= 0. Then N is an epimorph of some ei R in a Kupisch series for R, and ei R ∼ = e j J. Since E(e R) is uniserial by Corollary B.1.3, E(e R)/N contains a simple, necessarily essential, submodule isomorphic to e j R/e j J. But this k implies e R/ ∩∞ k=1 e J contains a minimal submodule, which is impossible.

142


k m Thus ∩∞ k=1 e J = 0, and if 0 "= L ≤ e R, then L = e J where m is minimal such that e J m ≤ L . The last statement follows.

Corollary B.1.5. Every indecomposable noetherian serial ring that is not artinian is hereditary. Proof. Let e1 , . . . , en be a basic set of primitive idempotents for R. The preceding discussion shows that since R is not artinian, neither is any ei R. But by Corollary B.1.4, every right ideal I of R is the direct sum of a projective right ideal and an artinian right ideal. In particular, I R = P ⊕ L with P projective and L a right ideal of finite length. But then L = 0 and so every right (similarly left) ideal is projective. It follows that if e1 R, . . . , en R is a Kupisch series for an indecomposable noetherian serial ring R that is not artinian, then ei−1 J ∼ = e1 R for i = 2, . . . , n and en J ∼ = e1 R. Proposition B.1.6. Let M R be a finitely generated (artinian) module over a noetherian serial ring. Then S = End(M R ) is a semiperfect noetherian (artinian) ring. Proof. By Theorem B.1.2 and Corollary B.1.4 M R is isomorphic to the direct sum of a finite number of modules of the form e R/eI where e is a primitive idempotent and I is an ideal of R. Since End((e R/eI ) R ) ∼ = e Re/eI e we see that the endomorphism ring of every finitely generated indecomposable module is a semiperfect noetherian (artinian) ring. Hence S is semiperfect by [1, Corollary 27.7]. Similarly, if e R/eI and f R/ f K are finitely generated indecomposable with e, f idempotents and I, K ideals of R, then the f R f e Re bimodule Hom R (e R/eI, f R/ f K ) is isomorphic to a submodule of an epimorphic image of Hom R (e R, f R) ∼ = f Re, and so is finitely generated over e Re and over f R f and hence also over the noetherian (artinian) rings End((e R/eI ) R ) ∼ = e Re/eI e and End(( f R/ f K ) R ) ∼ = f R f / f K f . It follows that S is a noetherian (artinian) ring. Proposition B.1.7. Let M R , N R be finitely generated modules over a noetherian serial ring R. Then Hom R (M, N ) and Ext1R (M, N ) are finitely generated both as right S = End(M R ) modules and as left S = End(N R ) modules.

B.2. Injective Modules

143

Proof. By Proposition B.1.6 S¯ = End(M ⊕ N ) is noetherian. If e, f ∈ S¯ are ¯ and f S¯ f are noetherian, as are e S¯ e e S¯ f and e S¯ f f Se idempotents, then e Se ¯ . Thus, with appropriate choices for e and f we see that Hom R (M, N ) is finitely generated as a right S-module and as a left S -module. Since R and S are noetherian, taking a projective resolution of M in mod-R yields easily, via the first assertion, that Ext1R (M, N ) ∈ S -mod. To see that Ext1R (M, N ) ∈ mod-S we may assume that N R is indecomposable. Then N is uniserial and, since indecomposable injectives are uniserial by Corollary B.1.3, N has an injective resolution, 0 → N R → I0 → I1 → · · · where each I j is uniserial. Suppose M R has finite length. Since each I j is uniserial, the trace of M in I j is a finite length submodule K j of I j , so Hom R (M, I j ) = Hom R (M, K j ) is a finitely generated S-module by the first assertion. Since S is noetherian it follows that Ext1R (M, N ) ∈ mod-S. In general, applying Corollary B.1.4, if M R = P ⊕ M1 where P is projective and M1 has finite length, one obtains easily that Ext1R (M, N ) ∈ mod-S via Ext1R (P, N ) = 0 and the preceding argument. B.2. Injective Modules Suppose that e1 R, . . . , en R is a Kupisch series for an indecomposable noetherian serial ring R that is not artinian, and let Si = ei R/ei J for i = 1, . . . , n. Then by Corollary B.1.4 the submodules of ei R and Rei must be ei R > ei J > ei J 2 > · · ·

and Rei > J ei > J 2 ei > · · · .

Thus the composition factors of ei R are, from the top down, Si , Si+1 , . . . , Sn , S1 , S2 , . . . , Sn , . . . . It follows from Corollary B.1.3 that the indecomposable injective R-modules are also uniserial. There are just n + 1 of these on each side. In Mod-R they are E 1 = E(S1 ), . . . , E n = E(Sn ) and E 0 with Soc(E 0 ) = 0. Each E i with 1 ≤ i ≤ n is artinian, its submodules are 0 < Soc(E i ) < Soc2 (E i ) < · · · , where Sock (E i ) = Ei (J k ), and the composition factors of E i are, from the bottom up, Si , Si−1 , . . . , S1 , Sn , Sn−1 , . . . , S1 , . . .

144


whereas the composition factors of E 0 are, in ascending order, . . ., Sn , Sn−1 , . . ., S1 , Sn , Sn−1 , . . ., S1 , . . . . From these observations we glean Proposition B.2.1. Let R be an indecomposable noetherian serial ring that is not artinian. Then every proper factor of an indecomposable injective R-module is the injective envelope of its socle, and R has a unique indecomposable non-artinian injective right (or left) module E 0 , all of whose proper submodules are indecomposable projective modules. Moreover, every indecomposable injective right (or left) R-module is isomorphic to a factor of E0. Corollary B.2.2. Let R be an indecomposable noetherian serial ring that is not artinian, and suppose that R has n isomorphism classes of simple right modules and indecomposable injective right modules E 1 , . . . , E n , E 0 with all but E 0 artinian. If i ∈ {1, . . . , n}, then for each set A there are sets B, C such (B) (C) that E iA ∼ = Ei ⊕ E0 . Proof. Since R is semiperfect and J is finitely generated, it follows that Soc E iA = EiA (J ) = Ei (J ) A = Soc(E i ) A . (B) Thus since Soc(E i )e j = 0 if j "= i, we see that Soc(E iA ) ∼ = Si for some set (B) B. But then (see [1, Proposition 18.13]) E iA ∼ = E i ⊕ E where Soc(E) = 0, (C) so (see [1, Theorem 25.6]) there must be a set C with E ∼ = E0 .

Proposition B.2.3. If R is a left (equivalently, right) linearly compact indecomposable non-artinian noetherian serial ring, then R has self-duality. Proof. Assume that R is self-basic and left linearly compact, and let E = E 1 ⊕ · · · ⊕ E n be the minimal left cogenerator. Then E is artinian and hence linearly compact ([81, Corollary 3.2]). Thus, letting S = End( R E), the bimodule R E S induces a Morita duality and SS is linearly compact (see Section 4.4). Now, letting Ak = r E (J k ), it is apparent that Ak is the minimal cogenerator over R/J k and that the bimodule R/J k Ak S/rS (Ak ) induces a Morita duality. But R/J k is a basic QF-ring (see [1, Section 32.6]), so we must have k ∼ R/J k Ak = R/J k R/J . Thus, as rings S/r S (Ak ) ∼ = End( R/J k Ak ) ∼ = R/J k .

B.2. Injective Modules

145

Now both {J k | k ≥ 1} and {r S (Ak ) | k ≥ 1} are downward directed sets of ideals with, by Corollary B.1.4, ∩k J k = 0, and hence ∩k r S (Ak ) = 0. Therefore, since R R and SS are linearly compact, ∼ lim S/r S (Ak ) ∼ ∼ lim R/J k = R= = S. ← − ← −

The structure of noetherian serial rings, as well as several of the results in this appendix, were given by Warfield in [79]. In particular it is proved there that every indecomposable noetherian serial ring that is not artinian is Morita equivalent to an n × n [D : m] upper triangular matrix ring over a local noetherian uniserial ring D with maximal ideal m, that is, a full ring of n × n matrices over D whose entries below the main diagonal belong to m. Moreover, it follows from Proposition B.2.3 and [81, Theorem 4.3, Lemma 4.9 and Proposition 3.3] that [D : m] has self-duality if and only if D is linearly compact.

Bibliography

[1] F. W. Anderson and K. R. Fuller. Rings and Categories of Modules. Springer– Verlag, Inc., New York, Heidelberg, Berlin, second edition, 1992. [2] L. Angeleri Hügel. Finitely cotilting modules. Comm. Algebra, 28:2147–2172, 2000. [3] L. Angeleri Hügel, A. Tonolo, and J. Trlifaj. Tilting preenvelopes and cotilting precovers. Algebras and Representation Theory, 4:155–201, 2002. [4] I. Assem. Torsion theories induced by tilting modules. Can. J. Math., 36:899–913, 1984. [5] I. Assem. Tilting theory – an introduction. Topics in Algebra, Banach Center Pub., 26:127–180, 1990. [6] G. Azumaya. A duality theory for injective modules. Amer. J. Math., 81:249–278, 1959. [7] S. Bazzoni. Cotilting modules are pure injective. Proc. Amer. Math. Soc., 31:3665– 3672, 2003. [8] K. Bongartz. Tilted algebras. Springer–Verlag LNM, 903:26–38, 1981. [9] S. Brenner and M. C. R. Butler. Generalizations of the Bernstein–Gelfand– Ponomarev reflection functors. Springer–Verlag LNM, 832:103–170, 1980. [10] W. D. Burgess and X. Du. On the torsion parts of a torsion theory counter equivalence. (preprint). [11] H. Cartan and S. Eilenburg. Homological Algebra. Princeton University Press, Princeton, New Jersey, 1956. [12] S. U. Chase. Direct products of modules. Trans. Amer. Math. Soc., 97:457–473, 1960. [13] P. M. Cohn. Morita equivalence and duality. Mathematical Notes, Queen Mary College, Univ. of London, London, 1966. [14] R. R. Colby. A generalization of Morita duality and the tilting theorem. Comm. Alg., 17(7):1709–1722, 1989. [15] R. R. Colby. A cotilting theorem for rings. In Methods in Module Theory, pages 33–37. M. Dekker, New York, 1993. [16] R. R. Colby, R. Colpi, and K. R. Fuller. A note on cotilting modules and generalized Morita duality. Venizia 2002 Conference Proceedings. (to appear). [17] R. R. Colby and K. R. Fuller. QF–3 rings and Morita duality. Tsukuba J. Math., 8:183–188, 1984. [18] R. R. Colby and K. R. Fuller. Tilting, cotilting, and serially tilted rings. Comm. Algebra, 18:1585–1615, 1990.

147

148

Bibliography

[19] R. R. Colby and K. R. Fuller. Tilting and torsion theory counter equivalences. Comm. Algebra, 23(13):4833–4849, 1995. [20] R. R. Colby and K. R. Fuller. Hereditary torsion theory counter equivalences. J. Algebra, 183:217–230, 1996. [21] R. R. Colby and K. R. Fuller. Costar modules. J. Algebra, 242:146–159, 2001. [22] R. R. Colby and K. R. Fuller. Weak Morita duality. Comm. Algebra, 31:1859– 1879, 2003. [23] R. Colpi. Dualities induced by cotilting modules. Venizia 2002 Conference Proceedings. (to appear). [24] R. Colpi. Some remarks on equivalences between categories of modules. Comm. Algebra, 18:1935–1951, 1990. [25] R. Colpi. Tilting modules and ∗–modules. Comm. Algebra, 21(4):1095–1102, 1993. [26] R. Colpi. Cotilting bimodules and their dualities. Interactions between ring theory and representations of algebras (Murcia). In Lecture Notes in Pure and Appl. Math., volume 210, pages 81–93. Marcel Dekker, New York, 2000. [27] R. Colpi and G. D’Este. Equivalences represented by faithful non-tilting ∗– modules. Canadian Math. Soc., Conference Proc., 24:103–110, 1998. [28] R. Colpi, G. D’Este, and A. Tonolo. Quasi–tilting modules and counter equivalences. J. Algebra, 191:461–494, 1997. [29] R. Colpi and K. R. Fuller. Cotilting modules and bimodules. Pacific J. Math., 192:275–291, 2000. [30] R. Colpi and C. Menini. On the structure of ∗–modules. J. Algebra, 158:400–419, 1993. [31] R. Colpi, A. Tonolo, and J. Trlifaj. Partial cotilting modules and the lattices induced by them. Comm. Algebra, 25:3225–3237, 1997. [32] R. Colpi and J. Trlifaj. Tilting modules and tilting torsion theories. J. Algebra, 178:614–634, 1995. [33] G. D’Este. Reflexive modules are not closed under submodules. In Lecture Notes in Pure and Appl. Math., volume 224, pages 53–64. Marcel Dekker, New York, 2002. [34] G. D’Este and D. Happel. Representable equivalences are represented by tilting modules. Rend. Sem. Mat. Univ. Padova, 83:77–80, 1990. [35] P. C. Eklof and A. H. Mekler. Almost Free Modules (Set Theoretic Methods). North-Holland, Amsterdam, New York, Oxford, Tokyo, 1990. [36] L. Fuchs. Infinite Abelian Groups, volume II. Academic Press, Inc., New York, London, 1973. [37] L. Fuchs and L. Salce. Modules over Non–Noetherian Domains. American Mathematical Society, Providence, 2001. [38] K. Fuller. Density and equivalence. J. Algebra, 29:528–550, 1974. [39] K. R. Fuller. A note on quasi–duality. In Abelian Groups, Module Theory, and Topology, volume 201 of Lect. Notes Pure Appl. Math. Marcel Dekker, New York, 1998. [40] K. R. Fuller and W. Xue. On quasi–duality modules. Comm. Algebra, 28:1919– 1937, 2000. [41] J. L. Gómez Pardo. Counterinjective modules and duality. J. Pure and Appl. Algebra, 61:165–179, 1989.

Bibliography

149

[42] J. L. Gómez Pardo, P. A. Guil Asensio, and R. Wisbauer. Morita dualities induced by the M-dual functors. Comm. Algebra, 22:5903–5934, 1994. [43] P. Griffith. On a subfunctor of Ext. Arch. Math., XXI:17–22, 1970. [44] D. Happel. Triangulated Categories in the Representation Theory of Finite Dimensional Algebras, volume 118. London Math. Soc. Lecture Notes Series, 1988. [45] D. Happel, I. Reiten, and S. Smalø. Tilting in Abelian Categories and Quasi-tilted Algebras, volume 120. Mem. Amer. Math. Soc., Providence, 1996. [46] D. Happel and C. M. Ringel. Construction of tilted algebras. Springer–Verlag LNM, 903:125–144, 1981. [47] D. Happel and C. M. Ringel. Tilted algebras. Trans. Amer. Math. Soc., 274:399– 443, 1982. [48] M. Hoshino. Tilting modules and torsion theories. Bull. London Math. Soc., 14:334–336, 1982. [49] M. Hoshino. Splitting torsion theories induced by tilting modules. Comm. Algebra, 11:427–441, 1983. [50] N. Jacobson. Lectures in Abstract Algebra, volume 2. Van Nostrand, New York, 1953. [51] J. P. Jans. Duality in Noetherian rings. Proc. Amer. Math. Soc., 12:829–835, 1961. [52] C. U. Jensen and H. Lenzing. Model Theoretic Algebra, volume 2. Gordon and Breach Science Publishers, New York, 1989. [53] H. Krause and M. Saorín. On minimal approximations of modules. Contemp. Math., 299:227–236, 1998. [54] H. Lenzing. Endlich präsentierbare Moduln. Arch. Math., 20:262–266, 1969. [55] R. N. S. MacDonald. Representable dualities between finitely closed subcategories of modules. Can. J. Math., 31:465–475, 1979. [56] S. MacLane. Homology. Academic Press, Inc., Publishers, New York, 1963. [57] S. Maclane. Categories for the Working Mathematician. Graduate Texts in Mathematics. Springer–Verlag, New York Heidelberg Berlin, 1971. [58] F. Mantese. Generalizing cotilting dualities. J. Algebra, 236:630–644, 2001. [59] F. Mantese, P. Ruzicka, and A. Tonolo. Cotilting versus pure-injective modules. Pacific J. Math. (to appear). [60] E. Matlis. Reflexive domains. J. Algebra, 8:1–33, 1967. [61] C. Menini and A. Orsatti. Good dualities and strongly quasi-injective modules. Ann. Mate. Pure Appl. (IV), 127:187–230, 1981. [62] C. Menini and A. Orsatti. Representable equivalences between categories of modules and applications. Rend. Sem. Mat. Univ. Padova, 82:203–231, 1989. [63] B. Mitchell. Theory of Categories. Pure and Applied Mathematics. Academic Press, Inc., New York and London, 1965. [64] Y. Miyashita. Tilting modules of finite projective dimension. Math. Z., 193:113– 146, 1986. [65] K. Morita. Duality for modules and its applications to the theory of rings with minimum condition. Sci. Rep. Tokyo Kyoiku Daigaku Ser. A, 6:83–142, 1958. [66] B. J. Müller. Linear compactness and Morita duality. J. Algebra, 16:60–66, 1970. [67] N. Popescu. Abelian Categories with Applications to Rings and Modules. Academic Press, Inc., New York, 1973. [68] G. Puninski. Serial Rings. Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001.

150

Bibliography

[69] J. J. Rotman. An Introduction to Homological Algebra. Academic Press, Inc., New York, San Francisco, London, 1979. [70] F. L. Sandomierski. Linearly compact modules and local Morita duality. In R. Gordon, editor, Ring Theory, pages 333–346. Academic Press, Inc., New York, 1972. [71] S. O. Smalø. Torsion theories and tilting modules. Bull. London Math. Soc., 16:518–522, 1984. [72] B. Stenstrom. Rings of Quotients. Springer–Verlag LNM vol. 76, New York, Heidelberg, Berlin, 1970. [73] R. G. Swan. Algebraic K–Theory, volume 76. Springer–Verlag LNM vol. 76, Berlin, Heidelberg, New York, 1968. [74] H. Tachikawa. Representations of trivial extensions of hereditary algebras. Springer–Verlag LNM, 832:579–599, 1980. [75] A. Tonolo. Generalizing Morita duality: A homological approach. J. Algebra, 232:282–298, 2000. [76] A. Tonolo. On a finitistic cotilting–type duality. Comm. Algebra, 30:5091–5106, 2002. [77] J. Trlifaj. On ∗–modules generating the injectives. Rend. Sem. Mat. Univ. Padova, 88:211–220, 1992. [78] J. Trlifaj. Every ∗–module is finitely generated. J. Algebra, 169:392–398, 1994. [79] R. B. Warfield. Serial rings and finitely presented modules. J. Algebra, 37:187– 222, 1975. [80] R. Wisbauer. Foundations of Module and Ring Theory. Gordon and Breach Science Publishers, Philadelphia, 1991. [81] W. Xue. Rings with Morita Duality. Springer–Verlag LNM vol. 1523, Berlin, Heidelberg, New York, 1992. [82] K. Yamagata. Extensions over hereditary artinian rings with self-dualities, I. J. Algebra, 73:386–433, 1981. [83] J. M. Zelmanowitz and W. Jansen. Duality modules and Morita duality. J. Algebra, 129:257–277, 1989.

Index

∗-module, 18 ∗-module weak, 15 [D :m] upper triangular matrix ring, 56

equivalence, 12, 13, 15, 17–19, 22, 24, 25, 135 ≈, 12 induced by a progenerator, 25 representable, 12

abelian subcategory, 86 Add(V ), 8 adjoint functor arrow of adjunction, 12 quasi-inverse, 12 adjoint functor, 132 arrow of adjunction, 133 quasi-inverse, 133 almost disjoint, 95 artin algebra, 9, 33

!, 86, 92 Gen, 2 gen, 2 generalized Morita duality, 109 GMD, 109 Grothendieck group on mod-R, 49 on proj-R, 54

balanced faithfully, 22 bimodule tilting, 33

Kupisch series, 141

ci (M), 52 Cogen, 10 cogen, 10 Copres, 14 copresented, 14 finitely copresented, 68 copresented semi-finitely copresented, 68 costar module, 69 cotilting bimodule, 97 cotilting module, 92 finitistic, 115 cotilting theorem, 113 , 66, 92 dense, 26 duality artin algebra duality, 33 duality, 66 representable duality, 66 duality module, 84

injective relative to, 75

linearly compact, 78 Mod, 1 mod, 1 module cotilting module, 92 quasi-duality module, 75 Morita duality, 84, 103 generalized, 109 weak, 109 Morita equivalence, 22, 24 partial tilting module, 36 generalized, 37 ⊥ ⊥ U , 91 ⊥ V ⊥, 1 Pres, 14 presented, 14 Prod(V ), 8 progenerator, 22 proj.dim., 1 projective dimension, 1

151

152 projective relative to, 22 pure-exact, 94 pure-injective, 94 quasi-duality module, 75 quasi-injective, 75 quasi-progenerator, 22 quasi-projective, 18, 22 quasi-tilted, 49 reflexive, 67 η-reflexive, 13 ν-reflexive, 13 reject, 10 representable equivalence, 15 ring extension, 26 scopres, 71 self-small, 6, 17, 21 serial ring, 139 small module, 6, 7 self-small, 6 solvable, 81 finitely solvable, 81 split extension, 63 subcategory, 12, 86 abelian subcategory, 86 finitely closed, 75

Index tilted ring, 48 tilting bimodule, 33 tilting module, 23, 31 generalized, 28 Tilting Theorem, 40 tilting torsion class, 34 tilting torsion theory, 34 torsion class, 10 torsion submodule, 11 torsion theory, 9 split, 46 torsion theory counter equivalence, 58, 62 torsion-free class, 10 torsionless, 67 trace, 2 trivial extension, 63, 64 U -dense, 82 U -linearly compact, 82 U -torsionless linearly compact, 119 V ∗ , 14 weak Morita duality (WMD), 109 WMD, 109 Yoneda Lemma, 131

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