Buchsbaum rings and applications: an interaction between algebra, geometry and topology

JU:rgen StUckrad Wolfgang Vogel

.Buchsbaum Rings and Applications An Interaction Between Algebra, Geometry and Topology

•

Springer-Verlag

Jiirgen Stiickrad

Wo1fgang Vogel

Buchsbaum Rings and Applications An Interaction Between Algebra, Geometry and Topology

Springer. Verlag Berlin Heidelberg New York London Paris Tokyo

/

Dr. Jurgen Stuckrad KarI-Marx-Universitat Leipzig Department of Mathematics D DR - 7010 Leipzig

Prof. Dr. Wolfgang Vogel Martin-Luther-Universitat Halle- Wittenberg Department of Mathematics DDR-4020 Halle

L( 1 E With 3 Figures

Sole distribution rights for all non-socialist countries granted ,to Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Mathematics Subject Classifikation (1980): 14M05, 13H1O, 13H15, 05A20, 55U99

ISBN 3-540-16844-3 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-16844-3 Springer-Verlag New York Berlin Heidelberg

Library of Congress Cataloging-in-Publication Data Stiickrad, Jiirgen, 1948Buchsbaum Rings and Applications. Bibliography: p. Includes index. 1. Buchsbaum Rings. 2. Geometry, Algebraic. 3. Algebraic topology. 4. C.ommutative algebra. 1. Vogel, Wolfgang, 1940II. Title. QA 251.3.877 1986 512'.24 86-17880 ISBN 0-387-16844-3 (U.S.) by VEB Deutscher Verlag der Wissenschaften, Berlin 1986 Printed in the German Democratic RepUblic

@

Bindearbeiten: K. Triltsch, Wiirzburg 2141/3140-543210

Preface Da die algebraische Geometrie weaentIich yom Fundamentalsatz der Algebra ansgeht, den man nur deshalb in der gewohnten allgemeinen Form aussprechen kann, weil man dabei die Vielfachheit der LOsnngel'll in Betracht zieht, so muB man anch bei jedem Resultat der algebraischen Geometrie beim Znriickschreiten die gemeinsame QneUe wiederfinden. Das ware aber nicht mehr moglich, wenn man auf dem Wege das Werkzeug verlore, welches den Fundamentalsatz fruchtbar nnd bedeutnngsreich macht. Francesco Severi Abh. Math. Sem. Hansischen Univ.

15 (1943), p. 100

This book describes interactions between algebraic geometry, commutative and homological algebra, algebraic topology and combinatorics. The main object of study are Buchsbaum rings. The basie underlying idea of a Buchsbaum ring is a continuation of the well.known concept of a Cohen-Macaulay ring, its necessity being created by open questions of algebraic geometry and algebraic topology. The theory of Buchsbaum rings started from a negative answcr to a problem of David A. Buchsbaum. The concept of this theory was introduced in our joint paper published in 1973. In presenting our treatment of algebraic geometry, it is a pleasure to acknowledge the help and encouragement which we have had from all sides. Some decisive results eame from the applications of homological algebra to derived categories, an approach we learned in joint discussions with Reinhardt Kiehl. A further development of these ideas, with a view towards their topological applications, came in our long collaboration with Peter Schenzel; to both colleagues go our special thanks. In addition, without using the theory of derived categories, we will describe a different approach to these applications with respect to liaison and combinatorics. This book has profited from the research investigations of a number of doctoral theses. In particular some ideas used here first appeared in the theses of Markus Brodmann, Juan C. Migliore, Peter Schcnzel, Philip W. Schwartau and Ngo Viet Trung. They are presently widely available in standard mathematical journals. Our treatment has also profited from many publications of our Japanese colleagues. ~'or example, we have gained greatly from the work of Shiro Goto, Yoichi Aoyama, Shin Ikeda, Yasuhiro Shimoda and Naoyoshi Suzuki. Among others whose suggestions have served us well, we note David A. Buchsbaum, David Eisenblld, Heisuke Hironaka and Balwant Singh. To all these and others who have helped us, we express our sincerest thanks.

6

Preface

In the late stages of polishing the manuscript wc received valuable suggestions from Henrili; Bresinsky and GUnther Eisenreich. We would like to thank the staff of Deutscher Verlag der Wissenschaften, especially Erika Arndt for her enormous patience and skill in converting a very rough manuscript into book form .

•

Leipzig and Halle, Spring, 1986

JUrgen StUckrad Wolfgang Vogel

Table of Contents

Preface . . . . . . . . . . . .

5

introduction and some examples .

9

Chapter 0 Some foundations or commutative and homological algebra . §t. § 2. § 3. § 4.

§1. § 2. § 3. §4.

45 52

Chapter I Characterizations of Buchsbaum modules

62

Characterization of Buchsbaum modules by systems of parameters Cohomological characterization of Buchsbaum modules Graded Buchsbaum modules. . . . . . . . . . Segre products of graded Cohen.Macaulay modules

Chaptern Hochster·Reisner theory for monomial Ideals. An Interaction between algebraic geometry, algebraic topology and comblnatoric8 §t. § 2. § 3. § 4.

§t. § 2. § 3. §4.

§t. § 2. § 3.

21 21

Local algebra and homological algebra Graded modules and Kiinneth formulas . I..ocal duality . . . . . RelilOlutions and duality. . . . . . . .

33

62 70 95 99

106 107

Foundations. . . . . . . . . . . . . . . . . . . . The homological Cohen-Macaulay criterion of Reisner. The topological Cohen· Macaulay criterion of Schwartau . Further applications to algebraic topology and combinatorics

123 132

Chapter m On liaison among curves in projective three space

155

On liaison among arithmetical Buchsbaum curves in p3 On liaison addition and applications . . . . . On curves linked to lines in p3 and applications On self-linked curves in p3 . . . _ . . . . .

174 181 192

Chapter IV Rees modules and associated graded modules of a Buchsbaum module

199

Some preliminary results . . . . . . . . . . . . . . . . . . . . The Buchsbaum property of Rees modules and associated graded modules . Blowing-up characterization of Buchsbaum modules . . . . . _ _ . . _

201 207 223

115

156

8

§1. § 2. § 3. § 4. § 5.

Table of Contents

Chapter V Further applicatiollll and examples

229

A Buchsbaum criterion for affine semigroup rings . . . . . . Some examples related to problems of Hironaka and Seidenberg On Buchsbaum rings obtained by glueing . . . . . . . . . . Construction of Buchsbaum rings with given local cohomology . Some examples of Segre products. . . . . . . . . . . . . .

229 234 238 242 245

Appendix On generalizatiollll of Buchsbaum modnles

252

Bibliography.

268

Notations

282

Index . .

284

Introduction and some examples

The aim of this introduction is to preEent the idEaS of our theory of Buchsbaum modules with a wealth of background material. The different viewpoints which are treated in the works of Laskcr-Macaulay-Grobncr and Severi-van der Waerden-Weil concerning the multiplicity theory for Bezout's theorem are basic to the understanding of the theory of Buchsbaum modules. The simplest case of Bezout's theorem is the following very simple but fundamental principle in the field of complex numbers. Fundamental principle. The number of roots of a polynomialj(x) in one variable, counted with their mUltiplicities, equals the degree of j(x). The definition of this multiplicity is well-known and clear. The next simple case to consider is that of plane curves. The problem of the intersection of two algebraic plane curves was already tackled by Newton; he and Leibniz had a clear idea of "elimination" processes which describe the fact that two algebraic equations in one variable have a common root. Using such a process, Newton otlerved in his "Geometria analytica", published in 1680, that the abcissas (for instance) of the intersection points of two curves of respective degrees m, n, are given by an equation of degree m· n. This result was gradually improved upon during the 18th century, until Bezout, using a refined elimination process, was able to prove in general that the equation giving the intersection had exactly the degree m· n; however, no general attempt was yet made during that period to attach to each intersection point an integer measuring the "multiplicity" of the intersection, in such a way that the sum of multiplicities would always be m . n. Therefore the classical theorem of Bezout states, that two plane curves of degree m and n, intersect in at most m . n different points, unless they have infinitely many points in common. In this form the theorem was also stated by Maclaurin in his "Geometrica organica'" published in 1720 (see p. 67/68) but the first correct proof was given by Bezout. An interesting fact, usually not mentioned in the literature, is that Bezout proved in 1764 not only the above-mentioned theorem, but already the following n-dimensional version: Let X be an algebraic projective variety of a projective n-space. If X is a complete intersection of dimension zero then the degree of X is equal to the product of the degrees of the polynomials defining X. The proof can be found in the papers of Bezout [1,3] and r2]. In his book "Theorie generale des equations algebriques", published in 1779, a statement of this theorem can be found already in the foreword. We quote from page XII: "Le degre de l'equation finale resultante d'un nombre quelconque d'equations complettes, renfermant un pareil nombre d'inconnues, & de degres quelconques, est egal au produit des exposans des degres de ces equations. Theoreme dont la verite n'etoit connue et demontree que pour deux equations seulement."

10

Introduction and 80me examples

The theorem appears again on page 32 as Theorem 47. The special cases n = 2, 3 are interpreted geometrically on page 33 in Section 3° and it is being mentioned there, that these results are already known from geometry. (For these historical remarks see also Renschuch [1], Dieudonne [1], Vogel [6].) To-day we have the following modern statement of Bezout's Theorem, where the degree of a variety X P" of dimension d, denoted by deg(X), is the number of points in which almost all linear subspaces L c: P" of dimension n - d meet X. Bezout's Theorem. Let X, Y be unmixed (i.e. each component of X and r h.as the dimensiA:m of X or Y, resp.) varieties of the projective n-space P'k over an algebraically cWsed field K 8uch that dim(X n Y) dim(X) dim( Y) - n. Letting 0 run over all proper components of the intersection X n Y (i.e. (~rreduc~"ble) components with dimension equal dim(X n Y» we get that there exist '~1Uersection multiplicities', 8ay i(X, Y; 0), of X and Y along 0 such that the following 'Anzahl-Formel' i8 true:

+

deg(X) • deg( Y)

= E i(X,

Y; 0) • deg(O). c Here the number i(X, Y; 0) itself measures the degree of contact of X and Y along O. It represents fairly sophisticated concepts in full generality and it has taken a century or two and a lot of work to be developed, see also S. Kleiman [1, 2]. To get equality in the above equation one may follow different approaches to arrive at several different multiplicity theories. It is well-known that there is no loss of generality in assuming for projective varieties that one variety is a complete intersection in order to define local intersection multiplicities. This statement does indeed follow from Samu~l's book [1], p. 81. However, it is not clear to us that it is possible to prove a global statement like Bezout's Theorem by reducing in a simple fashion to the case in which one of the two intersecting varieties is a complete intersection. Therefore we would like to present the high points of the proof of the following theorem: Theorem 1. Let X, Y be arbitrary varietie8 of Pic with dim(X n Y) :;::: O. Then there exi8t varietie8 X', Y' of p~+l such that one variety, say X', i8 a complete intersection with

deg(X) . deg( Y) = deg(X') . deg( Y') and there i8 a 1-1 corresporulR:nce between the components 0 of X n Y (in Pic) with dim(O) = dim(X n Y) and the components 0' of x' n Y' (~n p~+1) with dim(O') = dim(X' n Y') and that this correspondence preserves d~mensions and degrees. Proof (see also Vogel [6]): We will apply idealtheoretic methods. Let X, Y c: Pic be our projective varieties with defining ideals a and b in K[xo, ... , xnJ =: Rit and dimensions d and 0, resp. We introduce a second copy K[yo, •. 0' Y.. ] =: Ry and denote by b' the ideal in Ry corresponding to b. We consider the polynomial ring R := K[xo, ... , Xn> yo •. Y.. ] and the ideal c = (xo - Yo, Xn - Yft). Let ho( ... ) be the (rectified) leading coefficient of the Hilbert polynomial of the homogeneous ideal (... ). 0

0,

0

•• ,

Claim. ho(a . R:r) ho(b . R It ) = ho(a R + b' . R), where a . R is the extension ideal of a in R. 0

0

This follows from R/(a

+ b') . R '" R:r/a • Rit ®K RyJb' . RII ,

Introduction and some examples i.e. the Hilbert function H(n, (a+o')· R) of (a+o'). the Hilbert functions of a . R" and 0' . R II , that is

I:

H(n, (a+o') . R)

R can

11

be expressed in terms of

H(i, a· R z )' H(1, 0'· R II ).

l+j=n

The degree and the leading coefficient of the Hilbert polynomial of (a + 0') . Rare given by d + ~ + 1 and k o( (a +0') . R), resp. We choose an integer r such that the Hilbert functions H(i, a· R",) =: Hi and H(i, 0' . RII ) =: Hi are given by their Hilbert polynomials hi and hi, resp. for i > r. Then we can decompose for n 0 (n > .27): n

I:" Hi' H~_i

r

hi' k~_i

i=O

+i=O I: (Hi -

Some calculations involving the coefficient of

ita

Hi .

H~_i

ho(a) . ho(o') [ita

= ho(a) • hoW)' (

II

hi) . h~_i

nd+d+l

+i=n-r I: hi(H~_i

therefore yield:

(!). (n ~ t)] + n

d+~+1

)

n:,.-i)·

(other terms)

+ (lower order terms),

and our claim follows. It is well-known that the degree of a projective variety is equal to the (rectified) leading coefficient of its Hilbert polynomial (see, e.g., D. Mumford [5], p. 112, Theorem 6.25). Therefore we get our first statement of Theorem 1 where X' and Y' are defined by c and (a + 0') in R, resp. The second statement follows from the fact that there is a 1-1 correspondence between the isolated prime ideals a + 0 in R", and the isolated prime ideals of (a +0') . R + c . R in R. This correspondence is given by

Rz ::J ~

H>

(~

+ c)

R

and it therefore preserves dimensions and degrees, q.e.d. S. Kleiman [3] pointed out to us that the reduction of Theorem 1 may be accomplished by replacing the original varieties X, Y of P" by the varieties X', Y' of P2f1+1 which may be described as follows. Take three n-planes in P2"+1 in general position. Embbed X in the first, Y in the second and take Y' to be their join. Take X' to be the third plane. The claim in Theorem 1 is apparently proven by showing that the degree of the join Y' is equal to the product of the degrees of X and Y. We note that the reduction of Theorem 1 is not only to the case in which one of the varieties is a complete intersection but even a linear f!pace. At the beginning of this century one investigated the notion of the length of a primary ideal in order to define another intersection multiplicity. This multiplirity is defined as follows: Let X, Y c: P" be arbitrary varieties. Let 0 be a component of X n Y such that dim(O) = dim(X n Y). Denote by A(X, 0) the local ring of X at O. We set /L(X, Y; 0):= length of A(X, O)jI(Y)· A(X, 0),

where I(Y) is the defining ideal of Y. This length /L(X, Y; 0) is well-defined and is called the idealtkeoretic intersection multiplicity of X and Y at O. For instance, this

12

Introduction and some examples

multiplicity is the intersection multiplicity as set forth in the beginning in the case of projective plane curves. Prior to 1928 most mathematicians hoped that this multi- ' plicity would provide for Bezout's Theorem always the correct intersection multiplicity. In 1928 B. L. van der Waerden studied Macaulay's famous space curve (see Macaulay [1], p. 98) to show, that this idealtheoretic intersection multiplicity does not yield the correct multiplicity for Bezout's Theorem to be valid in projective spaces P" with n;;:: 4. We quote van der Waerden [1], p. 770: "In these cases we must reject the notion length and try to find anothcr definition of multiplicity". Nowadays it is of course well-known that p,(X, Y; 0)

=

i(X, Y; 0)

if and only if the local rings A(X, 0) of X at 0 and A(Y, 0) of Yat 0 are CohenMacaulay rings for all proper components 0 of X n Y where dim(X n Y) dim (X) + dim(Y) - n, see J.-P. Serre [2], p. V·20. Without loss of generality we may now suppose by applying our Theorem 1 that one of the two intersecting varieties X and Y is a complete intersection, say Y. With this assumption we get that p,(X, Y; 0)

i(X, Y; 0)

for each proper component O. Let Y be a complete intersection. Then there arises another problem posed by D. A. Buchsbaum [1] in 1965, as follows: Problem (from the viewpoint of the theory of intersection multiplicities). Is it true that p,(X, Y; 0) i(X, Y; 0) is independent of Y; that is, does there exist an invariant, say I(A), of the local ring A := A(X, 0) of X at 0 such that p,(X, Y; 0)

i(X, Y; 0)

I(A)?

From the viewpoint of local algebra we get the problem in its original foml: Let A be a local ring of dimension d 1 with maximal idealll't. Let q be an m-primary ideal which is generated by a system of parameters. Denote by eo(q; A) the multiplicity of the ideal q (see, e.g. Zariski-Samuel [1], Vol. II, Chap. VIII, § 10) and by l(Alq) the length of Alq over A. Is it then true that l(Alq) eo(q; A) is independent of q? For instance, is l(Alq)

eo(q; A)

= dim(A) -

depth(A)?

We will show that this is not always the case. From the viewpoint of the theory of intersection multiplicities we want to study our first counter-example. Example 1. Let X' c

Pk be the non-singular curve given parametrically by

Let Xc Pk be the projective cone over X'. We consider the surfaces Y and Y' in projective 4-space defined by the two hypersurfaces

Introduction and some examples

Let t:' be the defining ideal of X in K[xo,

Xl' X2, Xa, Xf].

13

Then

and Hilbert's characteristic polynomial of t:' is given by H(n,t:')

= 5· (;) + 6· (~)

- 3

(see, e.g. Renschuch [2], 8.2); that is, the degree of X is 5. Now, we see that X n Y and X n Y' intersect at the vertex 0: (0,0,0,0) of X, and Bezout's Theorem therefore gives us i(X, Y; 0)

= 5

and

i(X, Y'; 0)

10.

Since (t:' + (Xl' X,») = (Xl> X" X2Xa, x~, x~) it is easy to see that ,u(X, Y; 0) It remains to calculate the length p,(X, Y'; 0). We have

(t:'

+ (X2' xi + x~») = (X2' xi + x:, XIX., xix;, XIX;, x~)

ql

and we can construct the following chain of primary ideals belonging to ql c:: (ql' xix,) =: q2 c:: (q2, x;x,)

qa

(qa, XIX;)

c:: (q" xixa) =: qs c:: (qs, Xa X4) =: q6

(q6' XIXa)

c:: (qs, X~) =: q9 c:: (q9' Xa) =: qlO

(qIO' xi)

= 7.

(Xl> X2,

Xa,

X4) :

q, : q7 c:: (q7' xi) =: qs ql1 c:: (qw Xl) =: q12

c:: (Xl' X2, xa, X,) =: q13' It follows that p,(X, Y'; 0).= 13; that is, 2

=

p,(X, Y; 0) - i(X, Y; 0) 9= p,(X, Y'; 0)

i(X, Y'; 0)

3.

The theory of Buchsbaum modules started from such a negative answer to the above problem of D. A. Buchsbaum (see Vogel [4]). The concept of Buchsbaum modules was introduced in Stiickrad-Vogel [2] and [3], and the theory is now developing rapidly; see, for example, the following Symposium: Study of Buchsbaum rings and generalized C-Ohen-Macaulay rings. Proceedings of a Symposium held at the Research Institute for Mathematical Sciences, Kyoto University, Kyoto 1982. The basic underlying idea of a Buchsbaum module generalizes the well-known coneept of a Glhen-Macaulay module, its necessity being created by open questions in Commutative Algebra and Algebraic Geometry. For instance, such a necessity to investigate generalized Glhen-Macaulay structure occurs when one classifies algebraic curves in p3 or when one studies singularities of algebraic varieties. Furthermore S. Goto and Y. Shimoda [1] discovered that the Cohen-Macaulay property of Rees algebras of parameter systems can be described by certain Buchsbaum rings. Also it was shown that interesting and extensive classes of Buchsbaum rings do exist. We now introduce the definition of a Buchsbaum module and we describe a first geometrical interpretation of Buchsbaum modules. Let A be a Noetherian local ring with maximal ideal m. Let M be a finitely generated A-module.

14

Introduction and some examples

Definition 1. A sequence ai' ... , af of elements of m is a weak M-8eq:uence if for each i = 1, ... , r

(for i

=

1 we set (al> ..., ai_I)

=

(0) in A).

We denote by eo(q; M) the multiplicity of M relative to a parameter ideal q of M, i.e., an ideal of A generated by a system of parameters for M. Then as our first important result which explains the notion of Buchsbaum modules in connection with the above problem of D. A. Buchsbaum (see Chap. I, § 1) we get Theorem 2 and definition. The 10Uowing conditiOnB aTe equivalent: (i) Jf is a BucMbaum module. (ii) The diflerence 01 length lA(M/q . ~W) and multiplicity €o(Q; M) 01 M is an integer,

say I(M), 01 M not dependt'ng on the choice of the parameter ideal q of M. (iii) Every system 01 parameters lor M is a weak M-sequence. Note that a Noetherian local ring is said to be a BucMbaum n'ng if it is a Buchsbaum module over itself. We want to give some simple examples. Example 2. A finitely generated module M is Cohen-Macaulay if and only if M is Buchsbaum and I(M) 0, that is the class of Buchsbaum modules contains the CohenMacaulay modules. Example 3. Let A be a local ring with maximal ideal m. Let M be a Cohen-Macaulay module over A of dimension d > 1. Let N be a submodule of M such that the factQr module MIN is a finite-dimensional vector space over A/m, say EB A/m. Then N is a I

Buchsbaum module over A with the invariant I(N) = (d

1) . t.

Prool: Let q c A be an ideal generated by a system of parameters with respect to N. Take the exact sequence 0 -+ N -+ M -+ MIN -+ O. Since M is a Cohen-Macaulay module and m . MeN we get the following exact sequenee by applying the functor Aiq ®A: 0-+ Tort(A/q, MIN) -+ N/q . N -+ M/q . M -+ 1W/N -+ O. This is an exact sequence of A-modules with finite length. We obtain therefore that q is also a parameter ideal of M and that

lA(Njq· N)

l(Mjq .•W)

(d - 1)· t

since lA(Tor1(Ajq, MIN)) t .lA(Torl(AIQ, Aim)) = t· lA(qjq . m) = t . d. On the other hand the additivity property of the multiplicity symbol (see Cbap.O, § 1) implies 1) . t does not eo(q; N) = eo(q; "Wi. Thus the difference lA(Niq . N) - eo(q; N) = (d depend on the choice of q since eo(q; M) = l(Mjq . M). Therefore N is a Buchsbaum 1) . t, q.e.d. module with I(N) = (d This example has some useful applications. (3a) Let A be a regular or Cohen-Macaulay local ring of dimension d 2. Then the maximal ideal m of A is a Buchsbaum module which is not Cohen-Macaulay.

Introduction a.nd some examples

15

(3b) Take Macaulay famous curve X in p3 given parametrically by {s', sSt, S/3, 14} (see Macaulay [1], p. 98). This curve was studied by F. S. Macaulay as early as 1916. His purpose was to show that not every prime ideal in a polynomial ring is perfect. We will show that X is arithmetically Buchsbaum; that is, the local ring of the (affine) cone over X at the vertex is a Buchsbaum ring. Let K be a field and R K[ s, t] a formal power series ring in s and I. We put S = K[s', s3t, s&, t4] in R. Take the normalization T of S. Then T is a Cohen-Macaulay K[S4, s3t, s2t2 , st3, t4]. It is easily seen that the ring of dimension 2 and we have T conductor of S in T is the maximal ideal of Sj that is, TIS is a vector space generated by the element s2t2 • Hence we get that S is a Buchsbaum ring with invariant 1(S) 1. (3c) (See also Herrmann-Schmidt [1].) Let K be a field with char K =l= 2 and S = K[x, y] a polynomial ring. We put R = {f E S If(l, 0) = f( -1, O)}. Then R is the finitely generated subring K[1 x2, xy, y, x - x 3 ] of S. So X = Spec(R) is realized as a surface in the affine space Ai- which is non-singular in codimension 1, but with an isolated singularity at the origin. Applying the above statement we see immediately that the local ring of X at the origin is a Buchsbaum ring. Note that this example (3c) is a very simple example of Buchsbaum rings obtained by glueing (see (Joto [4]).

Example 4. 111 is a Buchsbaum module over A if and only if $I is a Buchsbaum module over .A in which case 1(111) I($I). Here denotes /\the m-arlic completion. (See also Lemma 1.1.13,) In order to describe examples from the viewpoint of the theory of intersection multiplicities we need some special results on Buchsbaum modules. We summarize these assertions in the following theorem. The proofs are given in Chapter I under even more general conditions. Let A be a local ring and let a A be an ideal of A. Denote by U(a) the intersection of the primltry ideals q belonging to a with dim(q) = dim(a).

Theorem 3. Let A be a local ring of dimension d;::: 1 with maximal ideal m. The following two conditions are equivalent: (i) A is a Buchsbaum ring. (ii) For each part all "., ak of a system of parameters of A we have

m· U{(al' ,." a~J) r: (at> ... , ak) for every k = 0, ... , d - 1. (notice the case d I!) Furthermore, we have the folkrwing statements. (iii) Let d > 2. Lei a, 0 be ideals of A 8uch that the intersection a n V is the zero ideal of A. Assume that A/a and A/v are Oohen-1I1acaulayrings of dimension dand dim{AI(a +0)) = 0, then m a + 0 if and only if A is a Buchsbaum ring. (iv) Let d > depth(A) 1. A is a Buchsbaum ring if and only if there exists in A a non-zero divisor x E m2 such that the ring A/(x) is a Buchsbaum ring. (Lifting Buchsbaum mod a non-zero divisor.) As an application of the statements (i), (ii) of Theorem 3 we get the following examples:

Example o. Let K be any field and set A:= K[x, y]/(x) n (x 2 , y). Then A is a Buchsbaum non-Co hen-Macaulay ring. If A := K[ x, y]/(x) n (xS, y) then A is not a Buchsbaum ring.

16

Introduction and some examples

From the vantage point of the theory of interseetion multiplicities we can construct the following examples by use of the statements (iii) or (iv) of Theorem 3. Example 6. Let X be the union of two planes in four-dimensional affine space A4 meeting at the point P: Xl X2 Xa X4 0; that is the ideal of X is (Xl' X2) n (Xa, X4)' Let A be the local ring of X at P; i.e.

Our statement (iii) implies that A is a Buchsbaum ring. It is well-known that A is not a Cohen-Macaulay ring. Example 7. Let X be the rational twisted cubic curve in p3 with defining ideal p (XoX2 - xi, XoXa XIX2, XlXa x~) in K[xo, Xl> X2' xaJ for any field K. It is wellknown that p is perfect. We will show that p2 is Buchsbaum; that is, we claim that thc local ring of the affine cone over X, counted with multiplicity 3 at the vertex, is a Buchsbaum ring with invariant 1. Prool: Note that p2 is the defining ideal of the curve X counted with multiplicity 3. Furthermore, we have that p2 is a primary ideal (see, e. g., Achilles-Schenzel-Vogel [1}). Therefore we can apply the statement (iv) of Theorem 3. By using localization it is not hard to calculate the intersection U(p2, x~) of the prhnary ideals belonging to p2 + (x~) of dimension 1. It then follows that (xo, Xl> X2, xa) . U(p2, xi) C p2 (x~). The statements (ii) and (iv) yield our assertion. It is also not too difficult to calculate the in"\(ariant by using a system of parameters.

+

Having this example we can construct new irreducible and reduced arithmetical Buchsbaum curves by using the theory of residual intersection. Therefore we will apply liaison among curves in p3 coupled with our deep result from Theorem III.1.2, on liaison, obtained by applying the theory of dualizing complexes. This r~ult, or the theory of residual intersection for the special case of curves, results in the following. Theorem 4. Assume that the scheme theoretic unicm 01 two curves Xl' X 2 01 P::" over an algebraically closed lield K i8 the CfJmplete t'ntersection 01 two hypersurlace8, and that the curve8 have no CfJmponent8 ~n CfJmmon (Xl and X 2 are then said to be hnked geometrically, see C. Peskine, L. Szpiro [3J and A. P. Rao [11). Xl is arithmetically Buchsbaum (i.e. the local ring 01 the alline cone over Xl at the vertex is a Buch8baum ring) il and only il X 2 is arithmetically BW'kbaum. Take, for example, Macaulay's curve X Xl and the curve X 2 with defining ideal (xo, Xl) n (X2. xa). Then it is not hard to show that

a=

pna=

(xoXa -

X1X2' xo~ -

x~xa).

Applying Theorem 4 and Example 6 it therefore follows again that Macaulay's curve is arithmetically Buchsbaum. Notice, that we have used only a simple calculation on p n a. This liaison was discovered by G. Salmon [1], p. 40, already in 1848 and a little later again by J. Steiner [lJ, p. 138, in 1857. Now, consider our Example 7. Let Y be the curve X counted with multiplicity 3. It follows from K. Rohn [21 that Y is linked for instance, to an irreducible and reduced curve Ci~a of degree 43 and genus 213 by two hypersurfaces of degree 4 and 13. We thus

Introduction and some examples

17

get that the liaison equivalence class corresponding to a vector space of dimension 1 contains also the curve 0i~3. Using the theory of liaison and Buchsbaum rings we also obtain new statements concerning the classification of algebraic curves in projective 3-space P~. For instance, let O~ be an irreducible and non-singular curve in p~ of degree 6 and genus 3 with defining ideal 1(0:) in 8 := K[xo, Xl> x 2 , xa]. It follows from M. Noether [1], p. 87, (aa) and (a~) and Theorem 4 that either is arithmetically Cohen-Macaulay, or is arithmetically Buchsbaum. We note that in 1881 F. Schur [1] discovered a first difference between the curves to distinguish between them. Nowadays the resolution of the curve is well-known if is arithmetically Cohen-Macaulay. It is (see also G. Ellingsrud [1] or L. Gruson and C. Peskine [1]):

0: 0:

0:

0:

0->- 83( -4)

->-

0:

8 4( -3) ->- 8 ->- 8/1(0:) ->- O.

0:

If is arithmetically Buchsbaum then we have obtained in addition the resolution of O!: 0->- 8( -6) ->- 8 4( -5) ->- 8( -2HB 8 a( -4) ->- 8 ->- 8f 1(0:) ->- O.

(See also Chap. III, § 1.) For arithmetically Buchsbaum curves 0 in p3 we get that the invariant l(A) of the local ring A of the affine cone over 0 at the vertex is given by l(A) = dim (~H1(P3,

Jdv)})

where J c is the ideal sheaf of the curve O. Having this arithmetically Buchsbaum property of a variety in P" we want to describe another geometric interpretation of the invariant l(A) of the corresponding (local) Buchsbaum ring A of surfaces. If X is any projective variety, the finite-dimensional vector spaces HI(X, Ox) are important invariants of X. One of the most interesting is the alternating sum of their dimensions, obtaining the so-called arithmetic genus:

P.(X)

dim H"(X, Ox) - dim H"-1(X, O.d

+ .. , + (_1)11-1 dim H1(X, Ox),

where n is the dimension of X. One drops the term dim HO(X, Ox) advantage that for X a curve,

Pa(X)

= dim

H1(X, Ox)

=

1, which has the

usual genus of X.

But, when X is a surface, we get

The point here is that the Italian geometers regarded dim H2(X, Ox) as the dominant term, and called it, for non-singular surfaces, the geometric genus; while dim H1(X, Ox) was considered a "correction" term, and was called the irregularity. Now, consider any arithmetical normal irregular (I.e. the irregularity =1= 0) surface F, such that we have H1(F, OP(p») =1= only for P O. Let A be the local ringofthe vertex of the (affine) cone over F. Applying our cohomological investigations from Chapter I, § 2, we get: A is a normal non-Cohen-Macaulay-Buchsbaum ring where the invariant l(A) of A is given

o

2

Buchsbaum Rings

18

Introduotion and some examples

by the irregularity of F, that is I(A)

=

dim Hl(F, OF)'

It is not 'hard to construct an arithmetically normal irregular surface F free of singularities. For instance, this will be the case if F is the Segre embedding of pairs of points EX G of any two plane curves E, G free of singularities, where at least one of them has a positive genus. For example, let E pI, and let G be the cubic defined by x~ x~ x~ in pl!. Let F be EX G in its Segre embedding in po. For such surfaces F we get that the vertex of the affine cone over F is a" normal Buchsbaum singularity which is not a Cohen-Macaulay singularity. This gives an answer to a question posed by H. Hironaka in a discllssion (at the University of Halle in 1974), who asked whether we can construct normal non-Cohen-Macaulay-Buchsbaum singularities, and which aroused our interest in the subject of singularities (see also Chap. V, § 2). To motivate the study of other Buchsbaum singularities wc quote the following remark made by D. Mumford in [2) on p. 42: "Incidentally, one should regard the depth of 0 (a local ring) itself, for example, as a measure of the topological complexity of the singularity at the closed point of Spec(O): if the depth is maximal, i.e., equals the dimension of 0, then 0 is in a weak sense, nonsingular, while if the depth is much less then the dimension, the singularity is very bad." As mentioned before, D. A. Buchsbaum (see the above problem) and also A. Seidenberg [2], on p. 620 considered the difference between the dimension and depth of any local ring taking it as a measure of the deviation from the Cohen-Macaulay property. With our theory of Buchsbaum rings we will study examples of projective varieties in Chapter I (see Examples 4.14) and Chapter V, § 2, which show that this measurc does not describe the non-Cohen-Macaulay property satisfactorily. Shiro Goto and Yasuhiro Shimoda [1] have introduced another aspect to the study of Buchsbaum singularities. Here certain Buchsbaum rings are characterized by the beha:viour of Rees algebras relative to parameter ideals. We conclude our considerations by briefly discussing a main result from Goto-Shimoda [11.

+ +

Theorem o. Let A be a local ring with maximal ideal m. Let H~(A) denote the ith local cohomology module oj A with respect to m. The jollowing two conditions are equivalent: (i) A is a Buch.ibaum nl/{/ and H~(A) (0) jor i =1= 1, dim(A). (ii) The Rees algebra R(q) = q" is a Oohen-Macaulay rt'ng jor every parameter ideal q oj A. fl~O This striking theorem gives a complete answer to a problem of Rees algebras of powers of parameter ideals. It is stated as follows (see Goto-Shimoda [1]):

Corollary 6. Let A be a local ring and assume that depth(A) 1. Then A is a Cohenil1acaulay ring ij and only ij the Rees algebra R(q") is Cohen-Macaulayjor every parameter ideal q oj A, jor all integer n > O. Also, Shiro Goto [5] obtains the following result. Let A be a local ring of dimension d> O. Then A/H~(A) is a Buchsbaum local ring if and only if Proj R(q) is a CohenMacaulay scheme for every parameter ideal q auf A. In a letter to one of the authors, dated February 8, 1980, he underlines its significance: "I believe that the above blowingup characterization of Buchsbaum rings really clarifies .the importance of the concept

Introduction and some examples

19

of Buchsbaum singularities." Therefore the main object of our study of Chapter IV is the stability of Rees rings and form rings with respect to the Buchsbaum property of parameter ideals. This chapter has its origin in the effort to extend Hironaka's desingularization to a more general situation due to G. Faltings [1] and M. Brodmann (1] (see also the introduction of Chapter IV). We conclude our introduction by briefly discussing an application of the theory of Buchsbaum modules to the so-called Upper Bound Conjecture which establishes some interesting connections among algebraic topology, commutative algebra and combinatorics (see Chap. II). Let A denote an abstract finite simplicial complex with vertices Xl, ••. , X•. That is a family of subsets of {Xl' ••• , X.} such that if a E ,,1 and 1:' C a then 1:' E A, and such that the vertices are in A. We call the elements of A laces. If the largest face of A has d elements, then we say dim A = d - 1. The I-vector of LI is I (f -1> 10, ... , Id-d where exactly Ii faces of A have i + t elements and 1-1 = 1. The Upper Bound Conjecture now states that the number of i-dimensional faces Ii of A is less than or equal to a certain number ci(n, d). In 1975 R. P. Stanley (1] solved the Upper Bound Conjecture for spheres and, more generally, for all simplicial COIllplexes A such that the assoeiated graded k-algebra k(Al for an arbitrary field k is a Cohen-Macaulay ring. In 1976 G. A. Reisner described those simplicial complexes for which k[A] is a Cohen-Macaulay ring. We now define A to be a Buchsbaum complex if k[ A] is a Buchsbaum ring. For instance, if the geometric realization X = iA i of A is a connected manifold, then L1 is a Buchsbaum complex. I.et IAI, for example, be the torus. We know that a major objection to the liRe of simplicial complexes in computing topological invariant of compact polyhedra is that the dissection of the polyhedron may require an uncomfortably large number of simplexes. Thus the r.lOst obvious dissection of the torus, which is pictured as

(see Hiiton.Wylie [1], p. 49)

requires 9 vertices, 27 edges and 18 triangles. But we do get from such triangulations our graded k-algebras k[Al. We obtain for the torus: k[Al =

k[xI' • '"' xg]/a,

where a

= (XIX"

X1X9, X2X6, X 2X" Xa X 4' XaXg, X4 X S, X 5X 9, XsX" X1X2Xa, X4X5XS' x,XSXg, X 1X 6 X S,

X2X4X9, XaX,x" X 1X 4X 7' X 2X 5X 9, XaXsXg) •

We can see that the torus is a (non-Cohen-Macaulay) Buchsbaum complex. For the invariant J(k[A]) of the Buchsbaum ring k[A] we have J(k[A]) = 2 since there are 1-cycles z~, zi in any simplicial decomposition of the torus, and as it turns out that {z~} + 0 and {zn 9= 0 generate the homology group Hl(A) freely. 2*

20

Introduction and some examples

Applying some fundamental principles of Buchsbaum rings our investigations now result in a particular solution of the Upper Bound Conjecture, which was extended to arbitrary manifolds by V. Klee [1] in 1964. For instance, we will prove the following statement in Chapter II, § 4: If the geometric realization X = ILII of LI is a connected manifold then we have:

E .v ( +d)1 .-0 z+ 1 v-1 (

)

dimk jji(LI ; k)

t'

for v = 0, 1, ... , d 1, where jji(LI; k) denotes the reduced simplicial cohomology of LI with coefficients in an arbitrary fixed field k.

Chapter 0 Some foundations of commutative and homological algebra

§ 1.

Local algebra and homological algebra

Chapter 0 contains the fundamental tools needed for the following chapters. In § 1 we will assume familiarity with the baeic techniques of local algebra and homological algebra. Since notation and terminology vary from one source to another, we will assemble in this paragraph (more or less without proofs) the basic definitions and results needed. More details can be found in standard sources such as: Atiyah-Macdonald [1], Matsumura [1], Zariski-Samuel [1], Cartan-Eilenberg [1], Grothendieck [3] and Serre [2]. All rings are tacitly assumed to be commutative and Noetherian with unit element. Let A be a "ring. By an A -module we mean a unitary module over A. A Noetherian A-module is then a finitely generated A-module.

1.

Associated primes

Let M be an A-module. We say that a prime ideal p of A is an as.,ociated prime of M, if one of the following equivalent conditions holds (i) there exists an element x E M with Ann xA = p where Ann N {a E A I aN = O} is the annihilator of the A-module N. (ii) M contains a submodule isomorphic to Ajp. The set of associated primes of M is denoted by ASSA III or by Ass M. If M is an A-module, the 8upport of M, written SUPPA III or Supp M, is the set of prime ideals \J of A such that the localization Mp of Mat p is 9= O. The Krull dimension of M, written dimA M or dim lll, is defined to be the supremum of length of chains of prime ideals of Supp lJI if it exists, and 00 if it does not. We have Ass M C Supp M, and any minimal element of Supp M is in Ass M. For example, let a be an ideal of A. Then the minimal associated primes of the Amodule A/a are precisely the minimal prime over-ideals of a. If M is a Noetherian A-module then Ass M is a finite set. An A-module M is said to be co-primary if it has only one associated prime. A submodule N of M is said to be a primary submodule of ill if MjN is co-primary. If Ass MjN = {p}, we say N is p-primary or that N belongs to \J (as a submodule of M). The connection with the classical definition of a primary ideal q of A is given by the following equivalence: (i) the module ill is co-primary,

*

0, and if a E A is a zero divisor for M then a is locally nilrotent on 1ll; that (ii) M is, for each x E M there is an integer n > 0 such that n"x = O.

22

O. Some foundations of commutative and homological algebra

I.et N be a su bmodule of 111. A primary decomposl:tion of N is an equation N Ql n ... (1 Qr with Qi primary in llf. Such a decomposition is said to be irredundant if no Qi can be omitted and if the associated primes of M/Qi (1::;: 2'::;: r) are all distinct. Any primary decomposition can be simplified to an irredundant one. If N = Ql n ... (1 Qr is an irredundant primary decomposition and if Qi belongs to ~i' then we have Ass MIN {\'l> .•. , \'r}. If ~i is an embedded prime of MIN, that is ~i is not minimal in Ass MIN, then the corresponding primary component 'Qi is not necessarily unique. On the other hand if Vi is minimal in Ass IlfjN then the primary component Qi is uniquely determined by N and by ~j' \Ve have the following well-known main result: If ~M is a Noetherian A-module then any submodule N of M has a primary decomposition. Having a primary decomposition N = Ql n ... n Qr we put U(N) = n Qi such that dim lll/Q; dim MIN. Finally, we set Spec A {prime ideals of A}. X Spec A is an affine scheme:

Spec A

=

As a point set, the set of primes of A. As a topological space a basis of open sets is given by the subsets XI {p E Spec: A I / ~ ~} for all / E A.

I

As a locally ringed space, its structure sheaf is defined by F(Xf , Ox) localization of M at the multiplicatively closed set (1, /, /2, ... ).

2.

AI

Systems of parameters and multiplicity

A ring A which has only one maximal ideal m is called a local ring, and Aim is called the residue field of A. If A is a local ring then the Krull dimension of A is finite. Let A be a local ring; m its maximal ideal. Let M be a Noetherian A-module of dimension d:2:: O. A family (xl> "" Xd) of elements Xl' "., Xd of m is said to be a system 0/ parameters 0/ M if dim M/(xl>" " Xd) ill O. If no confusion is possible wc denote a system of parameters (Xl> ... , xdl of M simply by its elements Xl> ... , Xd' We call an ideal q of A a parameter ideal of ~M if there is a system of parameters Xl> ••• , Xd of ill contained in q such that q . M = (Xl> ., " Xd) • ill. If d = 0, any ideal contained in Ann M is a parameter ideal of M; espeeially, the zero ideal is a parameter ideal. We have the existence of systems of parameters in any loeal ring and for every Noetherian A-module. Notice that if Xv .,., Xd is a system of parameters for M, then the dimension of M j(Xl> ... , Xi) ill is d - j for all j 1, ... , d. A system of elements Xl> ••• , Xj, j ~ d, of m is said to be a part of a system of parameters for M if dim M/(x l , ... , Xi) M = d - j. Let xl> ... , Xn be a sequence of elements of A. Reeall that if jlf is an A-module, then Xl' ••• , x. is an M-sequence if 1) Xi+l is a non-zero divisor on Mj(Xh ... , Xi) M for i 0, ... , n 1, and

2) M oF (Xl' ••• , xnl M. This property does not depend on the order of Xl' .• " X.' Every ~M-sequence is a part of a system of parameters.

§ 1. Local algebra and homological algebra

23

Let M be a Noetherian A-module. The depth of M, denoted by depth M, is defined as the supremum of all integers r such that there exists an "H-sequence Xl> ••• , X,. If X E Tn is a non-zero divisor on M then depth M ixM = depth M

1.

Furthermore, depth ilf::;: the infimum of dim Ai+:! as +:! runs through Ass M. In particular we get depth M

dim M

if

M =1= O.

111 is said to be Cohen-Macaulay if depth M = dim M. The ring A is said to be Cohen-Macaulay if it is a Cohen-Macaulay A-module. A is a regular local ring if and only if m is generated by an A-sequence. Renee a JOegular local ring is Cohen-Macaulay. A local ring A is called a (lowl) Gorenstein n'ng if A is Cohen-Macaulay, and whenever Xl' ••• , Xn is a maximal A-sequence, then the ideal (Xl' .•• , xn) A is irreducible. We have the following useful

Lemma 1.1. Let M be a Cohen-Macaulay A.module. Then: (il M is equidz"mensional (i.e. dim Alp = dim 1'1 for all mznz"mal prz"mes p in Supp M) wz"thout embedded primes (i.e. 0 M has no embedded primes). (ii) Let X be an element 0/ m such that dim Mjx.H = dim M - 1. Then X 1:~ a non-z!:TO dz'visor on M and MJxM is Cohen-Macaulay. Because of our investigations in Chapter I we need to characterize the C',ahen.Macaulay property by using local multiplicities. For this we examine the HilbertSamuel function. Let M be a Noetheri.an A-mo'dule of dimension d O. Let q be an ideal of A sueh that the length, IA(Mlq4Y), of MjqM over A is finite. Then we define the so-called Hilbert-Samuel function, denoted by PQ,M(n), as follows: PQ,M(n)

IA (1lfjqfl+ 1M)

for all integers n

O.

It is well-known that there is a polynomial in n, denoted by PQ,ltl(n), such that P q,M(1I pq,M(n) for all large n. The polynomial Pq,M(n) is the so-called (characteristic) Hilbert Samuel polynomzal of the ideal q with respect to M. There exist integers eo eo(q,lY (> 0), e1 := e1(q, M), ... , Cd := ed(q, M), where d = dim M, such that Pq,M(n)

€o(q, M)

(n : d) + el(q, 1'1) (n ; ~ 11) + ... + ed(q, M).

The leading coefficient €o(q, M) of Pq,M is called the mulHpUcity 0/ q 'U'z'th respect to M. When q m and M = A, we simplify the notation as follows: eo(q, A) = eo(A) and eo(A) is called the multipl1'ciiy of A. For our purposes we obtain the following main result:

Theorem 1.2. Let A be a local ring, let M be a Noetherian A.module, The following properties are equimlem: (a) M is a Oohen-Macaulay module. (b) There exists a parameter ideal q (c) For every parameter ideal q

eo(q, M)

=

l(lY jqM).

0/ M

0/ M

in A such that eo(q, M) in A,

IUlfJqM).

24

O. Some foundations of commutative and homological algebra

M. Auslander and D. A. Buchsbaum [1] used the methods of homological algebra to give an explicit expression for a general multiplicity in terms of the Euler-Poincare characteristic of the graded homology module of a certain Koszul complex. In particular, they gave an axiomatic description of multiplicity. This development has opened the subject to a much simpler treatment. An example of this may be found in D. J. Wright's paper [lJ. Here an inductive definition is given for the so-called general Illultiplicity symbol for n elements, relative to an arbitrary module over a commutative ring, which is suggested by Auslander-Buchsbaum [1], Theorem 3.3. We will have to use some well-known properties of this multiplicity symbol. It should be noted that the Recount of the general multiplicity theory in this section is restricted to what is required for our immediate application. However, the theory is more extensive than indicated here, and some of the results are valid under wider hypotheses, see, for instance, D. G. Northcott [lJ. Let A be a local ring and let M be a Noetherian A-module. Let Xl, ••• , Xn be elements of A such that the A-module M/(x I , ... , x,,) M has finite length. When n = 0 this condition is to be understood as meaning that lA(M) is finite. The definition of the mul1iplicity symbol of Xl> ••• , Xn with respect to M, denoted by e(xI,"" x.IM), useR induction on. n. First suppose that n O. In this case, by our convention, lA(M) is finite. We may therefore put e(0plf) Now assume that n>- 1. We set O:M Xl {m E M I mX I = O}. Since lA(M/(xl> ... , x,,) M) < (Xl it follows that lA((O :.u XI )/(X2, •.• , x n )· (O:M Xl») Accordingly, by our assumptions, e(x2 , ••• , x" I M/XIM) and e(x2'"'' x. ! 0 are both defined and so we may put


- Ext"(A/a i , ill). Also, the Ext"(A/a i , ill) and ;'l;jj(ill) form a direct system of A-modules and A-homomorphisms over I, and so the direct limit lim Ext"(A/ak , Jf) can be formed. It is k now easy to check that ~ Ext"(Ajak , ) becomes a covariant, additive, A-linear k

functor on the category of all A-modules; one can show that the functors Il~()

and

lim Ext"(Aja k , k

are naturally equivalent for each n

)

O. Therefore we get isomorphisms for all n

0:

lim Ext"(Ajak , ill).

k

In particular, we obtain canonical maIlS cp~w:: Ext~(A/a, 1'1-1) ->- H~(ill)

for each i

O.

Now, we' want to examine how this relates to the cohomology of the Koszul complex. Let A be a local ring. For an element x of A we define the K08zul complex K(x; A) generated over A by x as follows: Kj(x; A) = 0

Ko(x;

A)~

for all i

K1(x;

0,1,

A)~ A

and a map d l ' Kj(x; A) ->- Ko(x; A)

defined by dj{a)

-=-.

xa for all a E A;

that is, K(x; A) is the complex 0 ->- A A ->- O. I,et ill be an A-module and let Xl> ,. " Xr be elements of A. We define the Koszul complex K(Xb ••• , Xr ; Lll!) generated over A by Xl> •• " Xr with respect to ill! by: K(Xj; A) @ ... @ K(xr; A) @ ill.

§ 1. Local algebra and homological algebra

Its homology we denote by H;(xj, ... ,

Xr ;

27

J11). We now put

and Hi(Xh ... , Xr ; J11):= Hi(K'(Xh ... , Ir; 111)) = Hr_i(XI, .•. , Xr ; M).

Clearly, Ki(Xb"" Xr ; A) is a free A-module of rank

el•... I" 1

II

< '"
and hence lJI is an isomorphism of complexes. We define K'(a; 111) := K"(Xl' ... , Xr ; J11) and Hi(a; J11):= Hi(xv , .. , Xr ; J11) for all A-modules J11, If we choose another set of generators of a, the Koszul complex and hence its cohomology is unchanged (up to an isomorphism). Since HO(a; J11) HomA(A/a; J11), we get from Cartan-Eilenberg [1], Chap,IIJ, Proposition 5.2, canonical homomorphisms 1j!~: Ext~(A/a; J11) -'>" Hi(a; J11) for all i 0,

28

O. Some foundations of commutative and homological algebra

2. Consider the sequence of ideals

from which a direct system of complexes K(Xl' ... , Xr ; ~M)

-:>- K"(x~,

•.. , x~; M)

-'r •••

(M an A-module)

is obtained. The direct limit of this system is denoted by K;"'(Xl' ... , X,; M).

Since direct limits commute with exact sequences, we get Hi(K;"'(XlJ .•• , Xr ; M)) ~ ~ Hi(x~, ... , x;; M),

"

in particular

Hr(K;"'(Xl' ... , xr ; ~M)) ~ ~ M/(x~, ..• , x;) M,

" where the maps of the last right hand direct system are given by multiplication by Xl' ••.• X r •

a

Furthermore, HO(K;"'(xlJ •.. , x,; M)) !i!r: (0:1U (x~, ..., x~) A) = H~(M), x,) A. n Now, for r = 1 it is easy to verify that K;"'(x; M) is the complex

where

:== (Xl1 •. "'

O-+M!:.. M",-'>-O,

where M", is the localization of JI. with respect to the set II, x,

X2, ••• J

and h",(m)

m

(Observe K;"'(Xl' ... , x,; M) ~ K;"'(Xl; A) ... (8) K;"'(xr ; A) M, since tensor products commute with direct limits.) If M is an injective module results of Matlis [1] show that hr is always surjective, Le. IJ1(K;"'(x; ~M)) = O. Using an exact sequence defined by Corollary 1.7 below an easy inductive argument (on r) shows that Hi(K;"'(xlJ ... , x,; M)) = 0 whenever M is injective and 1. Hence by Cartan-Eilenberg [lJ, Chap. III, Proposition 5.2, and Chap. V, Proposition 4.4, we get:

for every A-module M and with a

(Xl' ... , X r )

A.

1.et us denote by Ak- the canonical homomorphism (defined by direct limit) JJi(a; M) -+H!(M), where a is an ideal of A. Then we have by the above remarks

Lemma 1.5. For all i (i)

:2: 0

we

have commutative diagrams

§ 1. Local algebra and homological algebra

(ii) II 6 is another ideal

01 A

29

with a C b then there are lor all i commutative diagrams

H'(6; M) ---+ H'(a; M)

1

1

H~(M)

---+ H:(M)

where the vertical maps are the canonical ones and the other homomorphisms are ~'nduced by 'P.

Next, analyzing the proof of Proposition 1 of Serre [2], IV-2, we get the following Lemma 1.6. Let L be a complex of A -modules and let x be an element of A. Then we have fot all p 1 commutative diagrams with exact rows:'

0 ....... Hl(x;H1H(L)) ....... HfI( K'(x; A) @ L) ....... HIl(X; HP(L)) ....... 0

'1

pI

r1

0 ....... H!A(HfI-1(L)) ....... HfI(K;,,(x;A)@L) ....... H~A(HP(L)) ....... O

where IX and y are ~'nduced by the canonical map K(x; ) ....... K~(x; ) which also defines a map K(x; A) @ L ....... K;"'(x; A) L which specilies p.

From Lemma 1.6 we obtain a very useful corollary. Corollary 1.7. Let A be a local ring and M an A-module. Let XI' •.• , x" r;;::: 2, be element8 of A. Then we have for all p 1 commutative diagrams with exact rows:

o . . . . Hl(X

1;

HP-l(X2 ,

''',

x r ; M)) ....... HP(xl> ••. , x r ; M) ....... HO(x 1 ;HP(x2 ,

t II x l M

t

x r ; M)) ....... 0

t

= 0, we get the following commutative diagram

o . . . . /IP-l(X2' .••, xr ; M) ....... HP(x}> ... , x r ; M)

t 0.......

••• ,

0

....... Hp(X2' .•. , x r ; M) ....... 0

t .......

t

Hr"', ...%rlA(M).......

Hrz ....ZrlA(M) ....... 0.

Prool: The second commutative diagram is a consequence of the first since for an A-module N with XIN = 0 Hl(Xl; N)

f

HO(Xl; N) = N,

H;',A(N)

= 0,

H~,A(N) = N

follows. Thus we need to prove the commutativity of the first diagram. We set L = K'(X2> ••• , x r ; M) and apply Lemma 1.6, Then the top row of the commutative diagram of Lemma 1.6 agrees with the top row of the diagram under consideration. Now we set for n 1

Ln := K(X;, .. ,' x;; M) and

30

O. Some foundations of commutative and homologieal algebra

and denote by (En) the bottom row of the commutative diagram of Lemma 1.6 if iJ := L. and by (Bex,) for L := Loo. Then we have a direct system of exad sequences (E l ) (E 2 ) -i>- ••• Since direct limits commute with exact sequences and with the local cohomology functors, we get (En) = (Eoo) and the map (E l ) -i>- (Eoo) (into the direct limit) describes a commutative diagram (with top row (E I ) and bottom row (Eoo». But (Eco) is just the bottom row of our commutative diagram. Thus, combining the commutative diagram constructed above and this commutative diagram we find a commutative diagram of the desired type, q.e.d.

Lemma 1.8. Let a c A be an zaeal and S A a multiplicaUvely dosed set with 1 E S, 0. Let M be an A-module and lor all i?:::. 0, in abreviated notation, Hi(M) l~ Ext~(a"; M). Then we have:

an S

(i)

"There

o

is an exact sequence H~(M)

and lor all

-i>-

M

i?:::. 1 there

are z80morphisms

Hi(M) "'-' H~+l(M). (ii) The lollowing condiHon8 are equz'valent: (a) Supp M C Via), (iii)

H~(HO(M»)

=

(b) HO(M)

H~(HO(M»)

H~(HO(M»)~ H~(M)

=

=

0,

(c) Hi(M)

=

0

lor all

i?:::. O.

0,

lor all i

2.

(iv) There is a (natural) homom()rphism (01 A-modules) g: HO(M)

-i>-

Ms

such that the composition gl is just the nautral m(.tp M

-i>-

Ms.

(v) g is z'njective if and only if S n l' = 0 lor aUl' E Ass M" V(a). (vi) (Formula of Deligne) There is a natural (A-)isomorphism H{J(M)::::::~Ma, Gal

where Ma denotes the localization 01 M at the multiplicatively dosed set {1, a, a 2 , •• •J. The (A- )modules Mao a E a, lorm an inverse .system: We deline jor a, b E a a honwmorphi8m lJa.b: Mb

-i>-

11fa

il and only il a E VbA, i.e. il at

me"

bc lor tEN, c E A by setting

(m E M, n EN).

Prool: Apply the functors Ext~( ; M) to the exact sequence 0 a" -i>- A -i>- A/a" 0 and take the direct limit of the resulting long exact cohomology sequence for n 1,2, ... This is again an exact sequence and (i) follows by virtue of HomA(A; JJJ) M and Ext~(A; M) = 0 for 1.

§ 1. Local algebra and homological algebra

31

(ii) (a) =? (b): Let Supp M C V(a) and take a E IlO(1I1). Choose s E Hom(a fl ; 111) representing a. Then for all a E an there is a p > 0 with aPs(a) = O. Since an is finitely generated, there is a q 0 with aqs(a) = 0 for all a E an. Therefore sian'. 0, that is, a O. (a) =? (c): Take a minimal injective resolution of 111

o

M

--'>-

1o --'>- 11

--'>- ...

Since Supp 111 V(a), Supp Ii module of the complex

o

IlO(/o)

--'>-

HO(/ 1 )

t;;; V(a)

for all

~.~

O. Now Hi(11f) is the £th cohomology

--'>- •••

But by the first part of the proof HO(/i ) 0 foralli ~ 0 which proves this assertion. (b) (a): If 1l0(lIf) 0 the exact sequence of (i) yields an isomorphism 1l~(M) ~ 111. 'fherefore each element of M is annihilated by some power of a, i.e. Supp iJ-f t;;; V(a). The implication (c) (b) is trivial and (ii) is therefore proven. (iii) From (i) and (ii) we get: Supp .tI

V(a) if and only if 1l~(M) '"'" M and HWlf) = 0

for all i~ 1.

We split the exact sequence of (i) into two exact sequences: (a) 0

Ilg(M)

~M

111'

--'>-

0

and (b) 0--'>-11/' IlO(M) 1l~(M)--,>-O. Since Supp 1l~(M), Supp 1l~(1I1) V(a) we get H~(M)

1l~(1I1')

for all

m(M')

1l~(IlO(1I1))

for all

1l~(M)

1l~(HO(.M))

for

1

and i~

2,

alll:~

2.

hence Also (al yields H~P{') (H'!,.(H!(M)) ~ llWW)):

o --'>- /l~(HO(1I1))

0 and thus (b) gives rise to the following exact sequence H~(M)

--'>-

HWW)

--'>-

m(IlO(M))

--'>-

O.

Now it is not difficult to see that the middle homomorphism is nothing but the isomorphism llk(.llf) H~(M/) obtained froll! (a) and this proves (iii). (iv) Leta E HO(M). 'fake 0 and s E Hom,((a n ; .:tIl representing a. Choose a E a nS. We define

g(a) Straightforeward calculations show that g is a well-defined A-homomorphism. Finally, for all m E M the homomorphism h: a M defined by h(al am for all a E a represents j(m) E HO(M). Therefore, if a E a n S, proves (iv).

gj(m)

h(a) am = -

a

a

m

= -. 1

This

o.

32

Some foundations of commutative and homological algebra

(v) Assume g is injective. For Ass M ~ V(a) let lJ E Ass M" V(a). Then there is a monomorphism A/lJ ...... M giving rise to the following commutative diagram

Alp -4 HO(AllJ) 4 (Alp)s

t

t

HO(M) 4

Ms

where all homomorphisms are injective (HO( ) is left exact and H~(Alp) 0 since lJ ~V(a». Therefore (Alp)s =l= 0, Le.lJ n 8 = O. Assume now :p n S = 0 for all :p E Ass M" V(a). Let a E Ker g and choose 8(ult )

.

E Hom(alt ; M) representmg a. Then for some a E a n S we get - - = g(a) 0, a lt i.e. there is abE S with bs(a") = O. Then bE P n S foralllJ E Ass(A/Ann,8(a n ») Ass M. Therefore Ass(AjAnn s(alt ») ~ V(a), i.e. there is an m:2: 0 with a"'s(a") O. Assume without lOBS of generality that m n. Then alts(c) = ca(a") 0 for all c E am. Hence we have for all c E am:

8

for all:p E Ass(AjAnn 8(cl) ~ Ass M.

a E :p n S

This implies Ass(AIAnn 8(cl) ~ V(a), Le. there is a q 0 with aIl8(c) = O. Since a'" is finitely generated we can find an r 0 with a'8(c) = 0 for all c E am. Therefore 810'"+" = 0, which means (J 0, i.e. !I is injective. We now prove (vi). We see that the maps HO(M) ...... Ma (a E a) given in (iv) are compatible with the homomorphisms ea.b' Therefore we have a homomorphism !p: HO(M) ...... ~Ma· aea

If

(J

E Ker!p, choose an

8

E Hom(a", M) representing

8(alt)

(J.

Then - a lt

=

0 (in Mal

for all a E a, i.e. for fixed a E a there is an mEN with 8(amH ) am . 8(a") = O. Since a is finitely generated, we find apE N with 81al> 0, j.e. (J = 0 and !p is injective. Now let p, E ~ Ma : N and let £Pa: N ...... M a, a E a, denote the canonical maps. aEO m/. If a (at> ..• , at) A, write £Pa,(P,) - ' E Ma with m; E M, n E N. Then for all . 1, .. "' OJ t : ai f 't,

i.e. there is an lEN with (ajaj)1 (m;aj ml~l'l O. m· We put m.:= aIm; and p := n l. Then £Pa (p,) = - ' and miaf = mjaf for all , a~ 1, ... , t. Let L denote the submodule of M generated by mb ... , mi' Assume

+

riaf = 0 with rl> ..., rl E A. Then for all j = 1, ... , t: I

a~

E rimi i=1

=

E r,mia1= E r,afmj = 1

0,

•

I.e. E rim, E 0 :L (11). Choose a q EN v.ith aqL n (0 :L (11» 0 (which exists by the lemm!l of Artin-Rees since L is finitely generated). We define a map 8: (af+q, ... , afHJ) A ...... M

by

8(U~+q) := aTmj,

i

=

1, ... , t.

§ 2. Graded modules and Kiinneth formulas

33

If E 8ia~Tq = 0 with s}) •.. ,81 E A then by the preceding we get E sja?m, E (O:L (a») n aqL 0, i.e. s is a homomorphism. Choose n E N with a" ... , Xd of rnB are called a system oj parameters of M if dim M/(Xh ... , Xd) M = O. An ideal q for which there is a (homogeneous) system of parameters Xl>"" Xd of M such that q. M = (Xl> ... , xd) • M is called a parameter ideal of M. Homogeneous elements Yh ... , Yr of rnB are called an M-sequence if for i 0, 1, ... , r - 1 (Yl' •.. , Yi) M :M Yi+l

(YH"" Yi) M.

As before depth M is defined to be the supremum of all integcr~ r such that there is an M-sequence consisting of r elements. Clearly any M-sequenee is part of a system of parameters. Therefore we get depth M

dim M.

M is called a Cohen-Macaulay module if depth M = dim M. Now we define the Hilbert function of a Noetherian module. Let M be a Noetherian graded R-module where R is a graded k-algebra. We set HM(n) := rankk([MJ.) (note that for all n E Z[MJ. is a k-vector space of finite rank). HM is called the Habert junction of M. There is a numerical polynomial hM (E Q[TJ, T an indeterminate) such that HM(n) = hM(n) for all sufficiently large n. The polynomial hM is called the 3*

36

O. Some foundations of commutative and homological algebra

Hilbert polynomial of ill. We write hM = ho(.M)

(~') + h (M) (a 1

'1'

1) + ... +

hdUI1)

with d := deg hM (considered as a polynomial in '1'), and where the integers ho(~~f) > 0, hl(JI), ... , hd(M) are called the Hilbert coellicienl O. Let x := Xl ® X2 E [R]p. Each element Xi gives rise to an isomorphism

o

Therefore X defines an isomorphism I all n 2:: 0 isomorphisms

I(p) where I

a(I h 12 ), Hence we have for

!!;:'(I) ~ !!;:'(/(p)) "" !!;:'(I) (p). Since Supp !!~(/) ~ {m}, this implies H;:'(I) = 0 for all n

0, q.e.d.

§ 2. Graded modules and Kiinneth formulas

43

It is not true in general that the Segre product of two injective modules is injective. A counter-example is in Chapter V, § 5, Example 5.6.

Now we can prove the following statement: Proposition 2.10. Let Nl> n;:::: 0 we get: -

MI

and N 2, M2 be graded R I -, reap. R 2-module8. Then for all

(i) There are natural R-homomorphi8m8

E8

T":

p+q=n

a(Ext~,(Nl> MIl, Ext'k.(N2, M 2))

-+

Ext~(a(Nl' N 2), a(MI' M 2))

(generalized Ktinneth relations). (ii) There are natural R-isomorphi8m8

,,": E8 p+q=n

a(!!p(M I),Hq(M2))-+!!"(a(Ml>M2))

(Ktinneth relations). Proof: We first prove (i): For i = 1,2 let I; be injective resolutions of Mi and K; := Homn,(N i , IJ Since a is an exact functor we have by Cartan-Eilenberg [1], Chap. IV, Theorem 7.2 (see also Chap. IV, Prop. 6.1):

E8

a(Ext~,(Nl>MI),Ext'k.(N2,M2)) =

p+q=n

E8

a(HP(K~),Hq(K~)l~ H"(a(K~,K~))

p+q=n

where HI denotes the jth cohomology of the underlying complex. By Lemma 2.6 we have a natural homomorphism of complexes

Now a(I~, I;) is a (not necessary injective) resolution of a(Ml> M 2) (see, for example Cartan-Eilenberg [1], Chap. IV, Theorem 7.2) and therefore we have natural homomorphisms H"(a(K~, K;)) -+H"(Homn(a(N I,N2 ), a(I~, I;)))-+Ext~(a(Nl> N 2), a(Ml> M 2)).

Putting together everything we obtain (i). Now we prove (ii). There are an integer p and for i = 1,2 mi-primary ideals qi which are generated by their homogeneous elements of degree p. For example, take the ideals generated by suitable powers of the basis elements of m I and m 2, resp. Then q: will be generated by homogeneous elements of degree pt for i = 1,2. Now we set in (i) Ni = q: for i = 1,2, t;:::: 1 and take the (direct) limit over all t. Since a(q~, q~) = (a(qI' q2))1 we get natural homomorphisms

,,": E8

a(!!P(M I), !!q(M2 ))

p+q=n

-+

!!"(a(M I, M 2)).

To prove (ii) it is sufficient to show that ,,0 is an isomorphism for any modules Ml> M2 and that ,," is an isomorphism for all n > 1 whenever M I, M2 are injective (see CartanEilenberg [1], Chap. V, Prop. 4.4). But the last statement is true by Lemma 2.9 (both sides are zero). Therefore we have only to verify that ,,0 is an isomorphism.

44

O. Some foundations of commutative and homological algebra Assume first that H~Pli)

=0

for i

1,2. Then by Lemma 2.8 we have for each

t'? 1 isomorphisms (q:= a(qb q2)}: a(Homn,(qL Mil, Homn.(q~, M 2 )} -+ Homn(ql, a(Mb

Mi»)'

Therefore in this case U O is an isomorphism. M;/!l~,(M;) for z· Let now lVI' M2 be arbitrary modules. We put Mi Then !l~(Mi) = 0 f~r i = 1,2 and since !l°(!l~,(Mi)} = 0 (note that Supp !l~,(Mi)

1,2.

{m,})

we obtain isomorphisms

!l°(Mi) ~ !l°(Mi)

for i

1,2.

Let;' denote the natural projection a(Mb M 2 ) commutative diagram:

-'>-

a(M~, M~).

Then we get the following

a(!l°(M1 ), !l°(M2 )} ~ !1°(a(M}, M 2»)

V

tirO)

a(HO(M~), HO(M~») ~ HO(a(M~, ~V~») We also have an exact sequence (see Cartan-Eilenberg [1], Chap. IV, Prop. 4.3(c»:

a(!1~(.iWI)' M 2 }

e a("Vv H~.(M2») -+ a(M}, M

2)

~ a(M~, M~)

Therefore Supp Ker 1 ~ {m}, i.e. H'(Ker 1) = 0 for all morphism, i.e. UO is an isomorphism, q.e.d.

-+ O.

O. Hence HO(l) is an iso-

Using the natural maps

ifk,: Ext'R,(m., M i )

-+ !18 (M i)

for i

1,2

and we get Corollary 2.11. Let M 1, M2 be graded R I " resp. R 2,moaules. Then there are for each 0 ccnnmutative diagrams n

e O'(Ext~,(m!> M

p+q~

I ),

Ext~.(m2' M 2 )} -+ ExtMm, O'(MI' M 2 )}

..

tIl

...1

p-tq=n

a(ipP

e a(!1"(M}), Hq(M ») 2

p+q~

~q

Ml' IW •

_" 1

)

~a(M"MtI

...

~ HfI(a(Mb M i )}

..

Finally we prove Corollary 2.12. Let Mb M2 be Noetherian graded R1"resp. R 2,moaules. Then

depth a(M!> M 2 ) '? inf (depth M I , depth Mil.

§ 3. Local duality

Proof: Using the canonical maps Mi --+ HO(M i ) (i tative diagram with exact bottom row:

45

1,2) and ,,0 we obtain a commu-

a(M!> M 2 ).!...t. a(HO(M 1 ), HO(M 2 ))

t"·

V

0--+ H~(a(1!fu 1112 )) --+ a(M 1> M 2 ) --+ HO(a(Ml' M 2 )) --+ H'm(a(Ml> M 2 )) --+ O.

If depth Ml 0 or depth M2 0 there is nothing to prove. Hence we assume depth Mi ~ 1 for i 1,2. Then p, is injective and therefore H~(a(M1' M 2 )) 0, i.e. depth a(M}, M 2 ) 1. Therefore if depth "'!f1 = 1ordepthM2 1 the conclusion follows. Assume now depth Mi > 2 for i 1,2. Then p, is even an isomorphism since M, =+ HO(M i) for i 1,2. Therefore l!~(a(Ml> "'!f2 )) = l!.k(a(M 1, M 2 ») = 0 and we get:

1 + inf{n E N I It''(a(M}, M 2 ))

depth a(M!> M 2 )

1 + inf{n E N

=

I

O}

a(HP(M 1 ), l!.Q(M2 )) =1= O}

p+q"~ll

~ 1 +inf{inf{n EN ll!.tI(M}) =l=0}, inf{rnE N ll!.m(M2 ) =l=0}}

inf{depth Mb depth M 2 }, q.e.d.

§ 3.

Local duality

In this paragraph we shall review further basic facts on homological algebra, In particular we shall introduce the notion of the dualizing complex which is a very useful concept of homological algebra. For our purposes here we shall need the dualizing complex to give a homological description of Buchsbaum modules. First of all we need to define some terminology. By a complex we shall understand a complex of modules over a fixed (commutative and ~oetherian) ring A. Let X': ... --+ X" --+

XfHl

--+ ...

denote a complex. We write X" for its nth cochain module and 8": X" --+ nth differential. If X' and Y' are complexes we define complexes Hom'(X', Y')

resp.

X'

Y'

given by Hom"(X', Y')

= [J Hom(Xi,

YH,,)

iEZ

with differential 8"(/)

= 8f -

(-1)" f8,

resp,

with differential o"(x' n E

yi) = ol(X') @ yi + (_1)i xi @ 8i (yl) ,

Z.

Recall that Xi @ yi --+ (-1)# yi X'@Y'=+Y'

X'.

Xi induces an isomorphism

X"~l

for its

46

O. Some foundations of commutative and homological algebra

Also, for n E Z

Y"[n] denotes the complex whose ith cochain module is yH" and whose ith differential is given by (-1)" 81+", that is Y"[ n) denotes the complex Y" shifted n places to the left. A morphism of complexes f' : X' -?> Y" is called a qua.si-isomorphism if the induced homomorphism on the cohomology

is an isomorphism for all i E Z. A complex X' is called bounded below (resp. bounded abo've, resp. bounded) if X" = 0 for all n ~ 0 (resp. n }> 0, resp. n 0 and n }> 0). All results which we list in the following are well-known. For proofs we refer to R. Hartshorne's lecture notes [2), R. y, Sharp's more elementary exposition in [1, 3,4), or B. Iversen's preprint [I), Let E' be a complex of injective modules which is bounded below. Then for any quasi-isomorphism j' : X' -?> Y" Hom'(f', 1): Hom'(Y", E')

-?>

Hom'(X', E')

is a quasi-isomorphism, This may be reformulated as follows: Let j': X' -?- Y" be a quasi-isomorphism and g': X' -'>- E' a morphism into a bounded below complex of injective modules, Then there exists h': Y" -'>- E' such that the following diagram is homotopy commutative

,.

Moreover, h' is unique up to homotopy, From this it follows that any quasi-isomorphism between bounded below complexes of injective modules is a homotopy equivalence. For any bounded below complex Z' there exists a bounded below complex of injective modules E' and a quasi-isomorphism f': Z' -?- E', Furthermore, E' may be chosen as a minimal injective complex, i.e. Ker(8f1 )

-?-

E"

is an essential extension. The above-mentioned results have the following "dual" form. Let P' be a bounded above complex of projective modules. A quasi-isomorphism X' -?- Y" induces a quasiisomorphism

A quasi-isomorphism between bounded above complexes of projective modules is a homotopy equivalence, For bounded above complexes X' there exists a quasi-isomorphism P' -'>- X' where P' is a bounded above complex of projective modules. In the case X' has finitely generated cohomology, P' can be chosen as a bounded above complex

§ 3, Local duality

47

of finitely generated projective modules, I..et F' be a bounded above complex of flat X- --+ yo of bounded above complexes, modules, Then for any quasi-isomorphism

r:

r ® 1: X' @ F' --+ yo ® F' is a quasi-isomorphism, Here we are mainly interested in complexes X' such that their cohomology modules H'(X'), ~'E Z, are finitely generated A-modules. Regarding this we remark that if X' is a bounded above complex and E' is a bounded below complex Qf injective modules, and assuming that both complexes have finitely generated cohomology modules, then Hom'(X', E") has finitely generated cohomology modules, too, Given a complex E" of injective A-modules and an A-module M, we set for i E Z Exti(M, E')

H'(Hom(M, E')).

I..et X', E' be complexes of modules over a fixed ring A. For n E Z consider the map X" --+ Hom"(Hom'(X', E'), E")

which assigns to x" E X" the element (It)'e% E Homtt(Hom'(X', E'), E") where It:

n Hom(XI, EH/)

= Hom'(X', E') --+ E"H

j€%

is defined by li( (gj)jE%)

(-1)i"g,,(x,,)

for all (gj)j€%

En Hom(Xi, EHi), jE%

In fact, this defines a map of complexes e: X' --+ Hom'(Hom'(X', E'), E') which is called the evaluation map. In particular, if X' is the complex Xi and XO = A we get a map

=

0 for i::f: 0

e: A --+ Hom'(E', E').

Now we shall state the definition of a dualizing complex.

Definition and Theorem 3.1. Let D' be arbounded complex 0'1 injective modules with linitely generated cohmrwlogy modules, Then the lollowing conditions are equivalent: (i) The evaluation map X- --+ Hom'(Hom'(X', D'), IT) is a qua8i-isomorphism lor any bonnded complex X' with I~"nitely generated cohomology. (ii) The canonir-al homomorphism A --+ Hom'(D', D') is a qua8i-i80m0rphism, If D' satisfies one of these equivalent conditions we call it a dualizing complex for A.

In particular it follows that if A has a finite injective resolution E" then E" is a dualizing complex, This means that if A is a Gorenstein ring then it possesses a dualizing complex since A has finite injective dimension,

48

O. Some foundations of commutative and homological algebra

For the proof of this result see Hartshorne [2J, Chap. V, § 2, or Sharp [3J. Let A be a ring and D' a dualizing complex for A. For any ideal a in A we have that Hom~(Ala, D') is a dualizing complex for Ala. Let A be a local ring which is a quotient of a local Gorenstcin ring. Then A possesses a dualizing complex. By the theorem of Cohen we know that any complete local ring is a quotient of a regular ring. Hence it follows that any complete local ring admits a dualizing complex. On the other hand there are local rings which don't have a dualizing complex, compare Hartshorne [2], Chap. V, Proposition 10.l. Let A be a local ring with maximal ideal m. If D' and D are dualizing complexes then there exists an n E Z such that D' and D'"[n] are homotopy equivalent, see Hartshorne [2J, Chap. V, Theorem 3.1, or Sharp [3]. Since k = Aim itself is a dualizing complex there exists an integer d E Z such that t

•

Hom'(k, D')

= k[ -d)

for the local ring A admitting a dualizing complex D'. We call D' rwrmalized if the integer d = 0, Since the translate of a dualizing complex is again a dualizing complex we can normalize by translation. In the sequel a dualizing complex is assumed to be normalized,

Proposition 3.2. Let A be a ring admitting a dualizing complex D', (a) For any prime ideal.).) the locah"zation A~ has a dualizing complex, (b) Let D~~ denote the normalized dualiZing complex

0/ AI>' Then there

'18 a homotopy

equivalence D' ®A AI'

-l>-

D~I>[dimAI.).)].

Proof: Let X' be a bounded above complex and E' a bounded below complex of in-

jective modules and suppose that both complexes have finitely generated cohomology modules. Then the canonical map

is a quasi-isomorphism, Furthermore, D' AI> is a complex of injective A~-modules whose cohomology modules are finitely generat~d over AI>' The quasi-isomorphism (*) and the definition of the dualizing complex imply that D' C8JA AI> is a dualizing complex for AI>' Hence we have proved (a). Since D' C8JA AI> is a dualizing complex there exists an integer s E Z such that Hom~~(k(.).»), D'

®A A I1 ) = k(.).») [s]

where k(.).») = AIJ/.).)A~. The assertion (b) follows if we can show that s = dim AI.).), Our last condition gives . ExtA~(k(.).»), D' C8JA AI') 4=

o.

By slightly modif.ying results of Bass [1] to the case of a bounded below complex of injective modules having finitely generated cohomology we get ExtA8+tllmAflJ (k', D')

4= O.

Since D' is a dualizing complex of A we have Hom'(k, D') -s + dim AI.).) 0, This concludes the proof, q.e.d,

k and we therefore obtain

S 3. Local duality

49

Corollary 3.3. Swppose that A has a dualizing complex, Then A admit8 a dualizing complex D' with Di =

E9

E(A/'p).

IlESpecA dimAill=-i

Proof: We take a minimal injective complex which is dualizing and normalize it. Since D' A Il [ -dim A/'p] is homotopy equivalent to the normalized dualizing complex of Ap it suffices to show for a local ring A with maximal ideal m the following: J)O = E(A/m) and E(A/m) does not occur in Di for i =l= O. Since D' is chosen to be minimal Hi(Hom'(k, DO))

Hom(k, Di)

follows. Hence Hom(k, Di) = k if i = 0 and 0 otherwise. Since Di is isomorphic to a direct sum of E(A/'p), 'p E Spec A, we obtain the required result, q.e.d.

In the sequel we will assume a dualizing complex D' of this form. For a bounded below complex X' we denote by Rrm(X') the complex obtained as ' follows: Choose a quasi-isomorphism X' -i>' E' where E" is a bounded below complex of injective modules, and let

This complex is unique up to homotopy and we define

In particular, for a module M, considered as a complex, we recover the local cohomology modules considered in § 1. Now we state and prove the main result of this paragraph. Local Duality Theorem 3.4. Let A be a local ring with maximal ideal m, Swppose that A has a dualizing complex D'. Let X' be a bounded below complex wz~h ftnuely genemted cohomology modules, Then there exists a qua8i-ismnorphism Rrm(x')

where E

=

-i>'

Hom(Hom'(X', D'),

E),

E(k) denotes the injective h1tll of the residue field k.

Proof: First we remark that for any bounded above complex L' of finitely generated free modules and any bounded below eomplex of injective modules E" we have an isomorphism of complexes

H we now start with a bounded above complex Y' with finitely generated cohomology modules we can choose a quasi-isomorphism L' -i>' Y' where L' is as above. This induces. a quasi-isomorphism Hom' ( Y', E')

-i>'

Hom'(L', E'),

We observe that the complex on the right consists of injective modules whence

4 Buchsbaum Riugs

50

0, Some foun~ations of commutative and homological algebra

Note also that the quasi-isomorphism L' Hom'( Y', rm(E'»)

-)0-

-)0-

Y' induces a quasi-isomorphism

Hom'(L', rm(E',)

by using that rm(E') consists of injective modules. Putting this together we obtain a quasi-isomorphism Rrm(Hom'(Y', E'»)

-)0-

Hom'( Y', rm(E'») ,

For the normalized dualizing complex D' we obtain rm(D') and therefore we have Rrm(Hom'( Y', D'»)

-)0-

=

E (see R-B. Foxby [1])

Hom'(Y', E).

If now X' is a bounded below complex with finitely generated cohomology modules we get for Y' Hom'(X', D') Rrm(Hom'(Hom'(X', D'), D'»)

-)0-

Hom(Hom'(X', D'), E).

By using the quasi-isomorphism e: X'

-)0-

Hom'(Hom'(X', D'), D')

we have proven the statement, q.e.d. The particular case of a local Gorenstein ring is of some interest since in this situation the local duality is of quite a simple form.

Corollary 3.5. Let A be a, fa,ctor of the local Gorenstein ring B with dim B ha,ve for a,ll i E Z na,tura,l isomorphisms H~PI1.) '" HomA(Ext~-i(M, B),

: n, Then we

E),

where E denotes the injective hull of A/rnA (consider M a,s a, B-module a,nd Ext~-i(M, B) as an A-module), Proof: Since B is a Gorenstein ring the minimal injective resolution E' of B is a dualizing complex. Hence ' B[ -n]

-)0-

E'[ -n]

is a normalized dualizing complex. Then D' Taking the cohomology in Rrm..(M)

-)0-

HomB(A, E') is a dualizing complex of A.

HomA(Hom~(M, D'), E)

and taking into account that Ext~-i(M, B) '" H-i(Hom~(M, E'[ -nD)""'" H-i(Hom~(M, D'[ -11,]))

we get the corollary, q,e,d. Let M be a finitely generated A-module. Then we obtain by the Local DU!1lity Theorem H!(Hom'(1lf, D'») 9= 0 if -dimA1lf i, and

§ 3, Local duality

This follows since i of M, We call

51

" is the highest non-vanishing local cohomology module

= dim" M

the allwnical module of the A-module M, Next we examine the complex Hom'(llf, D'). Since Di

=

e

E(A/p)

for 0

i

dim A

Il€Spec" dim"/Il=-i

we have

e

Homi(M, D')

Hom,,(M, E(AM),

I1€SPCC" i E Z, the set of primes in Z of dimension i, We collect some useful data which we need later, Proposition 3.6. Let M be a finitely generated A-module, Then we get the following properties: (a) AssAKM = {AssAM)d' where d dimAM,

0 z' dimA.U, (b) (ASSAM); = (Ass"Ext"A'(M,D'))i' (c) dimAExt7(M,D')S:i, OS:i inf{2, dim"KM ). Proof: The assertions (a) and (c) follow immediately from the definition of the dualizing complex D', Next we shall prove (b), l!"or this let p E (Ass"MJi, It follows that

.»AiJ E ASSAil Mil

and

Hg"p(M v )

The Local Duality Theorem for AI> hnpJies HO(Hom"IlUlfll' D~Il)) =\= 0, 4*

0,

52

o.

Some foundations of commutative and homological algebra

Since HomA;.(M;.,

D~;.) ~

(HOlllA(M, D') [-dim A/V]) @A A;. we get

H-dlmAII1(HomA(M, D')) ®A A;.

*' 0,

that is V E Supp ExtAdlmA/;,pl, D'). From (c) we get the inclusion (AssAM)j

(ASSAExt'Ai(M, D'));.

The other inclusion can be proved similarly, It follows by (a) that dilllAM = dimAK3[ and that depthAKJ[?: 1. Without loss of generalit,y we can assume the existence of an ,lll-regular element x which is therefore also K,l["regular. The short exact sequence O--l>M":""'M --l>M(xM -,..0

induces an exact sequence

o --l>K

M ":"'" KM

KM/%M --l> •••

by applying the functor Hom( ,D'). Therefore KM/xKM is isomorphic to a submodule 1 we obtain the property (d), q.e.d. of K M /zM • Since K MlzM has depth Corollary 3.7. Let A be a local complete n'ng. Let ill be a Noetherian A-module. Then we have

(ASSA1'1l)j

for i

(AssAHomA(H:"(M), E))i

0, ... , dim M - 1.

Proof: Using local duality we get the corollary from Proposition 3.6(b), q.e.d.

§ 4.

Resolutions a.nd duality

Let R denote a Noetherian graded k-algebra (k a field) with maximal ideal m, i.e. we have [R]j = 0 for all i < 0, [R]o = k and m = EB [R]i (althought many ofthe following ;21

facts remain true in more general situations). For the notation compare § 2. Our aim is to recall basic facts and some applications regarding the following topics: - free resolutions of (Koetherian) graded R-modules; - duality similar to the local duality studied in the previous section. For the proof of the graded version of the Local Duality Theorem 4.14 we use the methods of R. Y. Sharp [2]. This may be considered as an alternative for proving these results without using dualizing complexes.

1.

Graded modules of finite length

We shall often be concerned with graded R-modules of finite length. Recall that a module ill has finite length if it has a finite composition series; then the length lR(M) is the common length of all composition series of M. (If M is a graded module then for any composition series there is a composition series of the same length consisting of

§ 4. Resolutions and duality

53

graded submodules of M.} For graded R-modules we have the following characterizations of finite length:

Lemma 4.1. Let M be a graded ii-module. Then the following conditions are equivalent: (i) M has finite length. (ii) M isfinite-dirnensional as a k-vector space. (iii) J.lf is Noetherian and m"M = 0 for some n. (iv) M ~8 Noetherian and Ass M P and Ext~(M, R) =F O. Proof: Let

F: 0

-'>-

Fp ~ F p_1 -'>-

~ Fo ~ M

...

-'>-

0

be a minimal free resolution of M. Let Gt:= HomR(F;, R) and 'ljJi:= HomR(tpi+l' il): Gt -'>- GHl for all O. Then Extk(M, R) is the ith cohomology of the complex

G: 0

-'>-

GO~

G11

QI-,>- •••

-'>-

0,

i.e. Extk(M, R) 0 for all i > p. Since 1m P11 ~ m· F p _ 1 , 1m 1pp-l ~ m· Gp. But Gp =F 0 and we get an epimorphism Ext~(M,

R) cy Gp/lm 'ljJpl

~

Gp/m· Gp =F 0,

by the Lemma of Nakayama, q.e.d. We now want to describe an application of Proposition 2.3 for curves in P~. Definition 4.6. By a curve C c PZ we mean a closed subscheme of P~, given by the homogeneous ideal l(C) of S := k[XQ' " ' J X a], satisfying any of the following equivalent conditions: (i) C is a one-dimensional scheme, (ii) Sll(C) is a two-dimensional graded ring, (iii) l(C) has height 2. Furthermore, we will assume that C is locally Cohen-Macaulay and equidimensional.

Pro

Let C be a curve in Let A be the local ring of the vertex of the affine cone over C with maximal ideal m; that is A (Sjl(C)}n' where n (Xo, ... , Xa) S denotell the maximal homogeneous ideal of S. Then we get the following corollary: Corollary 4.7. Let C be a curve in P~. The following c(mditions are equivalent: (i) The local cohomology module lI1n(A) is annihilated by m.

(ii) The gruded S-module

ED Hl(p~, 1(0) (m)) is annihilated by n. m€Z

Proof: We have by Proposition 2.3 (and the isomorphisms given there):

ED Hl(pZ, l(C) (m)) '" !!~(l(C)). mEZ

It follows immediately from the exact sequence 0

~

l(C) ~ S ~ S/l(C) -+ 0 that

~(l(C)) '" lIA(S/l(C)).

Let M be a graded S-module with Supp ~lf ~ In}. Then for any mE ]I{ there is an n with n"m O. Hence we obtain an isomorphism between (non graded) S-modules:

56

O. Some foundations of commutative and homological algebra

J.lf =:::: Mn. Therefore we have: . H:n(A) '" (!t.MS/I(O)))n '" !t.MS/I(O)) =:::: E8

Hl(p:, 1(0) (m))

meZ

and this proves our Corollary 4.7. q.e.d. Using the notation from Chapter III we therefore have given a new proof of the following corollary (see our Theorem III.1.2 and the rlOte in the preface):

C.orollary 4.8. The arithmetical Buchsbaum property for curves in

P2 is preserved under

liaison. Proof: The property of Corollary 4.7(ii) is preserved by shifting degrees and dualization. Therefore we get our assertion from our observations on liaison among curves in pi of Chapter III, Proposition I.2.12 and Corollary 4.7, q.e.d. 4.

Dualization

If M is a graded R-module, the k-vector space J.lfv

E8 Homk([llf]_n, k) is in a natural neZ

way a graded R-module with [MO]. := HOID/c([M]_n, k). We say that M" is the dual (or sometimes the k-d·ual) of M. If f: ilf ~N is an R-homoIDorphisID (of degree zero) of graded R-modules, then f defines an R-homomorphism f": N" ~ M". The functor llf H> M", f H> f" is an exact contravariant functor from the category of graded RIDodules to itself. We have:

Lemma 4.9. Let M, N be graded R-modules. Then (i) In the subcategory of all graded R-modules with rankk["lf]"

< 00 for all n E Z (especially, in the 8ubcategory of Noetherian or Art~'nian graded R-modules) the duality is perfect: There is a canonical isomorphism M"" M. (ii) There i8 a natural isomorphism of graded R-modules )'M.N:

HomR(M, N') ~ (M ®n N)·.

Proof: (i) and the construction of ;'M.N in (ii) follows from standard techniques of linear algebra. (For p E Z the k-vector space [M ®n N]p has a basis of elements of the form m (8) n with m E [M);, n E [N)j) ~. + i p and therefore we can define for all p E Z / k-linear maps

Ap: HomR{M, NV(p)) ~ [(ilf

N)v)p = Homk([J.lf ®R N)_p, k)

in the usual manner. Some straightforeward calculations sbow that

)'A/,N

E8 }'p PEZ

is a natural R-bomomorphism.) It is easy to see that A"t.N is an isomorphism whenever )'11 is a graded free R-module, i.e, if M ~ E8 R(ni) where I is a set (of indices) and

n, E Z. Then consider an exact sequence F

iff

~

G ~ J.lf ~ 0, where F, G are free graded R-modules, This gives rise to a (;ommutative diagram with exact rows

o ~ HomR(llf, NV) ~ Homn(G, N°) ~ Homn(F, N°) Sin(;e

).G,lV, ).r,lV

are isomorphisms,

;'M,lV

is an isomorphism too, q.e.d.

§ 4. Resolutions and duality

57

Corollary 4.10. R" is an injedive objed in the category of g·raded R-modules. More precisely, R" is the £njective hull of ~ (or better, of ~V ~ ~). Proof: By Lemma 4.9(ii) we have for all graded R-modules M: HomR(M, R") ~ M", i.e. the functor HomR( ,R") is exact and therefore R" is injective. The epimorphism R ~ ~ gives rise to a monomorphism !;V ~ R" and it is easy to

eheck that this is an essential extension, q.e.d. R-module.~ M, N and all i morphisms (of connected sequences of derived functors):

Corollary 4.11. For all graded

0 there are natural iso-

ExtkOlf, N°) ~ Torf(M, N)". Proof: Note that {Torf(M, N)"liEN are the right derived functors of the left exact contravariant functor ( ®R N)", N fixed. For i = 0 we take the isomorphism of Lemma 4.9(ii). Since Torf(M, N)' = 0, Extk(M, N°) 0 for all i> 0 whenever M is projective, the result follows from Cartan-Eilenberg [1], Chap. III, Prop. 5.2,

and Chap. V, Prop. 4.4, q.e.d. The local version of the following very useful lemma can be found in R. Y. Sharp [2], Lemma 3.2:

Lemma 4.12. Let T be a right exact, covariant additive R-linear functor from the category of graded R-modules to itself which respects shifts of degrees. Then there is for every graded R-module M a natural homomorphism ;,v: M which

~8

®n T(R)

~

T(M)

an isomorphism if M is f%ilitely generated. If T commutes with dired i£mits,

;M is always an isomorphism. Proof: Let M be a graded R-module. For any homogeneous element m E M let h m : R ~ M(deg m) denote the homomorphism defined by hm(r) := rm for all r E R. Then T(h m }: T(R) ~ T(M(deg m») = T(M) (deg m) is an R-homomorphism. Define ;M: M ®R T(R) ~ T(M) by ;M(m e) := T(h m) (e) for all m E [MJi> f! E [T(R)]J' It is

easy to verify that ;M is a natural R-homomorphism.

Now;M is an isomorphism if M R(p) with p E Z. Therefore ;.u is an isomorphism if M is a finitely generated free graded R-module, since T commutes with finite direct sums. If T commutes with direct limits, ;M iF; an isomorphism if M is an arbitrary free graded R-module, since tensor products also commute with direct limits. Now we can take an exact sequence F h G.f!....t, M ~ 0, where F, G are (finitel~' generated) free graded R-modules. Then we obtain a commutative diagram with exact rows T(R} g0id • M

F

l'(F}

Since

~F' ~(;

~'G T(G)

T(R}

~O

~ '-" T(M}

~O.

are isomorphisms, ;.u is an isomorphism by the 5-Lemma, q.e.d.

58

O. Some foundations of commutative and homological algebra.

Using the definitions of Section 3 of this paragraph we define:

Definition 4.13. A graded k-algebra R is called a graded Gorenstein k-algebra, if inj dimR R < 00. The graded versions of the result of H. Bass [1] yield that inj dim R = dim Rand that R is a graded Cohen-Macaulay algebra, i.e. dcpth R dim R. Now we are able to state and prove the following. Theorem 4.14. (Duality Theorem). (i) 8uppose that R i8 a graded Gorenstein k-algebra of dimension n. ]1'or any graded R-module ~W and all i E Z there are natural R-isomorphisms !!..!n(M)"", Ext~-i(M, R) (r

where r

1),

r(R) is the index of regularity of R (d. Def.2.2).

(ii) If R is arbitrary and if 8 i8 a graded Gorenstein k-aigebra of dimension n such that R i8 a factor of 8, then we have for all graded R-module8 M and all i E Z natural

isomorphism8 !!..!n(M)" "-' Ext~""'(M, S) (8 -

1)

where s r(8), Mis con8idered as an 8-module and Ext~-i(M, 8) a8 an R-module. (iii) If under the assumptions of (ii) ~W is Noetherian, we get !!..!n(M)~ Ext;-i(M, S)· (1

8).

Proof: By Corollary 4.11 we get with N Rv (Rv. ~ R by Lemma 4.9(i): ExtiRUtf, R) Torf(M, RO). for all j. To prove (i) we establish the following claim:

~

Claim. Let R be a graded Gorenstein k-algebra with n:= dim R. Then we have R" ~ !!..~(R) (r - 1), where r := r(R). We use induction on n. If n 0, !!..~(R) = R and therefore this is an injective graded e(R) (d. Del. 2.2). R-module. Thus we have to show that R" '" R(r - 1). Now r = 1 Take a E [R]r-l'" (O}. Then m . a 0, i.e. we can find a monomorphism !f --+ R(r - 1). Since R(r - 1) is injective, this induces a monomorphism Rv --+ R(r - 1) (R" is the injective hull of !f). Therefore we obtain for all i E Z:

+

rankk[R"li s:;; rankk[R]r-1+i>

rankk[Rl-i

and replacing i by - r + 1 i we find rankk[Rlr_i-i-l rankk[R(r I)J; for all i E Z and this gives

rankk[Rl_i, i.e. rankk[R"J,

R·~R(r-l).

If n > 0, we take a homogeneous non-zero divisor x E m with d : = deg x. Since R := RjxR is again a graded Gorenstein k-algebra of dimension n 1 with r(R) = r(R) d, we have

+

!!..~-I(R) (r

+d -

The exact sequence 0

--+

0--+ !!..':n-1(R) (r y

1) '"

H':n-:I{R) (r

R( -d) ~ R --+

+d

+d -

1)~

R".

R --+ 0 induces a diagram with exact rows

1) --+ H':n(R) (r -

1)"':-+ H':n(R) (r

+d -

1) --+ 0

!

--+

O.

§ 4. Resolutions and duality

59

Since R" is injective there is a homomorphism f: H':n(R) (1' - 1) --+Rv such that the diagram is commutative. Now f(d): !1':n(R) (I' + d 1) --+ RV(d) makes the whole diagram commutative. Therefore Ker f (Ker f) (d) and· Coker f ~ (Coker f) (dl. Both Ker f and Coker fare Artinian graded R-modules and hence Ker f = ('A)ker j = 0 by these isomorphisms, i.e. f is an isomorphism and our claim has been proven. ~ow

we have with Lemma 4.12

(!1':n( ) is right exact):

(11f®R")"~(J1{®!1':n(R)(r

1))"

(M

!1':n(R))v(1-r):::::!1':n(J1{)V(I-r),

hence !1::'(J1{). ~ HomRC,1{, R) (I'

1).

The right derived functors of the left exact contravariant functor H':n( ). are just the functors !1::'-'( )", i> O. If F is a free graded R-module then !1':n-i (F) = 0 for all i> 0 since F is a direct sum of the Cohen-Macaulay modules R(m), m E Z. If P is a projective graded R-module then there ill a free graded R-module F such that P is a direct summand of F, hence !1~i(Pl 0 for all ~. > O. Since Extk(P, ill = 0 for all i> 0, we find by (,.,artan-Eilenberg [1], Chap. III, Prop. 5.2, and Chap. V, Prop. 4.4, for all i"20 natural isomorphisms !1::,~i(M)· ~ Extk(M, R) (r - 1) which proves (i). To prove (ii), let n denote the maximal homogeneous ideal of 8. Then !1~(iIf)" ~ Exts-i(M, 8) (s - 1) where M is considered as an S-module. But H~(M)V ~ !1:n(M)", considered as S-modules. (iii) follows from (ii) by dualization and Lemma 4.9(i) since H:n(M) is an Artinian graded R-module for all i, q.e.d.

Corollary 4.15. Let M be a Noetherian graded R·module. Then H:nUIf) is of finite length for all ~. =1= dim M if and only if M is locally Cohen-J~facaulay and equidimen.sional, i.e. M(1J) ~8 a Cohen-Macaulay R(1J)-module jar all l' E Proj R and dim 11{ = dim Rjl' for all m~'mmal primes l' E Supp M. Proof. Let 8 be a graded Gorenstein k-algebra suth that R is an epimorphic image of 8. If Mis lotally Cohen-Macaulay then we have for all l' E Proj 8 with M(1J) =1= 0:

o=

Ext~{l.lpfUl» 8(1J))

for all

~.

=1= dim 8(.\1)

(Extk(M,8))(1J) dim M(Il)

dim S - dim M.

Therefore Ext~(M, 8) is of finite length for all i =1= dim 8 - dim M by Lemma 4.1. Thus !1:n(M)" and hence !1:n(M) is of finite length for all ~. =1= dim M by Theorem 4.14. The converse is also true, q.e.d. For another important consequence of Theorem 4.14 we need the following notation: Let M be a Noetherian graded non-free R-module. Then take a finitely generated free graded R-module F and an epimorphism :n:: F --+ 1lf suth that Ker:n: ... , F s), where X o,.", Xm are variables (of degree 1) and F I, ... , F. are forms such that n:= dim R = m - s + 1 then R is a graded complete intersection, hence a Gorenstein k-algebra with r(R) = -m + deg FI + ... + deg F •. Using again the notation of Chapter III we get a further application of Theorem 4.14:

< m, locally Cohen-Macaulay subscheme which is linked by a complete intersection C to a 8ubscheme W c: Pt'. If C Hln ... nH., s=codimpZC, Hl>_ .. ,H. hypersurfaces of P~ of degree dl> ... ,d, (thus n = m s) we have for all i = 1, ... , n isomorphisms

Corollary 4.17. Let V c: PZ' be a pure n-dimensional, n

EEl H'(P';, P€Z

dy(p)) ~

(EEl Hn-.+I(pZ', dW(P)))" (m + 1 P€Z

where dy, dw denote the corresponding sheafs of ideals.

dl

§ 4. Resolutions and duality

61

Proof: Let S:= k[Xo, ... , X".], n:= (Xo, ... , x".) S, R:= S/(Fl> ... , F q ), m:= (Xo, ... , X".) R, Fi = 0 the equation of Hi for £ 1, ... ,8. Then by Proposition 2.3 and the isomorphisms given there: '

EEl H'(P,!:, 3 v(P)) ~ H~+1(I(V))

!.f.1t(S/I(V))~ H:n(RII(V) R)

l'E%

~ H~-;+l(RII(W)

~

+ 1-

d1

-

(EEl H ..-i+l(P'!:, 3 w(p)))" (m + 1 PE%

q.e.d.

R)" (m

... -

d1

d8 ) ... -

d.),

Chapter I Characterizations of Buchsbaum modules

§ 1.

Characterization of Buchsbaum modules by systems of parameters

We will always denote by A a local ring with (unique) maximal ideal m. First we will recall the problem of D. A. Buchsbaum expresscd in the language of local algebra (see Introduction). To do this let M be a Noetherian A-module and q a parameter ideal of M. We will use the notion of the Hilbert-Samuel polynomial Pq,M and its leading coefficient co(q, M), the multiplicity of q with respect to M (see Chap. 0, § 1, 2.). D. A. Buchsbaum's original questions was as follows: Does there exist a natural number I(M), such that the difference of "length" and "multiplicity" of every parameter ideal q of M is equal to I(M), Le., it is true that

lA(M/qM) - co(q, ilf)

I(lll)

for all parameter ideals q of M?

Already in the Introduction we gave examples which showed that this will not be true in general (see Examples 1 and 5). But we have also seen (see Examples 2-7 of the Introduction) that there are a lot of in this sense "good" examples, see also Chapter V. The main purpose of this paragraph is to give a first characterization of those Amodules M, for which the above qnestion has a positive answer. We start with some definitions and easy lemmas. Definition 1.1. Let III be a Noetherian A-module. A system of elements Xl' ... , X, E m is called a weak M-sequence, if for each i = 1, ... , r (Xl> ••• , Xi-I) •

III : Xi =

(Xl> ••• , Xi-I) •

III : m.

Remark 1.2. For r = 1 we have 0:MX 1 = 0 :Mm

or, equivalently, m· (O:M

Xl)

= 0

(since

Xl

Em).

Consequently, if Xl is a non-zero divisor of M, Xl also forms a weak M-sequence (consisting of one element). We therefore see that the weak M-sequences are a direct generalization of the well-known lW"-sequences (see Chap. 0, § 1, 2.). We know that every ilf-sequence forms a part of a system of parameters of ilf. For weak ilf-sequences we have: Lemma 1.3. Let III be a Noetherian A-module of positive dimension d. Then every wea,k M-sequence Xl> " ' j X, with r S d is a part 0/ a system of parameters 0/ M .•

§ 1. Characterization by systems of parameters

63

Proof: Clearly, X2' "., x, forms a weak M/xlM-sequence. Therefore, by an easy induction argument, it is sufficient to prove the lemma for r = 1. Let x E m be an element 0 :11{ m. Then x q p for all p E Ass M" {m}, particularly, x fi p for with 0:11{ x all p E Supp M with dim Alp = dim M (> 0). But this means dim Mix' M d - 1, Le. x forms a part of a system of parameters of M, q.e.d. d for every Remark 1.4. It was stated in Stiickrad-Vogel (2], Corollary 4, that r weak M-sequence Xl> ••• , XT • This is not the case, however, as was pointed out by Balwant Singh with the following example: Let k be a field and set A k kx with X2 = O. Then x is a weak A-sequence but dim A = O. Also, we note that two maximal weak M-sequences (Le. the number of elements in such a sequence is eqtial to or less then dim M and it is not possible to lengthen it) have not always the same number of elements. To see this, we refer to Proposition 2.1 of this chapter and Example 1.2.5 or Example V.5.4. These examples render modules M fulfilling statement (iii) of Proposition 2.1 which are not Buchsbaum modules. By Proposition 2.1 every system of parameters contained in m 2 is a weak M-sequence. On the other hand it is easy to see that every element x with dim MIx, M = dim M - 1 forms a weak M-sequence, i.e. every weak M-sequence of maximal length consists of at least one element. But not every system of parameters is a weak M-sequence since 111 need not be a Buchsbaum module, see our next definition.

e

Now we are able to define Buchsbaum rings, resp. modules. Definition 1.6. A Noetherian A-module M is called a Buchsbaum module if every system of parameters of M is a weak M-sequence. A is called a Buchsbaum ring if it is a Buchsbaum module as a module over itself. From the Introduction we already have quite a number of examples of Buchsbaum modules. Another easy consequence which enhances our knowledge of Buchsbaum modules is the following statement. Lemma 1.6. Assume that A is an epimorphic i'fTU1{Je of the local ring B. An A-module M is a Buchsbaum rnodule over A if and only if it is a Buchsbaum module considered as a B-module by "restricting scalars". . Proof: Clearly, dimB M dim,!. M and every system of parameters of M in A may be obtained by restricting a system of parameters of M as a B-module. Conversely, the restriction of any system of parameters of M in B to A is again a system of parameters of M as an A-module. Therefore the validity of the statement of our lemma becomes clear by the definition of weak M-sequences, q.e.d. Before continuing we state a seemingly technical yet in the sequel very useful result. To this end we define: Definition 1.7. Let a A be an ideal and M a Noetherian A-module with dim Mfa·.L11 O. Letd:= dim M. A system of elements Xl> ••• , XI of A is called an .LV-basiS of a if the following conditions are fulfilled: (i) Xl,. '" Xt form a minimal basis of n. (ii) For every system il>"" id of integers with 1:::; t'1 < ... < i d ::::- t the elements Xi" ••• , Xi. form a system of parameters of M.

640

I. Characterizatiolll! of Buchsbaum modules

Remark 1.8. 1. Since 0 dim Mia· jJl = dim M/(xl> ... , XI) . M dim M d. 2. If dim M = 0 any minimal basis of a is an M-basis.

it follows that

t

We now prove:

Proposition 1.9. Let a c A be an ideal and M l , ... , Mfl NoetMrian A·modules with dimA M;/aMj = 0 lor all i 1, ... , n.· Then there are a1> ... , al E a lorming an M;-basiB 01 a lor all i 1, ... , n. Prool: Clearly, we. can omit zero-dimensional modules from our collection, i.e., we may assume d j := dim M; > 0 for all i = 1, ... , n. We now prove by induction on m, 0-:;; m -:;; t, t := rank A/m alma, a (bl> ... , bl ) A: There are elements a1> ... , am of a with

(i) a (al> ... , am, bm+l> ... , bl ) A and (ii) for all I = 1, ... , n,all j 0, "', min(m, dl ) and all 1-:;; i l < ... < i j m,ai" ... , ai, is a part of a system of parameters of MI' This is trivialfor m = O•.Let therefore be 0 < m t and assume that aI' ... , am-l (E a) fulfil (i) and (ii). Define

I\)

L :=

E Spec A

I there are 1, j, i1> ••. , i j E N, 1 1-:;; n, 0-:;; j < min(m, dl)' 1 i l < ... < i j
•• _,

Xd-l)' M:m.

Xl> _._,

(Xl" _,

Xd 01 M we have

Xd-l) . M:x

lor all X E m

Xd-l> X lorm again a 8ystem 01 parameter8 01 M.

§ 1. Characterization by systems of parameters

(iii) For every system

0/

111: Xi+l

(Xl' 0", X;) •

(iii)' For every system (Xl' ',., Xd-l) •

(iv) For every part

parameters

0/

=

Xl, "" Xd 0/

(Xl' ... ,

M we have lor all ~. ~ 0, "" d - 1:

xJ 0 jlf: Xf+l '

0/ parameters Xl, 0", Xd 0/ M ilf :Xd =

(Xl' "" Xd-l) •

a sY8tem

0/

U((xv ... , Xi)' M)

=

65

parameters

we have

M: x~, Xl, .,., Xi 0/

M with i


"', Xi)' M:m.

Proof: The implications (i) =? (i)', (i) =? (ii), (il' (ii)', (ii) =? (iii), (Hl' =? (iii)', (iii) =? (iii)' are trivial. It remains to prove (iii)' =? (iv) and (iv) =? (il. (iv), To prove (iii)' =? (iv) we show (iii)' =? (iil' and (iiy Let Xl> •• " Xd be a system of parameters, (iii)' implies that (Xl> ... , Xd-l) ' Jf:Xd = (Xl>

"0,

Xd-l)oM:x~

(Xl' .. "

Xd_l)'~H:x~ = .'"

i,e, (Xl' ... , Xd-l)·M:Xd = U((XI' ''', Xd-I),M) since dimM/(xI, ... , X£1-l)' M = 1. Consequently, (Xl' .. ,' X£1-I)' M :X£1 does not depend on Xd and this implies (ii)'. Assume now that (ii)' is satisfied. Let Xl> ... , Xi be any part of a system of parameters of M, i < d. Then we choose Xi+I' •• " Xa such that Xl> Xd form a system of parameters. For all integers n ~ 1 the sequence Xl' ... , Xi> X~+l' ... , Xd-l' Xd is also a system of parameters. By Krull's Intersection Theorem we obtain for all X E m such that Xl> ••• , Xd-I> X is again a system of parameters of M the following: "'j

(Xl' ... , Xi)'

1H:Xd

= (n

(Xl' .. " Xi>

X~+1' .. " Xd_l) , M) :Xd

ft~l

= n ((Xl>

... ,

Xi, X7+1' ... , X;;_l) , M:Xd)

ftZl

=

n ((Xl> "., Xi, X!'+l' ... , X;;_l) , M:x)

(Xl' ... , Xi)' ~H:x.

ft~l

Amongst the elements X we find an (lH/(xt> .. " A (see Proposition 1.9) and this gives (Xl' ... ,

Xi)

,Jf :Xd

=

(Xl> ... , Xi)'

Xd-l) ,

M)-basis ofthe maximal ideal m of

llf:m,

On the other hand it is easy to see that Xd can be chosen in such a way that we have (Xl> ... , Xi) , M :Xd U((XI' , •. , xd' M) and therefore we obtain (iv), Finally, we verify (iv) =? (i). To do this let Xl, ,." Xd be an arbitrary system of parameters for M, Then we have for all i = 0, "" d 1; (Xl' ... ,

xd' M:m ~

= i.e. Xl' .. "

Xd

is a weak

(Xl' .'" Xi)'

M:Xi+l ~ U((XlJ

"'J

x;l' .H)

(xl".·,xj).M:m,

~H-sequence,

q.e.d.

Corollary 1.11. Let lrl be a Buch8ba,um module, Assume XI, "') Xr is a part 0/ a SY8tem 0/ M with r < dim ilf. Then (il M/(XI,'''' x r ) , M and (ii) MfU((xl> "') Ir) 'M) are Buch8baum module8 (0/ dimens70n dim M - r), 1Jarameters

5 Buchsba.um Rings

of

66

1. Characterizations of Buchsbaum modules

Furtherrrwre, the localizaticm8 M),) are Cohen-Macaulay module8 for all pnme ideals

:p =1= m of Supp M. Proof: (i) is a direct consequence of the definition of weak M-sequences.

Since MjU((XI, .•. , xr), M) = (_Mj(XI' ••• , xr)·M)jU(O) (U(O) in Mj(xl> ••. , x,). M) it is sufficient to prove (ii) for r = O. Let Yl' ... , Yi> i < dim MjU(O) = dim M be a part of a system of parameters of MjU(O). Then it is a part of a, systelu of parameters of Mas well and we have by Proposition 1.1O(iv) (U(O)

+ (Yl> ... , y;). M):m

U(U(O)

+ (YI' ..., Yi)' M)

U((Yl> ... , Yi)' M) ... , Yi)' M) :m.

Therefore MjU(O) is a Buchsbaum module by Proposition 1.1O(iv), The last statement follows from Proposition 1.1O(iv), q.e.d. We note that the converse of (i) or (ii) of Corollary 1.11 is not true in general. We will return to this topic later when we discuss the so-called "lifting property", see Proposition 2.19 and Proposition 2.23. Now we come to the main result of this paragraph. It shows the connection between the Buchsbaum modules defined above and the original problem of D. A. Buchsbaum (see Theorem 2 of the Introduction). Theorem 1.12. Let M be a Noether~'an A-module with d:= dim M > O. Then M is a Buch8baum rruxlule if and only if there is an integer J(M) 0 such that

l(Mjq . M) - eo(q, M)

J(M)

for all parameter ideals q of M.

Proof: For every parameter ideal q of M we set c(q, M) := l(Mjq . M) - eo(q, M)

0

(see Lemma O.1.3(ii».

Assume first that there is an integer J(M) such that c(q, M) = J(1lf) for all parameter ideals q of M. Let, Xl> •• " Xd be an arbitrary system of parameters for M, q := (Xl> .• ,' Xti) . A. ]'or i = 1, ... , d we let 1'1f. := (Xl' ... , Xi-l) • .M :Xij(Xl> ... , Xi-l) . M

and ei := e(Xi+1' • ,., XdIMi)'

Then by Lemma O.1.3(vi) c(q, M)

=

Now we replace Xti by

ei'

x~

and we let (1

q':= (Xl' ... , Xd-l, X~) • A,

=

2e;

1)

ei:= e(Xi+l' ... , Xti-l> x~IMi)'

Then by Lemma O.1.3(v) we have for i ei

d-

1, ... , d - 1:

§ 1. Characterization by systems of parameters

67

and this results in a-I

+ 1: e,

l((Xl> •.. , Xd-l)' M :X~/(Xl> . '" Xd-l)' M :Xd)

.=1

i=l

= c(q', M) - c(q, M) = I(M) - I(M)

d-l

d-l

+ 1: e;

l( (Xl' ... , Xd-l) . M: X~/(XI' "., Xd-!) . M: Xa)

l(Md) -

1: ei i=l

O.

Since all terms of the left sum are non-negative integers, l( (Xl' ... , Xd-l) . M :x~/(Xl> ... , Xd-l) . M :Xd) = 0,

i.e. (Xl' ... ,

Xd-l)' M :Xd

=

(Xl' ... ,

Xd-l)' M :x~.

Hence, by Proposition 1.10 (iii)', M is a Buchsbaum module. Now assume that M is a Buchsbaum module. Let Xl' ••• , Xd be a system of parameters for M. If we use the same notations as defined above, we have for i = 1, ... , d: m . Mi

0,

especially,

dim M;

O.

Therefore by Lemma 0.1.4 e(xi+l' ... , xdIM.) = 0

for i

1, ... , d

1

and we have c(q, M)

l(Md)'

We notice that the left part of this equation does not depend on the order of the elements Xl> ... , Xd and, consequently, the right-hand term does not depend on this order. Let now q':= (Yl, •.. , Yd) . A be another parameter ideal of M. We show by induction on d that c(q, M) = c(q', M). If d = 1, c(q, M) l(O:MXI) l(O:Mm) = l(O:MYI) = c(q', M) and we are done. Suppose now that d 2. 'Ve choose an element z E m such that Xl, •• ,' Xd-l, Z and Yt> •.. , Yo-I, z are again systems of parameters with respect to M. Then by Pro~sition 1.10 (ii)' and the induction hypothesis we have with q := (Xl> ••. , Xd-l)' A, q' := (Yl> ... , Yd-l)' A, Xl := M/z. M: c(q,M)

l(q.M:Xd/q·M) =

l(q·M:z/q·M)

l( (z, Xl' ... , Xd-2)' M :Xd-l/(Z, Xl, ... , Xd-2) . M) l((XI' .•. , Xd-2)' M :Xd-l/(XI ,

= c(q, M) = =

... ,

Xd-2)"

Xl)

c(q', M) = ... = l(q'·M :z/q' . M)

l(q'·M:Yd/q'·M)

=

c(q', M),

since M is a Buchsbaum module of dimension d - 1 by Corollary 1.11 and are parameter ideals for M, q.e.d.

q resp. if

Now we are able to.state the following

Lemma 1.13. Let M be a Noetherian A-module 0/ positive dimension. M is a Budl.sbaum module if and only if tke m-adw completion this case I(M) = I(if). 5*

if of M

is a Buchsbaum module over

A.

In

68

I. Characterizations of Buchsbaum modules

Proo/: Let It! be a Buchsbaum module and let tJ denote a parameter ideal of :it. Then there is a parameter ideal q of M with tJ· if = q .:it and we have ' lA(:itjtJ·:it) - eo(tJ, if) = lACitjq . if)

eo(q, if)

= (t(MIq . M) -

eo(q, M)

--

lA(M/qM) - eo(q, if) l(M).

This is independent of the ehoice of tJ, i.e.,:it is'a Buchsbaum module with l(Jl) = liM) by Theorem 1.12. Conversely, if q is a parameter ideal with respect to M then q is a parameter ideal of :it and the same reasoning as before provides our statement, q.e.d. Next we prove two lemmas needed in the sequel. Lemma 1.14. Let M be a Noetherian A-module and a

A be an ideal8UCh that MJa .M is a Buchsbaum module 0/ positive dimension. Then /or every part 0/ a system 0/ parameters xl> ... , x, o/Mja . M we have with b := (Xl' ... , xr)·A:

Proo/: Assume the statement of the lemma is false. Then there is a maximal member among all ideals a for which the assumptions are fulfilled but the statement is not true. Let it be ao. Then there is an r 2 0, a part of a system of parameters Xl> ... , Xr of M lao·M and an integer k 1 with

We also may assume that k is minimal with respect to this property. We set

ao·M:m = U(ao' M)

U Clearly, r

(see Proposition 1.lO(iv)).

0 is impossible (since then b

= 0). If r

U n x~ ·M = x~ . (U:.I{ x~) = x~ . U

1,

Xt-l • Xl'

U ~ X~-l • ao . M,

a contradiction. Therefore r 2 2. Choose a E (U n b k .M) "- ao . b k - 1 • M. Then we can write a u. E (Xl' •.• , Xi)k-l. M. Notice that m . a ~ ao· M. Set b

Now, for all

Xi • Uj

X

E b'k . ill,

E m we have

bE ((a o

X •

b

,

= I: Xi

• Ui

with

;=1

where b'

+ X • x, . U

r

X •

a E ao . 1,{, i.e.

+ X,' A) .M:m) n b'k ·M.

Since l"'f/(a o + Xr ' A)· M ~ (M/ao ·M)ixr . (M/ao ·M) is a Buchsbaum module of positive dimension (r 2) by Corollary 1.11(i) and since b' is generated by a part of a system of parameters with respect to M/(a o X r ' A)·M, we have by the maximality of ao that b E (a o

+

Xr •

A) . b'k-l . M.

§ 1. Characterization by systems of parameters

Hence a

b

69

+ Xr ' Ur E (a o ' b'k-l + Xr ' b'k-l + Xr ' bk-l) ·ld n U ao • b'k-l . M + (x r • bk- ~~f n U) ao ' b'k-l . ~~f + x, . (bk - 1 • M n U :MX ao • b'k-l • M + (bk - l . M n U). 1 •

r)

Xr •

If k

2, by the minimality of k, we obtain

a E ao . b'k-l . M

+

Xr •

ao . bk - 2 • iII

=

ao . bk - 1 • M

which is not possible. Hence k 1 and this implies a E ao . M + Xr • (M n U) which is also impossible. This contradiction proves the lemma, q.e.d.

=

ao . M

An easy consequence of this is Lemma 1.15. Let.M be a Buchsbaum module over A 01 positive dimeMion. Then lor every system of parameters Xl' ••• , Xa of M we have for q := (XV"'' Xa) . A: (qk+I.M:x.)nq.M=qk·M If depth M

1.

forallk

1 then

qk+l . 11I : Xa Proal: Set q'

qll • M

~

1.

(Xl> " ., Xa-I) . A. Then we have by Lemma 1.14 (a

qhl '111 :Xa

(qk • Xa • M qk • 111 qk. M

If depthM If depth M

for all k

+ q'k+1 • M) :Xa =

qk • M

+ (q'k+1

+ ((Xd' M: m) n q'kTl . 111) :Xd + O:MXt/.

qk . M

Xa • A) : • M :Xd)

+ Xd' q'k • M :Xd

1,O:.I(Xd O:Mm = 0 and we have obtained the required equality, 0, we have by Lemma 1.14:

(qk+l.~lf:Xd)

n q. M

(qk. M qk, M

since O:Mm n q.111

0 (a

+ O:Mm) n q. M + (O:Mm n q. M) =

qk.

~Ilf,

0 in Limllna 1.14), q.e.d.

Remark. Our characterization of Buchsbaum local rings A resulted in the notion of weak A-sequences. Several authors have studied further generalizations of a regular sequence (for example, M. Fiorentini [1J, C. Huneke r1J, N. V. Trung [to], or P. Schenzel [3]). We will show that these generalizations coincide in a Buchsbaum local ring. First we recall some definitions: Definition 1.16. Let A be a local ring of dimension n system of parameters of A. Then: 1.

Xl> " ' ,

>

0, Suppose that

XI' •. " X"

is a

Xn is a weftk A-sequence if (Xl> "', Xi)' A :Xi+1

(Xl' ... , Xi)' A:m

for every i

=

0, "" n

I,

2, XI, "" Xn is said to be a d-sequence if for each subset {iI"'" ii} (possibly 0) of {I, ''', n} and all k, m Il, ..., n} " {iI' ... , i j } we have ((Xi", '"

Xij ) '

A :xk ' Xm)

(Xi" .. " Xi,)' A :Xk •

1. Characterizations of Buchsbaum modules

70

3,

Xl> " "

x. is a relative regular sequence if for every integer i

((xt " 4. An element

•• , Xi-t, Xi+t, .,., X

Xn) ,A :Xi) n (Xl> ••• , xn) . A

=

=

1, ... , n we have

(Xl> •.• , Xi-I'

Xtf-t> ••• ,

Xn) ,A.

in an m-primary ideal q is an absolutely superficial element for q in A if

(qk+t:X) n q

=

qk

for all integers k> 1,

XI, .,., Xn is said to be an absolutely superficial system of parameters if the element Xi is an absolutely superficial element for the image of (Xl> ., " x n )· A in A/(xt> .. " Xi-I)' A for all integers i 1, ... , n. 5. Xl> •• " Xn has property (F), if

((Xl' "', xi-d· A :Xi) n (Xl> ••• , x n)· A = (Xl> ••• ,

xi-d· A

for every integer i with 1 :s;; n. (Note for i 1 we obtain (O:xd n (Xl> ... , x.)· A = 0.)

Proposition 1.17, A is a Buchsbaum ring if ami only if one of the five conditions of Definition 1.16 hold8 for all BysterruJ of parameter8 of A. In this case all five conditions are equivalent. Proof: By definition of course A is a Buchsbaum ring if and only if L holds for every , system of parameters of A. The equivalence of 1. and 2. follows from Proposition LlO(ii) and (iii). Now we prove the implications 1. :::;. 4. :::;. 5. :::;. 3. :::;. 1. (for all systems of parameters of A). 1. :::;. 4. follows from Lemma 1.15. 4. :::;. 5.: Let B := A/(x!> ... , Xi-I) • A (1 :s;; n). Then (O:BX;) n (x;, "', x.)· B 5,;; ((x;, "', xn)k+l. B:x;) n (Xi> ... , x.)· B

(x;, •.. , xn)k. B by

4. Since n (Xi'

for all k

~

1

••. , xn)k. B = 0, we get for A :

k~1

((Xl' ... ,

XI-i)'

A :x.) n (Xl' .•. , Xn) . A

(Xl' ... ,

Xi-d·

~.

5. :::;. 3. is trivial. 3. :::;. 1.: For t' n we obtain from 3. : ((XI' ... , x._ I), A :xn) n (Xl> ... , x.)· A (XI' ... ,

x.-tl· A

= (Xl' ... , X.-l) . A

hence (Xl>"" Xn-l)' A ;x~ 1.lO(iii)', q.e.d.

§ 2.

(Xl> ''',

+ Xn . A n ((Xl> .. " X._l) . A :xn) + Xn ' ((Xl> ... , x n- 1) . A : x;),

x.-tl· A :X. and 1. now follows by Proposition

Cohomological characterization of Buchsbaum modules

The aim of this paragraph is to establish "parameter-free" cohomological criteria for the Buchsbaum property of modules over a local ring. The main result describes a cohomological characterization of Buchsbaum modules without the usc of systems of parameters.

§ 2. Cohomologicai characterization of Buchsbaum modules

71

An essential role in our investigations will be played by the local cohomology modules with support in the maximal ideal m of A, i.e. the modules H~(M) (see Chap. 0, § 1, 3.). Let A denote again a local ring with maximal ideal m, and residue field k A/m. We know that a Noetherian A-module M is a Cohen-Macaulay module if and only if H~(M) = 0 for all ~. with S i < dim M (see Chap. 0, § 1,3.). Related to this is our next result which gives a first indication that local cohomology is an appropriate tool for studying Buchsbaum modules.

°

Proposition 2.1. Let M be a Noetherian A-module oj positive dimension. The jollowing conditions are equivalent: (i) There it! a system oj parameters jor M contained in m2 which 18 a weak M-sequence. (ii) Every system oj parameters jor M in m2 it! a weak M-sequence. (iii) m· H~(M) = 0 jor all i with Os i < dim M. Furthermore, ij one oj these conditions it! juljilled, we have (iv) l(M/q· M) - eo(q, M) it! independent oj q jor all parameter t'deals q m2 • Prooj: The implication (ii) =? (i) is triviaL First we prove {i) =? (iii). Let d := dim M 1. Let Xl' ... , Xd be a weak M-sequence in m 2 • Since Xl E m 2 we have O:,I{m2 ~ O:MXl = O:Mm ~ O:Mm 2 , i.e. O:Mm = O:Mm2. This implies O:Mm O:Mm" for all n;:::> 1 and consequently H:ilM") = U O:Mm" O:,l{m, i.e. m·H:it(M) O. n:
1 the proof has been given. Suppose d

Now we use induction on d. For d have two exact sequences

2, We

and

o -+ Xl . M ~ M -+ M' -+ 0 with .lit' SUPpO:MXl = SuppH:it(M) ~ {m), l:fk(O:.lfXl)

Since Hk(p) is an isomorphism for all 1

0 for all j

1. Therefore

1 and we have O:Him IM)Xl'

The long exact cohomology sequence obtained from (E2 ) yields epimorphisms H{;l(JJ') -+ Ker Hk(i)""'" O:Hi (M)X l , m

By the induction hypothesis (dim M' d 1 and X 2 , ... , Xd form a weak M'-sequence 1. Thus m· (O:Hi (M)xd = 0 for all in m2) m· Hi;;;l(M') = 0 for all j with 1 S 1 d m j = 1, •.. , d - 1. Since Xl E m 2 we again obtain

s

Hk(M) = H:it(Hk(ll{»)

= U O:Hi (M)mn n~l

m

O:Hi

(M)m,

m

Le. m' H~(M) = 0 for all 1 = 1, .. " d 1. Since the case '1 settled, (iii) is proven. To prove the implication (iii) =? (ii) we need the following

=

° has already been

Lemma 2.2. Let M be a Nretherian A-module 01 positive dimension such that H~(M) are modules oj linite length jor all i < dim M. Then jor every part 01 a system oj parameters

72

1. Characterizations of Buchsbaum modules

Xl' ••• , Xr with re8pect to M the 8ubmodule (Xl> ••• , x,). M oj M i8 unmixed up to m-primary component8, i.e., ij p E AssM/(Xl> ... , x r) . ill, then either p = m or dimAlp dimM - r.

Prooj: Let d:= dim M. We use induction on r. If 'I' = 0 we have to show that (Ass M)j := {p E Ass M I dim Alp i} 0 for all i, 0 < i < d. Since lA(H~(.it))

--

lA(H!n(M») = lA(H!n(M») < 00 for all i < d we can assume (using Proposition 3 and Lemma 8(i) of the Appendix) that A is complete. Then Corollary 0.3.7 yields (Ass M); = (Ass Hom(H!nOlf),

E))i

for all i

< d.

Since H!n(M) are modules of finite length for each i < d, the same holds for the module Hom(H~(M), E), i.e., Ass Hom(H!n(M), E) 2 and all d 2 uniquely determined homomorphisms rli,.: Il:n(Mlxfl

•

M)

-l-

with

Il:n(M)

Il:n(YI) . fl.,.

=

1, we have for all

Il:n(h.).

It is not difficult to verify that with flH,. the top row of our last commutative diagram

becomes a "split sequence", i.e. Il:n(Mlx fl • M) '" Il:n(M)

for all

EB Il:;I(M')

Il:n(M)

EB Il:';I(M)

2 and all n > 2. This means

d

m . H:n(.J.1:fjx" . M)

=0

for all

2

d

(n::2:2).

Since dim Mix" • M d - 1, the induction hypothesis proves our statement for this case. (b) Xl is an arbitrary element (in m2). By an easy induction argument (induction on z") we may assume that Xl' ••• , Xi is already a weak M-sequence, in particular, (Xl> ... , Xi-I)' M:Xi

=

=

(Xl' ... , Xi-I)' M:m

(Xl> ... , Xi-I)' M:(m),

Xi E m 2 •

where the last equation follows since Now, let y E m, z E m 2 be elements such that system of parameters of M. Set

N Since N:m (1)

Xl> ••• , Xi-I,

y, z is again a part of a

(Xl> ... , Xi-I) . M.

N:y

N:m

~

N:y

N:z c N:(m) (Lemma 2.2) we get

N:(m) and N:m

= N:z = N:(m).

From this we obtain (y.M+N):(m)

(y.M+N:y):(m)

=

+ N) :(m»):y ((y2. M + N):y):m

((y2. M

+ N) :y) :<m) ((y2. M + N):m):y (by step (a)) (y. M + N:y):m = (y. M + N:m):m.

((y2 . M

Furthermore, we have (z E m2) (y. M

+ N:m):m ~ (y. M + N):m2 ~ (y.

M +N):z

~

From all this we obtain (2)

(y.M+N):z=(y.M+N):(m)

•

(y.M+N:m):m .

(y' M

+ N):(m).

74

1. Characterizations of Buchsbaum modules

Now, let mE (Xl> "" Xi)' ~y :Xi+l = ((Xl> .. ,' Xi-I)' M:m + Xi' M):m (using equation (2) with Y = Xi> Z Xi+l)' Choose an UW/(xt> ,.,' Xi) • M)-basis JIt, ... , Yt, of the maximal ideal m of A. Let y E {YI' ... , Yl}' Then

Y' m

u

=

+ Xim'

with

u E (Xl>

••• ,

Xi-I)' M:m,

m' E M

and

y2 . m y , u

+ Xj .y . m' ,

y . u E (Xl'

Hence, by step (a) and equations (1) and (2) with m' E (y2, Xl> .. " Xi-I) ,M :Xj) : y

Z

Xi-I) . M. Xi we get

«y2, Xl' .'" Xi-I) , ..elf :m):y

= (y2, Xl' ... , Xi-I)' M:y):m = = (y,

••• ,

(y. M

+ (Xl' .. " Xi-I) .M:m):m

Xl> "., Xi-I)' M:Xi'

Therefore u = y. m -

Xi'

m' E (y, Xl> ... , Xi-I)·M n (Xl' ... , Xi-I)' M:m) (Xl>""

Xi--I)' M

+ y. M n (Xl> ••• , Xi-I)' M :m)

(Xl> , •• ,

Xi-I) . AI

+- y. (Xl' ... , Xi-I) . M:y. m) +- y. ((Xl> ' •. , Xi-I)' M :m)

=(Xl> ••• , Xi-I)' M =

Thus y. m

u

(Xl' ... , Xi-I) . M,

+- Xj . m' E (Xl' ... , xd . M, and consequently

mE (Xl' ... ,

Xi) •

M:y

for all y E {YI' ... , y,}.

But this implies m E (Xl' ... , Xi) • M: m and Xl' ... , Xi+l is therefore a weak M-sequence. This proves (ii). Finally, the implication (ii) ::;> (iv) is obtained using similar arguments (with some obvious modifications) as was in the corresponding part of the proof of Theorem 1.12, q.e,d. Remark 2.3. Condition (iv) of Proposition 2.1 does not imply (i) or (ii) or (iii), general. For example, let k be any field and X, Y indeterminates. Set R := k[ X, Y], A := k[X, Y]/X, R n (X3, Y)·R,

III

m = (X, Y) ·A.

Then H~(A) X . RjX . R n (X3, Y) . R and therefore m· ~(A) =l= O. But for all parameters Z of A in m 2 we have (where Z is the picture of Z ERin A) l(AJz . A)

eo(z . A, A)

= l(O:AZ) = l((X . R n (X3, y). R) :ZjX . R n (X3, = l(X . RjX . R n (X3, y). R)

y). R)

and this last number does not depend on z.

•

Also, we note that N. Suzuki [5] and S. Goto [10] have studied the class of local rings for which condition (i) of Proposition 2.1 is satisfied. These rings are called q'uasi,Buchsbaum rings. S. Goto [10] has established the ubiquity of quasi-Buchsbaum rings which are not Buchsbaum rings.

•

§ 2. Cohomological characterization of Buchsbaum modules

75

Corollary 2.4. II M is a Buchsbaum module then rn· H~(M) = 0 lor all i =!= dim M. In particular, the local cohomology modules are modules oll~'m'te length ~n this case. Unfortunately, the following example will show that the converse of this statement is false. Nevertheless Corollary 2.4 gives it first necessary condition for Buchsbaum modules independent of systems of parameters. Example 2.0. Let k be a field and XI, X 2 , X a, X 4 indeterminates. Take

A := k[ Xl' ... , X4]j(Xlo X 2 ) n (Xa, X 4) n (Xi, X 2 , X 3 , X~). Then U(Oj = Xl' X 4 . A and therefore rnA' H?nA(A) = rnA . U(O) = O. The exact sequence

o -+ H~)A) -+ A

-+

B

-+

0

with B: = AjU(O) = k[ Xl' ... , X4]j(Xlo X 2 ) n (Xa, X 4) results for all

i>

1 in isomorphisms (H~A(H~)A)) = 0 for

i> 1):

H~A(A)~ H~A(B).

Therefore (see Example 6 of the Introduction or Proposition 2.25 which show that B is a Buchsbaum ring and hence a Buchsbaum module over A by Lemma 1.6) rnA . H~JA) = O. Notice that dim A = 2. We next show that A is not a Buchsbaum ring. Take z:= Xl X 4 mod A. Clearly, dim Ajz . A = 1 and we have

+

Therefore Xl . U(z· A) ~ z· A, i.e. rnA . U(z. A) we find that A is not a Buchsbaum ring.

~

z· A. Thus by Proposition 1.1O(iv)

It is possible to construct similar examples which will have depth zero. In Chapter V (see § 5, 3.) we will give an example with depth greater then zero (Example V.5.4). The usefulness of local cohomology for our purposes will next be demonstrated by the following result which enables us to give a "parameter-free" expression for the invariant I(M) of a Buchsbaum module M. Proposition 2.6. For any Buchsbaum module M with d : = dim M we have

I(M)

=.r (d -. 1) .l(H~(M)). d-I

,=0

~

Proof: We already know from the proof of Theorem 1.12 that

I(M)

= l((xl> ... , Xd-l)' M:Xdj(Xlo ... , Xd-l)' M) = l((XI' ... , Xd-l)' M:rnj(xlo ... , Xd-l)' M),

where Xlo "" xdis some system of parameters of M. Let us writex~:= xi, M' := Mjx~ ·M. We use induction on d. If d = 1, I(M) = l(O:Mrn) = l(H~(M)), since rn· H~(M) = 0 by Corollary 2.4 which finishes the proof. If d ~ 2 we have already seen in the proof of

•

76

I. Characterizations of Buchsbaum modules

Proposition 2.1 (impl1cation (iii) =? (ii), step (a)) that H~(M') ~ H:nUJI)

E8 H;;I(M)

and we oqtain with the induction hypothesis (dim M' I(lll) = l((x~, x2,

••• , Xd-l)'

lJI:m/(xi, x 2 ,

d - 1):

... , Xd-l)'

= l((X2' ... , Xd-l)' M' :mJ(xz, ... , Xd-l) . M')

M) I(M')

.X/ (d . 2) .l(H:n(M')) .};2 (d ~ 2). (l(H:nUJI) + l(H~:l(M))) =

.=0

'=0

t

=i~l ((~ =~) + (d =d-l

i

t

2)) .l(H:n(M))

(d i 1) .l(H:n(M)) ,

q.e.d. Now we want to state a result which shows first that for a Buchsbaum module the Hilbert-Samuel function Pq,M(n) and the Hilbert-Samuel polynomial Pq.M(n) coincide for each parameter ideal q of M and all n O. Secondly, the Hilbert-Samuel coefficients ei(q, M) are shown to be independent of q for all 1. We obtain an expression for I(M) using these ei(q, M). Finally we find for each t>- 0 non-negative integers It(M) such that

J}[

i>-

l(MJqt+l . M) for all t

0 and all parameter ideals q of 111, where d := dim M

>

Proposition 2.7. Let J}[ be a Buchsbaum module with d:= dim M parameter ideal,q of M (i)

l(M jqt+1 . M)

;=0

(il)

i)

= Ed (t+d . ' e.(q, M) d

for all

O.

>- O. Then lor every

t> O.

t

for all i = 1, ... , d,

ei(q,1}[)

for

for

P 4= -1, P = -1.

Ii.

(iii)

I(M)

=

ei(q, M).

Proof: (iii) is a consequence of (i) if we set t = O. We prove (i) and (ii) by induction on d. For d = 0 there is nothing to prove (q = 0). Assume d>- 1. Let q = (Xl' ... , xa) . A be any parameter ideal of M and let q'

•

§ 2. Cohomological characterization of Buchsbaum modules

:= (Xl' ... ,

t>

77

Xd-l) . A, M':= MjXd . M. Then we have an exact sequence for., each

1:

0--+ qt+l . M :Xdjqt . M

--+

f

Mjqt . M -+ Mjqt+1 . M

--+

M'jq't+l . 1',f'

--+

0

where t is obtained from multiplication by Xd' As in the proof of Lemma 1.15, we obtain

qt+l. M:Xd = qt. M

+ O:MXd =

qt. M

+ O:,Hm

and therefore

since O:Mm n qt. M = 0 by Lemma 1.14. Also l(qt. MjqH1. M) = l(Mj qt+l. M) - l(Mjqt. M)

= -l(H::'(M))

+ l(M'jq't+1. M').

By our induction hypothesis we have

=};l (t + d -. i - 1) .ei(q', M')

l(M'jq't+l. M')

i~O

d - z-

1

and

ei(q', M') =

d1-

l

j~O

(d ~1 -i -1 2) . l(H~(M'))

for all z' = 1, ... ,

d-

1.

From the exact sequence g

0--+ MjO:Mm -+ M

--+

M'

--+

0

where g is induced by multiplying the cosets by Xd, we find for all j S d - 2 short exact sequences (see also the proof of Proposition 2.1(iii) :::::} (ii), step (a)):

o --+ H~(M) --+ Hk(M') --+ H{;l(M) --+ O. Therefore for i = 1, ... , d - 1

=dlj~O

= =

l

(d ~1 -i -1 2) .l(Hk(M)) +j~l X/ (~~1 -i -2 2) .l(Hk(M))

X/ ((d ~1 -

i -1 2)

j~O

1j~i (d 7~ ~ l(H::'(M))

+ (d ~1 -z' -2 2)) .l(Hk(M))

1) .l(H{,(M))

+ l(H}.,(M))

for

z'S d- 2,

for i

=

d - 1.

78

1. Characterizations of Buchsbaum modules

Now for all t 2 0 I

I(Mjql+l . 11{)

I(Mjq. M)

+ .E l(qi. Mjqi+l . Ml 1=1

t

l(M/q· M)

+ .E l(M'/q'id. M')

t .l(H~(M))

;=1

= I(M/q' M) +;~ j~1 = I(Mlq' M)

(d ~ iii

+ £1 ((t + d --: i) d-1

j=O

= l{M/q· M) - iE1e/(ql, M')

+( ~

1).

(ea-I(q', M')

11).

ej(q', M') - t

.1(H~(M))

_ 1) .ei(q', M') _ t .l(H~(M))

+ I(H~(M)) +

7;: e~ ~ ~ i) .

ej(q', M')

-1(H~(M))).

For sufficiently large t this polynomial in t coincides with Pq,M(t) and comparing coefficients we obtain

ej(q, M) = ei(q', M')

for

i

0, ... , d

2,

ea-I(q, M) = ed-I(q', M') - I(H~(M)), a-I

ed(q, M)

=

l(M/q ..Jf)

.E ej(q', M') + l(H~(M)).

j=O

But this proves (i) since the above equation is true for all t for ej(q', M') and a-I l(M/q . M) = I(MI/q' . M') = .E ej(q', M')

O. From our expressions

j=O

(see (i) applied to M' and q', t

0) we also obtain (ii), q.e.d.

Coronary 2.8. Let M be a Buchsbaum module 01 dimension d > O. Then lor every t there i8 a natural number [,(M) 8uch that jor every parameter ideal q oj M '

l(MjqH. M)

(t ~ d) . eo(q, M) =

0

[I(M).

This follows immediately from Proposition 2.7(i) and (ii). Next we will prove a first sufficient "parameter-free!' criterion for the Buchsbaum property which allows us to find many examples for Buchsbaum modules and which is also a necessary condition if A is a regular local ring. First we need the following

Lemma 2.9. Let 0 ->- M' ->-llf -4 Mil ->- 0 be an exact sequence oj A·modules with m· Mfl O. Then the sequences

o ->- Ext~(k, M')

->-

Ext~(k, M).f.!...". Ext~(k, Mil) ->- 0

(k

A/m)

§ 2. Cohomological characterization of Buchsbaum modules

with Ii := Ext~(k, I) are exact lor each i.e. the sequence

i>

°

il and anly il we have exactness lor

79 t'

= 0,

0--+ HomA(k, M') --+ HomA(k, M) --+ M" --+ 0 is exact.

Prool: The only if part is trivial. Assume now that 0 --+ HomA{k, M') --+ HomA(k, M) ~ M" --+ 0 is exact (note:

HomA(k, ll-f") '"'":: O:M"m = M" since m· M" --+ 0 and get a commutative diagram

=

0). We apply HomA(k, ) to M.1.. M"

HomA(k, M) ~ Mil --+ 0

nl

II

M

- - t o M"--+O.

Next, by using Ext~(k, ), we obtain commutative diagrams Ext~(k, HomA(k, M)) -+- Ext~(k, M") --+ 0

I

.J.-

Ext~{k, Mil) --+ O.

Ext~(k, M)

Since M" is a direct summand of HomA{k, M) (by virtue of 10), the top row is exact. Therefore the bottom row is also exact, q.e.d. Theorem 2.10. Let M be a Noetherian A-module with d:= dim M maps (see Chap. 0, § 1, 3.)

1. II the canonical

IPk: Ext~(k, M) --+ H:n(M) are surjective lor all

t'

=f: d then M is a Buchsbaum module.

Prool: By the surjectivity of IPk the H:n(M) are modules of finite length and we conclude by Lemma 2.2 that for every part of a system of parameters Xl' ••• , Xr of M with r < d the submodule (Xl> ••• , x r )· M of M is unmixed up to m-primary components. Now, IP~ is the inclusion HomA(k, M)~ O:,itm c O:M(m) = H:h(M) and therefore we have for every parameter X (i.e. dim Mix. M = d - 1)

m· (O:MX) ~ m· (O:M(m»

m· H~n(M) '" m· HomA(k, M)

0,

i.e., x is a weak M-sequence. 1, we are already done. We use induction on d. If d Assume d>- 2 and let Xl' •.. , Xd be an arbitrary system of parameters for M. Suppose first that depth M > 0. In this case, since O:.IIIX I = 0, we have an exact sequence 0--+ M ~ M --+ Mlx l

This gives for i Mixl' M):

= 0, ... , d

•

M --+ O.

- 2 commutative diagrams with exact rows (set M'

o --+ Ext~(k, M) --+ Ext~(k, M') --+ Ext~+l(k, M) --+ 0

1. Characterizations of Buchsbaum modules

80

By hypothesis q;k, - Ilin(M) since the .other IPk's, i =1= dare .obvi.ously surjective and theref.ore Theorem 2.10 implies (i).

T.o this end we pr.ove the m.ore general statement:

f.,'laim. Let M be a N.oetherian A-m.odule with r: M-sequence Xl> ... , Xr in m 2 such that (Xl' ... ,

Xr )'

M:(m)

=

(Xl' ... ,

X r )'

depth M

< dim M.

If there is an

M:m

then IP~ is an is.om.orphism. F.or the proof we use inducti.on.on r. If r 0, IP~ is the embedding O:Mm ~ O:M(m). Als.o O:M(m) O:,ym by (iii) and IP~ is theref.ore an is.om.orphism. If r > 0 we have an exact sequence

o -'>- M

~ M

-'>-

M'

-'>-

0

with M':= MlxI . M.

This gives rise t.o a c.ommutative diagram

19'~-;1 0-'>-

Il';;"I(M'}

19'M --+

Ilin(M}

24 Ilin(M).

By the inducti.on hypothesis IPM-I is an isom.orphism and thus IPM splits into an is.om.orphism 9?~: ExtA(k, M} -'>- O:Hf (M)X I Il';;"I(M') and the embedding t: O:HT (M)X I ~ Il':n(M).

m

m

§ 2. Cohomological characterization of Buchsbaum modules

83

and consequently

O:H'm(M)X l

=

O:H'm(M)m

H';,,(M)

O:H'm(M)(m)

=

which shows that £ is the identy map. Hence

9'~

is an isomorphism, q.e.d.

We are now able to give our first class of examples for Buchsbaum modules. To do this we define: Definition 2.13. Let r, d be integers with 1::;; r ••• , Yd indeterminates. We set


1. Since xi, ... , x= is a weak M-sequence in m2 , m . H~(M) 0 for all i < d by Proposition 2.1. Assume first depth M > O. By Lemma 2.2, (0) is unmixed up to m-primary components and hence unmixed. Therefore we have an exact sequence 0 -* j1f ..:'4 M -* M' -* 0 with M':= Mlxl . M. This gives rise to commutative diagrams with exact rows 0-* HI-I(m, M) -* Hi-1(m, M') -.,.. HI(m, M) -* 0 (D j )

l,~l 0-.,.. H;;;I(M)

for all i < d. (Notice for all i < d.)

iA~l

-*

Xl'

H;;;I(M')

Hi(m, M)

iAk -* H~l(M)

-*

0 for all ~. and

0 Xl'

H~(M) ~ m . H:n(M)

0

§ 2. Cohomological characterization of Buchsbaum modules

85

Let A' A/Xl' A, m' := m· A. Then x2"'" XI is an M'-basis of m' (considered as A'-modules; we write 53 for the image of X E A in A') satisfying the hypothesis of (iii) (with respect to A' and M/). Therefore the natural homomorphisms ),~~,: Hi(m', M') -~ H:",(M')

are surjective for all i < d - 1 by the induction hypothesis. Let n := (X2' ••• , XI) • A. Then obviously as A-modules H'(m', M') ~ H'(n, M')

and (for instance by R. Y. Sharp [IJ, Theorem 4.3) H:",(M') ~ H~(M').

Hence the corresponding natural homomorphisms l~w : H'(n, M/)

-) O. Further N also fulfils the assumptions of (iii) too, since Xl> ••• , XI is of course also an N-basis of m. For each 1 ~ i] < ... < id ~ t, rl> ... , rd E {I, 2}, o~ j < d we have

+ (x[;, : .. , x~:). M):xi::: ~ ((O:Mm) + (x;;, ... , xi;) . M) :(m: (xi;, ... , xi;) . M :(m) (xi;, "', xi;) . M:m ((O:M m) + (xi;, ... , xi;) . M):m

((O:Mm)

(notice

(xi:,. 0., xi:) . M : m2 ~ (xi:, .. 0, xi;) 0M: xL

(x~;, ... , xi,) .

M:m implies

(xi;, ... , xi;) . M :(m) = (xi;, .0., x~:) . M :m).

Therefore ;.~v is surjective for aU i < d. We now prove that also g' (in the diagram (Di» is surjective for all ~. < d. Then the proof of (iii) ::::} (iI) will be complete by the diagrams (D/). ~ To this end we prove the injectivity of /i for all i d. We use the notation introduced in Chapter 0, § I, 3.

86

I. Characterizations of Buclulbaum modules

Let Ki Ki(x}> ... , X,; M), ili;= Ki(XI' ... , X,; O:Mm) and let Hi, Hi denote the cohomology modulcs of these complexes. Since (Xl"'" X,) (O:M m) 0, the differentiation of j(i is zero, i.e. fl· j(i for all i. Let d t denote the differentiation H of K. We have to show that itt" 1m d = 0 for all t''S d (considered as submodules of Ki). Let X E j(i :l 1m di - 1, i.e. x di-l(y) with y E K,-1: We write

Then we obtain X = di - 1(y)

E

=

(

lS;n1.< ... O. Is M then a Buchsbaum module~

... ,

nd

The following (unpublished) example due to S. Goto shows that this question has a negative answer. Example. Let R: k[X I , ••• , X a, Yv "', Ya], d:2: 3, be the formal power series ring in the indeterminates Xl' ... , X a, Yl> ... , Y a over an arbitrary field k. Put a:= (Xl' ... , Xa) R (j (Y I , ' ' ' , Y a)· R, q:= (Xi, X 2 ,

Fi := Xi

••• ,

+ Y.

A := R/((a

(j

q)

Xd,

Yi, Y

for i

+ Ff· R)

2 , ••• ,

Y d )· R,

1, .,', d, with n

3.

Then dim A = d 1 and A is not a Buchsbaum ring since m· U(O) =f: 0 in A. It is now easy to see that the images of F 2 , ••• , Fa in A form a system of parameters for A and have the required property. We note that the images of F 2 , .,', Fa are even a-part of an A-basis of the maximal ideal m of A since they form a part of a minimal basis of m. A first and important application of Theorem 2,15 is the solution of the so-called lifting problem for Buchsbaum modules, i.e. the possibility of lifting the Buchsbaum property by a non-zero divisor. M. Hochster asked the following related question:

88

I. Characterizations of Buchsbaum moduleS

Let A = Ria be a local ring where R is regular and a is an ideal of R. Suppose that: (i) All is a Cohen-Macaulay ring for all .\) E Spec A " {m}. (ii) there exists a non-zero divisor x of A such that A/x· A is a Buchsbaum ring. Is it true that then A is a Buchsbaum ring? The following example shows that the answer to this question is negative. Example 2.18. Take A : = k[ Xl, X 2 , X a, X 4 ]/(Xi, Xi) II (Xs, X 4 ) where k is an arbitrary field and Xl • ... , X, are indeterminates. Then we get the following: (i) All is a Cohen-Macaulay ring for all .\) E Spec A " {m}. (ii) A/(X l + Xa) . A is a Buchsbaum ring (of dimension one). (iii) A is not a Buchsbaum ring. (iv) m· H~(A) =l= O. The statements (i), (iii), (iv) are clear (see e.g. Proposition 2.25). To prove statement (ii) it is sufficient to show that (X 1,

••• ,

X 4 ) • U(a

+ F . R) ~ a + F

.R

(by Proposition 1.10) where

a

=

(~, Xi) . R II (Xa, X,)· R

F:= Xl

+ Xa,

(Xi' Xa, X~. X" X 2 • Xa,X i

•

X,) . R,

R:= k[XlJ X 2 , Xa, X,].

But this follows from U(a

+ F . R)

(Xl' X 3 , X 4 ) • R

II

(Xi, X 2 , Xi, F) . R

(Xi, X;, X l X 2 ' X 2 X" F). R. What we can prove is the following: Proposition 2.19. Let M be a Noetherian A·module with depth M > O. The /ollowi1UJ . candz'tions are equivalent: (i) M is a Buchsbaum module. (ii) There is a non-zero divisor x E m 2 0/ M such that Mix, M is a Buchsbaum module. (ii') M/x· M is a BucMbaum module lor every non-zero divisor x E m2 0/ M. (iii) Thue is a non-zero divisor x E m 01 M such that: a) MIx, M is a Buchsbaum module. b)x· H:n(M) 0 lor all i < dim M. (iii') For all non-zero divisors x E mol M a). and b) 01 (iii) are true. (iv) There is a non-zero divisor x E m 01 M such that: c) M /x . M is a Buchsbaum module. d) x· H:n(M/x 2 • M) = 0 lor all i < dim M 1. (iv') For all non-zero divisors x E m 01 M c) and d) 01 (iv) are true. Prool: The necessity of the conditions (ii), (ii'), (iii), ... is obvious by Corollary 1.11(i) and Corollary 2.4. Therefore it remains to prove the implications (ii) :=;, (i), (iii):=;, (i), (iv) :=;, (i).

§ 2. Cohomological characterization of Buchsbaum modules

89

We start with the exact sequence O-+M

M -+M' -+0,

where M':= Mjx. M.

From it one obtains a commutative diagram with exact rows 0-+ Hi(m, M) -+ H'(m, M') -+ HHl(m, M) -70

Now in each case ).,k. is surjective for all i < dim M' = dim M - 1 by Theorem 2.15. We want to show that we have always x . H{.lI.Y) = 0 for all j < dim M. Then by the commutative diagrams (Dj-l) ).k: is surjective for all j < dim 111 and (i) follows by Theorem 2.15. In case (iii) this is clear. In case (ii) we obtain by the 'exactness of the bottom row of (D j - 1 ) an epimorphism Hi;;;l(M') -70 :Hi (M)x. m

Since m . Hi;;;l(M') = 0 for all j < dim M (Corollary 2.4), we get m.(O:H~(M)X)

0

for all

i

1. If P is a Buchsbaum point of Vjk then P is a Buchsbaum point of V n FIIJk(u). The wnverse is true if grade (/1' ... , fT) . A > 1 and (/1' ... , fT) • A b;:; lJ2 . A.

The next result was proven by N. V. Trung, see [2], Theorem 4. It gives informations about the lifting property in case depth M = O. Proposition 2.22. Let M be a Noetherian A -module of positive dimension d and depthM = O. M ~'s a Buchsbaum module if and only ~'j the following conditions are satisfied: (i) m· H~(M) = O. (ii) MIH~(M) is a Buchsbaum module. (iii) There is an ~V-ba8is Xl, ••• , XI of m such that H~(M) n (Xi, ... , Xid)'"V

0

for all I S il

< ...
(ii).

Conversely, the proof of Theorem 2. 15(iii) =:;> (ii) shows that the canonical maps 41: Ht(m, llf) -i>- H~t(M) are surjective for all i < dim ~V, i.e. -LV is a Buchsbaum module by Theorem 2.15, q.e.d. The next proposition which has its origin with M. Brodmann [4], gives a further criterion for "lifting" the Buehsbaum property. It is closely related to the statements of Proposition 2.19. Proposition 2.23. Let M be a Noetherian A-module with d: dimM? 2 and depthM O. M is a Buchsbaum rnodule if and only ~1 mn:n(llf) = 0 and MjxM :(m) is a Buchsbaum module for all X E m with dim llfjx1lf d - 1.

§ 2. Cohomological characterization of Buchsbaum modules

91

Proof: If N: is a Buchsbaum module then mH:n(M) 0 by Corollary 2.4 and JfjxM :(m) = IlfjU(Xllf) is a Buchsbaum module by Corollary 1.11. Assume mH:n(M) 0 and that 1lfjxM:(m) is a Bm·hsbaum module for all x E m with dim MjxM dim MjxM:(m) = d - 1. If d 2, there is nothing else to· prove. Let d > 3. Choose x E m 2 with x It .):l for all .):l E Ass M. The exact se- 0 induces for all {epimorphisms

Since mHf;;l(MjxM) mH:;;l(ilfjxM:(m») = 0 for all i with 2 i < d (this follows from 111 jxM: (m) being a Buchsbaum module, compare Corollary 2.4) we have m(O:Hi (M)X) 0 for 2 i < d. Since x E m 2, this implies O:H i (M)m 2 ~ O:H' (M)X m

m

m

~ O:H' (M)m ~ O:H i (M)m 2 • Therefore

m

m

O:H:n(M)m

0:H~(M)m2

= ... =

O:H~(M)(m)

Htn(M)

and this shows that H:n(M) is annihilated by m for { 2, ... , d 1. Since mH:'t(M) 0 by our assumption (and H~(M) = 0), we have by Lemma 2.2 that for every system of parameters Xl' ••• , Xd of M the submodule (Xl> ..., xi)M is unmixed (in M) up to m-primary components for all i = 0, ... , d - 1. H x E m with dim MjX1lf = d - 1, X is a non-zero divisor of M (since 0 is unmixed in M). Therefore the exact sequence 0 -'>- M ~ M -'>- MjxM -'>- 0 induces an isomorphism H~(MjxM) ~ H:n(M), i.e. we have mH~(1ffjxM) = O. But this means m(xM:(m») xM or xM:(m) = xM:m. In particular for every y E m with dim M/(x, y) M d 2 we have xM:y xM:m. Let now Xl> ••• , Xd be an arbitrary system of parameters of M. 'Ve choose an x E m such that x, Xl> ••• , Xd-l is again a system of parameters of M. Since IJf/X"~Y :(m) is a Buchsbaum module for all n 1, it follows for all j = 1, ... , d I and all n 1 that (x",

Xl, ... ,

+ (Xl' ... , Xi-I) M):Xi = (x"M:(m) + (Xl' ••. , XH) M):m.

xH)M:Xj ~ (x"M:(m)

If we take the intersection over all n -;::: 1 we get by Krull's Intersection Theorem

in particular

+

Claim. x1M:m n (x2M:m (X3' ... , Xi) M) ~ xlM for all j For i 2 we have to prove: xIM:m n T2M:m TIM. Now x l x 2M:m XIX2.lf:X2 = XliII, hence x 1x 2M:m

(X 1x 2M :m) n xliff X1((X l X21lf:xl):m)

=

X l ((X I X 2 .1J.{ :m) :X I )

=

Xl(X2 M

: m ).

2, ... , d - 1.

92

1. Characterizations of Buchsbaum modules

Since dim M/X 1X2M = d

1, we conclude xIM:m nx2M:m 2

x1M:m n x2M:m

= (XI(x 2M:mx 1»):m X1X2M:m Let 3:;; j

(xiM n (x2M:m)):m

= (X 1(x 2 Jf:m)):m = (x1x 2M:m);m

XI(x2)1I:m)

xiM.

d - 1 and mE (xIM:m) n (x2M:m + (xa, ... , Xj) ill). Then m = U2 U 2 E x2M: m, ma, ... , mj E M. Therefore

+ X3ma + ... + xjmj with mj E (xIM:m

+ X2M:m + (X3' ..• , xj-l) M):x;

(XI1 X2, ••• , Xi-I) M: (m) = (Xl' ..• , x;-d M : m,

+ ... + mj_1Xi-1' But this implies m~xI = U2 + ~X2 + (ma + m~)xa + ... + (mj_l + mj_I)Xj_l E x 111I:m n (x2M:m + (xa, ... , xj-l) M) x1Jf,

, i.e. we have xjmj =m~xI

m

i.e. m E x1M which proves the contention. Now we have (M/x1M;m is a Buchsbaum module): m( (Xl' ... , Xd-l) M :Xd) ~ m( (xIM; m ~

and exchanging Xl and

xlM; m

+ (x

2 , ... ,

Xd-l) M) :Xd)

+ (X2' •.. , Xd-l) M

X2

m((xI' .•. , Xd-l) M: Xd) ~ x2M: m

+ (Xl' xa, ... , Xd-l) ill.

Hence

m((Xl' ... , Xd-l) M:Xd)

+ (X2' ... , xd-d M) n (x 2Jll:m + (Xl' X Xd-l) M) (X2' X3, ... , Xd-l) M + ((xIM :m) n (x2111:m + (Xl' xa, ... , Xd_1M») (XI1 X2, ... , Xd-I) M + ((xIM:m) n (x 2M:m + (xa, •.. , Xd-l) "If))

~ (xIM:m = =

3 , ••• ,

(Xl' ... , Xd-l) M.

Therefore by Proposition 1.10 M is a Buchsbaum module, q.e.d. Corollary 2.24. Let M be a Noetherian A-module with d;= dim M 3 and depth M > O. Assume lurtherrrwre that either A Z8 an epirrwrphic im,age 0/ a Gorenstdn n'ng or Hin(M) is a Noetherian module. Then M i8 a Buchsbaum module il and only il MJxM:(m) is a Buchsbaum module lor aU X E m with dim M/xM = d 1.

Prool: The "only-if-part" needs no further elaboration. Assume that M /xM: (m) is a Buchsbaum module for all X E m with dim 11l/xM d - 1. Suppose A is an epimorphic image of a local Gorenstein ring. If there is a ~ E Ass M with 1:;; dim Aj~ < d - 1 then (O:M.\.l)V O:Mv.\.lAv O. We choose an X E q with X ~ q for all q E AssM with dimAjq = dand xMv n (O:Mv.\.lAv) = O. Then

'*'

Hom(Av/)'Av, xMv) '" O;.Mv.\.lAv = xMv n (O:Mv.\.lAv) = 0

§ 2. Cohomological characterization of Buchsbaum modules

93

and the exact sequence 0 -+ xMlJ -+ MlJ -+ (M/xM)lJ -+ 0 induces a monomorphism Hom(AlJ/vAlJ' MlJ) -+ Hom(AlJ/pAlJ, (M/xM)lJ)' Since Hom(AlJ/pA lJ , MlJ)::::::: 0 :MlJVAlJ i= 0, it follows that Hom(AlJ/VAlJ, (MlxM)lJ) i= 0, i.e. VEAssM/xM. Since AssMlxM:(m) = AssM/xM'-.{m}, we obtain fJ E AssM/xM:(m). Also since dim M/xM d - 1, M/xM:(m), is a Buchsbaum module, hence dim Alp = d - 1 2. Let A be an epimorphic image of the local Gorenstein ring B. Then by the local duality theorem (cf. Corollary 0.3.5), H:n(M) Hom,4(Ext~-1(M, B), E), where E denotes the injective envelope of A/m (as an •. A-module) and n dim B. Let H:n(M) i= 0 and V E ASSAExt~-l(M, B). Then dim A/V :S: 1 by Sharp [2], Proposition (3.8) and Theorem (2.3), and we have with q denoting the inverse image of fJ in B): r-.J

Ext;-l(M, B)lJ ~ Ext~;l(2l!lJ' Bq) where Bq is a Gorenstein ring of dimension n 1 and AlJ is an epimorphic image of B q • If I denotes the injective envelope of the AlJ-module AlJ/VAlJ' then by local duality HOmAlJ(Ext~;l(MlJ' Bq), I)~ H~AlJ(MlJ)

0

if dim A/V

=

1

as was shown previously. Hence dim Alp 0, i.e. p m. Thus Ext~-l(M, B) is a module HomA(Ext~-l(M, B), E) is a module of finite of finite length and therefore H:n(M) length, i.e. a Noetherian A-module. Next assume that H:n(M) is Noetherian. Take an element x E m with dim M/xM = d 1 and xH:n(M) = O. Then from the exact sequence 0 -+ M":"'" M -+ M/xM -+ 0 we get a monomorphism H:n(M) -+ H:n(Mjdf) H:n(lrf/xM :(m»). The last module is annihilated by m since lrf/xM :(m) is Buchsbaum and 1 < d 1 dim M/df:(m). Hence mH:n(M) 0 and the Corollary follows from Proposition 2.23, q.e.d. r-.J

r-.J

Another application of Theorem 2.15 gives informations on the Buchsbaum property of a local ring whose zero ideal is the intersection of two "perfect" ideals. We note that the ideals "of type (r, d)" defined above (Definition 2.13) belong to this setting, see also Lemma 2.14. Additionally the following statement has some useful applications with respect to liaison among arithmetically Buchsbaum curves in pl!.

Proposition 2.25. Let A be a local ring with d dim A 2 2 and a, b ideals of A with a n b = 0, dim A/a + b < d. Assume A/a and A/b to be Oohen-Macaulay rings of dimension d. Then A i8 a Buchsbaum ring if and ooly if either a + b = m or B A/(a + b) ~'s a Buchsbaum ring of dimeru:;wn d - 1. Proof: As in the first part of the proof of Lemma 2.14 we have an exact sequence:

O-+A -+A/a(f)Alb-+B-+O. Since depth A/a

depth Alb

HH(m, B)

lA~-1

~

= d, we have for all i < d commutative diagrams:

H'(m, A)

lA~

H:;;-l(B) ~ H:n(A).

By Theorem 2.15 A is a Buchsbaum ring if and only if ).~ is surjective for all i

d - 2.

94

I. Characterizations of Buchsbaum modules

If dim B 0, i.e. RJ.n(B) = 0 for all j 1 this is equivalent to the surjectivity of ).~. But RO(m, B)~ O:Bm~ (a + 0) :m/(a + 0) and R?n(B) B = A/(a + 0) and thus A is a Buchsbaum ring if and only if a + 0 = m. If dim B > 0, ).ltm B cannot be surjective since R'!t:mB(B) is not a module of finite length. Since dim B < d - 1, A is a Buchsbaum ring if and only if dim B d - 1 and ).1 are surjective for all j < d - 1, i.e. if and only if B is a Buchsbaum module over A by Theorem 2.15 and hence a Buchsbaum ring itself by Lemma 1.6, q.e.d. •

Another application of Theorem 2.15 was proven by U. Daepp and A. Evans in [1]. In order to formulate this result we need to introduce some further notions. Let M be an A-module and X an indeterminate. Then we set M*:= 1'1:f[X]m[xj,

where m[X] denotes the kernel of the map A[X] -+ (A/m) [X] given by the canonical projection A -+ A/m, i.e. m[XJ is a prime ideal of A[XJ. Now, it is clear that M* M @AA* and that the natural map A -+ A* is a local .flat homomorphism. For every A-module N of finite length lA.(N*) = lA(N). This implies for example eo(q*, M*) = eo(q, M) for every Noetherian A-module M and every ideal q c A with l(Mlq· M) < 00. • For each ideal a c A we have a* = a A* a· A* and thus for the Koszul complex K.(a· A*, M*)

~

K.(a, M)@AA*

(as complexes).

Therefore Ri(a*, M*) ~ Ri(a, M)@AA* and R~.(M*) R~(M) @AA* since tensor products commute with direct limits. Consequently, (m* is the maximal ideal of A*) dimAM = dimA.M*. Now, we are in position to prove.

Lemma 2.26. Let M be a Noetherian A-module 0/ positive dimension. M is a Buchsbaum module over A if and only if M* is a Buchsbaum module over A. Moreover, [(M*) = [(M). Proof: We have ).it.:::::: ).it®AidA• and since ®AA* is an exact functor, ).it. is surjective if and only if ).it is surjective which proves (using Theorem 2.15) the first statement. Next, let M (and therefore M*) be a Buchsbaum module. Let q be a parameter ideal of M. Then q* is a parameter ideal of M"" and [(M"") = l(M*/q*' At*} - eo(q*, M*)

=

l(Mjq. M)

eo(q, M)

=

l(Mlq· M)*) - eo(q*, M*}

[(M) ,

q.e.d. The previous lemma has an interesting consequence which was first proven for a special case by Daepp and Evans in [1]. To establish this we need a lemma (for notations see Chap. 0, § 2, 1.):

Lemma 2.27. Let R be a graded ring and aS8ume that ~ c R is a homogeneoos prime ideal with [R]l g;; ~. Then Rp""",Rtp)

§ 3. Graded Buchsbaum modules

95

and hence we have for every graded R-module M: MlJ~MrlJ)'

Proof: This follows immediately from Lemma 0.2.1. (It is sufficient to prove RlJ ~ RrlJ)' From Lemma 0.2.1 we obtain RlJ ~ (RlJ.h)lJ.RlJ.h ~ (R(lJ)[X]x)m[x]x ~ R(lJ)[X]m[x] = Rrw where m is the maximal ideal of R(lJ)')

Corollary 2.28. Let R be a Noetherian graded ring and let M denote a Noetherian graded R-rrwdule. Then we have for every homogeneous prime ideal tJ c R with [RlI g;; tJ: M lJ is a Buchsbaum rrwdule if and only ~'j M(lJ) ~'s a Buchsbaum module. In this case, I(MlJ) =I(M(lJ))'

§ 3.

Graded Buchsbaum modules

The goal of this paragraph is to expand the results of both previous paragraphs to graded modules. The geometric background and the motivation for this is to get information about the Buchsbaum property of the local ring at the vertex of the affine cone over a projective variety. Throughout we use the concepts introduced in Chapter 0, § 2. In addition we always suppose that our graded k-algebras (k a field) are generated by their homogeneous elements-of degree one, i.e. they are of the form k[Xo, ... , Xn]ja where Xu, ... , Xn are indeterminates (of degree 1) and a is a homogeneous ideal of k[Xo, ... , X n]. Let R always denote such a graded k-algebra with the maximal (homogeneous) ideal m = ffi [R]n. n:2:1

Definition 3.1. Let M be a Noetherian graded R-module of positive dimension. M is called a Buchsbaum module if Mm is a Buchsbaum module (over Rm)' M is called an h-Buchsbaum module if every homogeneous system of parameters with respect to M is a weak M-sequence. Thereby weak M-sequences are defined analogously to the local case. It is clear that M is an h-Buchsbaum module if it is a Buchsbaum module. Our goal is to study the converse of this statement. We are able to prove it if k is an infinite field. This means, geometrically speaking, that the Buchsbaum property of the local ring of the affine cone at the vertex over a projective variety may be verified by regarding only homogeneous systems of parameters. In a similar way as in § 1,3f-bases consisting of homogeneous elements (M a Noetherian graded R-module) of homogeneous ideals a with dim Mja . M are defined. Therefore we omit an additional definition. But in contrast to the local case such bases ~ay not exist. We give two examples:

1. In a homogeneous basis of a appear elements of different degrees : We choose R = k[X, Y], a = (X, Y2) . R, M = RjX . R (X, Y indeterminates). 2. The field k is finite: We choose a = m and M := Rjp . R where p denotes the product of all elements of degree one in R. If we exclude these two cases we are able to prove the existence of M-bases. The proof is essentially the same as in the local case (Proposition 1.9). In addition we need here the following easy

96

1. Characterizations of Buchsbaum modules

Lemma 3.2. Assume that the ground lield k is inlinite. II Yb ... , Yt are elements 01 [RJ/, 1 > 0, and il there are homogeneous ideals hI> ... , h. z'n R with (Yt> ... , Yt) R g; h; lor all i 1, ... , s, then there are elements lXI' ••• , IX, 01 k with IXI •

Yl

+ ... +

<XI'

Yt

q hi

U ...

u h •.

Prool: Let V t;:;;;; [RJ/ be the vector space spanned by the Yi'S. We put Vi V n hi' i = 1, ... , s. Since (YI' ... , ytl . R hi for all i, Vi is a proper subspace of V for each i. Since k is infinite, this implies V n (hi U ... u h.) = VI U •.• u V. ~ fT, hence we get V hI U ... u h., q.e.d.

Now we can" prove: Proposition 3.3. Assume that the ground lield k is z'nlinite. Let a R be an ideal possessing a basis whz'ch consists 01 homogeneous elements 01 the same degree r and M I , " ' j Mn Noetherian graded R-modules with dimR MdaM; 0 lor all i = 1, "', n. Then there are all ... , at E a lorming a homogeneous M;-basis 01 a lor all i = 1, n. "'j

Prool: We can use the proof of Proposition 1.9 word for word with only one modification: when constructing am we simply choose an element of degree r" not contained in (at> ••• , am-I) R u U V (apply Lemma 3.2), q.e.d. PEL

Also we have: Proposition 3.4. Let M be a Noetherian graded R-module with d: dim M > O. The lollowz'ng conditz'ons are equivalent: (i) There Z8 a homogeneous system 01 parameters 01 M contained in m2 whwh is a weak M -sequence. (ii) Every homogeneous system 01 parameters 01 M contained in m 2 is a weak M-sequence. (iii) m· !l:n(M) 0 lor all i =F d.

Proal: (ii) =} (i) is trivial and (i) =} (iii) may be verified as in the proof of Proposition 2.1. We only have to pay attention to the necessary shifting of degrees. Finally, we obtain the implication (iii) (ii) by localizing at m and applying Proposition 2.1, q.e.d.

Likewise by localizing at m and applying Theorem 2.10 we obtain Theorem 3.5. Let M be a Noetherian graded R-module with dim M maps (! Rim)

> O.

II the natural

glk: Extk(!, M) .....,. !l~(M)

are surjectz've lor all i

< dim M then M

is a Buchsbaum module.

Corollary 3.6. Let M be as z'n Theorem 3.5. II in addition r:= depth M < dim M and !l~(M) = 0 lor all r, d then the lollowz'ng conditions are equivalent: (i) M is a Buchsbaum module. (ii) M is an h-Buchsbaum module. (iii) m· !l~(M) = O. Prool: (i) =} (ii) is clear, (ii) =} (iii) follows from Proposition 3.4 and (iii) if we localize at m and apply Proposition 2.12, q.e.d.

Now we prove the main result of this paragraph:

=}

.d

(i) we obtain

§ 3. Graded Buchsbaum modules

97

Theorem 3.7. Assume that k is an infinite field. If M is a Noetherian graded R-module w#h d := dim 1W > 0, the following corul#ions are equ~'valent: (i)

M is a Buchsbaum module.

(ii) M £s an h-Buchsbaum module.

(iii) Take a homogeneous llf-basz8 Xl, ••• , XI of m. Then for each system'il' ... , Zd of £ntegers until, 1 £1 < ... < ia - t the sequence x'\ . .. , X~d is a weak M-sequence for all ~ ~ rl, ... , rd E {I, 2}.


-ll~(M) are surject£ve for all i




k[Xh ... , Xn] over

O.

(ii) m· ll~(M)

0 for all £ < dim M. We note that such Rand M exist (see Example V.5.4). We have to distinguish the following cases: 1. Each linear form of R is not a part of a system of parameters of M, i.e. all homo geneous systems of parameters of M are contained in m 2 • Then M itself is an h Buchsbaum module by applying Proposition 3.4. 2. There is (at least) one linear form l of R with dim Mil· M < dim M. Let p be the square of the product of all these linear forms and set N := Mjp· M. We note that p is a non-zero divisor of M contained in m 2• Let Xl' "0' Xa be a homogeneous system of parameters of N. Then Xl> •• " Xd E m 2 and by Proposition 3.4 p, Xl, ••• , Xd is a weak M-sequence. Therefore Xl> ••• , Xa is a weak N-sequence and N is an h-Buchsbaum module. If N would be a Buchsbaum module, i.e. N m is a Buchsbaum module over Rm , the lifting property (Proposition 2.19) implies that Mm is a Buchsbaum module. This is a contradiction and we have found an h-Buchsbaum module which is not a Buchsbaum module. As an application of our last theorem we will state a new sufficient criterion for graded Buchsbaum modules using only local cohomology modules. To this end we need: 7 Buchsbaum Rings

,

98

1. Characterizations of Buchsbaum modules

Lemma 3.9. Let R:= k[Xh .••, X,,] (Xl' ... , Xnindeterm~'nate8) and let H be a graded R-module with m . H = O. Then

Extk(~, H):::: HomR(R(r)( -i), H)

for all i> O.

H(7)(i)

Proof: The graded Koszul complex K.(Xh ••. , X .. ; R):

o -+R(:)(-n) -+ ••• -+R(;)(-2) -+ R(7)(-l) -+R-+O provides a free resolution of (m.HomR( ,H)

~

Rim. Applying HomR( ,H) we get for all i> 0

= 0):

Extk(~, H)

HomR(R(r)( -i),

"-J

H):::: HomR(R(r), H(z,») '"'-' H(f)(£) ,

q.e.d. For abbreviated notation we define for each graded R-module M the following set of integers: g(M) := {i E Z I [M]j 9= O}. Proposition 3.10. Let M be a Noetherian graded R-module with d:= dim M > 0 and m . !!:n(M) = 0 for all i < d. If for each pazr of integers i, j with 0 ~ i < j < d and all p E g(lt:nCLlf»), q E g(l:!~(M»), (z'

+ p)

(j

+ q) 9= 1

then M is a Buchsbaum module. Proof:IFirst assume R = k[X}> ... , X .. ], Xl' ... , X" indeterminates. If 0< i then for q E g(ltfn(M» (by Lemma 3.9):

[Extk-i+l(!, It:n(M) )]q

[It:n(Mh-r+I)]q+j-i+l i + 1 Et g(lt:n(M»).

< j reap. m2' We set R a(RI' R 2 ) and m := a(ml> m2)' m is then the maximal homogeneous ideal of the graded· k-algebra R, see also Chapter 0, § 2, 4. First we state and prove some preliminary results: Lemma 4.1. Let R be a graded k-algebm. Suppose that M i8 a Noetherian graded R·module with d := dim M > O. We hwve: (i) If depth M > 1, then [M]" =t= 0 for all n a(M).

for all n < e(H::"UW)) (a(M), e(H::"(M)) are defined in Chapter 0, Definition 2.2).

(ii) m· £Hiit(M)]" =t= 0

Proof: (i) If [M]p

0, then we have for all q < p:

[R]p_q' [M]q

1 this implies !lifn(M) =F 0, i.e. dim M = d. Let d 1. If dim M = 0, then M and !lfn(M) = O. Therefore !l°(M) 0 which is impossible, q.e.d.

!l~(M) ~

Lemma 4.3. Let R be a graded k-algebra and M (( Noetherian graded R-module with d ;= dim M > O. 111W is a Cohen-l1facaulay module then

HM(n) - hM(n)

(-I)d rankk[!l'fn(M)]n. !

Prool: By Serre [1], Nr. 79, we have for arbitrary Noetherian graded R-modules M:

E

hii(n)

1)/ rankk(Hi(X, M(n))

for all n E Z

;:?i:O

where X := Proj Rand 1W denotes the sheaf associated to M. For all 0 and n E Z one has (see Proposition 0.2.3)

Hi(X, iJf(n))::::. H((M(n)) = [!l'OW)]" and therefore

hM(n)

hii(n)

=

rankk[!l°(M)]"

+E

1)' rankk[!l::l(M)],.

j~l

=

rankk[M]" -

E j~o

1)/ rankk[!l:n(M)],.

§ 4. Segre products of graded Cohen-Macaulay modules

is obtained. Since rankk[M]n = HM(n) and since by our assumption H:n(M) all i =l= d, the statement now follows, q.e.d.

101

=0

for

Together with Lemma 4.1(ii) this implies Corollary 4.4. Let B, M be a.s in Lemnw 4.3. Then (see Definition 0.2.2): (i) r(M) = 1 + e(ll~{M»). (ii) r(M) = inf{n E Z I hM{n) = HM(n)}.

Our next result enables us t{) calculate the index of regularity rIB) of a graded CohenMacaulay algebra B without knowledge of the Hilbert function Hn(n) of R.

Lemma 4.0. Let B denote a graded Oohen-Macaulay algebra over k with d:= dim B > O. Then we have for every system of parameters Xl, •.. , Xa with respect to R which i8 contained in [Blt: d

+ r{B)

inf{t E H I mt

(Xl' ••• ,

xa) . B} .

inf{t E M I mt Hi!:,(M2 )}

-+

H~(M)

1q(ldMl'~~.)

l~:r -+

O'(MI' Ext~:(~, M 2))

ll~(M)

1~~

if dl =1= d 2 and a similar diagram if dl = d2 • Now, M is a Cohen-Macaulay module if and only if H:n(M) 0 for all i < d l + d 2- 1, . i.e. if and only if O'(H;;:,(llfl }, M2) = O'(MI' Hi!:,(M2l) = O. But this is equivalent to (Lemma 4.1(i), (ii) and Corollary 4.4(i))

r(M1} = 1 + e(H~,(Ml})

a(M2)

and

r(M2 )

= 1 + e(Hi!:,(M2))

a(M I}.

Next assume that M is a Buchsbaum module. Then m . H:n(M) = 0 for all i 1 by Proposition 3.4 and, consequently, 2 -

+d

< dl

and Om· O'(Ml> Hi!:,(M2)} But depth

mj'

O'(m l • M I, m2 • Hi!:,(M 2)).

> 1 for i = 1,2 and by Lemma 4.1 1 + e(H;;:.(M I)) 1 + e(m Hf;,(M I)} ~ a(m2 . M 2) = 1 + a(M2)

.ilf i

r(M I) =

l •

and, by exchanging MI and M2

Conversely suppose that these both relations hold. We want to show that the canonical maps tpk are surjective for all i < d l + d2 - 1. By Theorem 3.5 this will prove our statement. According to the above commutative diagrams it will be sufficient to prove the surjectivity of O'(tp'j;., idM .) and O'(idM ., '1'1;,). If a(M2) > e(Hf;.(M I)), O'(H;;:.(M I), M2) 0 and there is nothing to prove. Assume therefore a(M2) = e(H;;:,(M1 )} =: e.

103

§ 4. Segre products of graded Cohen.Macaulay modules

Now, [0'(Hg;,(M 1), M 2)]" = mg;,(M1)]PQ9k[M2 ]p = 0

for all p =1= e,

I.e. [0'(9'~" idM,)]" is surjective. It is also easy to see that Ext'1i,(!,Ml)~Homll,(k,Hg;,(Ml))::::O:Hd'(M,)ml and -m, that 9'~1 is the corresponding embedding. On the other hand mI' [Hg;,(M I )]. 0, i.e. [Hg;,(MI)]e c [O:!!~I(M,)md•. Hence [9'~.l. is even an isomorphism and therefore [a(9'~" idM .)]. is also an isomorphism. Thus 0'(9'~" idM .) is surjective. Exchanging Ml and M2 we obtain the same for a(idM" 9'~.) and the proof is finished, q.e.d.

For dim Ml = 1 or dim M2 We do this in the following

=

1 we need another formulation of our statement.

Lemma 4.7. Let Mv M2 be as in Theorem 4.6, but dim Ml • hat'e for M O'(Ml> M 2): (i) If dim M2

=

1, dim M2

> 1.

Then we

1, M is a Cohen-Macaulay module.

2, M is a Cohen-Macaulay module if and only if r(M I ) M is a Bucksbaum module if and only q r(M I ) 1 a(M2)'

(ii) If dim M2

+

Proof: (i) By Corollary 4.2 we get dim M

=

a(M2) and

1 and by Corollary 0.2.12: depth M

1.

(ii) is obtained by the same methods used in the proof of Theorem 4.6, q.e.d. Next we state two corollaries of Theorem 4.6. The first is a main result of Chow (1] for our graded k-algebras and the second gives a very easy method for calculating the Cohen-Macaulay resp. Buchsbaum property of graded complete intersections.

Corollary 4.8. Let Rh R2 be graded Cohen·Macaulay algebras over k with d; : = dim Ri 2 for i = 1,2. O'(RlI R 2) is a Cohen-Macaulay algebra if and only if RI and R2 are proper k-algebras, ~·.e. there are systems of parameters Xl>"" Xd, of Rl and YI' ... , Yd, of R2 consisting of homogeneous elements of degree one such that m~' ... , Xd.) . Rl and mg· ... , Yd,) • R 2• Proof: By Lemma 4.5 we have for i proof is finished, q.e.d.

=

1,2: d i

+ r(Ri) ::;; d i •

Corollary 4.9. Let R1:=k[XI, ... ,Xn ], R 2 :=k[Y1 , ~1uJeterminates)

and let al = (/1> ... , fr) . Rl of the lJNrwipal cla88 r, re8p. 8 with n - r, m

(i) O'(RIJalJ R 2/a2) is a Gohen-Macaulay algebra if and only if r

8

E deg /; ::;; n

and

;=1

;=1

m.

j=l

r

E degJ.

E deg gi::;;

n

+1

s

and

E deggj j=l

0, the

Y m ] (X 1, ... ,Xn , Y 1 , ... , Ym (gll . '" g.) . R2 R2 be ideals 2. Then we have: .... ,

Rl> a2 8

Since a(R,)

m+ 1.

104

I. Charact.erizations of Buchsbaum modules

Proof: We apply again Theorem 4.6 and notice that for instance r(Hl/a l ) = -n

r

+ 1: deg /;

(see Grabner [1], 142.) ,=1 The modifications of these corollaries to the case dim Rl = 1 or dim R2 1 (Corollary 4.8) or n - r 1 or m s = 1 (Corollary 4.9) are easy to obtain (use Lemma 4.7) and are left to the reader. By reason of the following geometrical discussions we still prove the following

Proposition 4.10. Let llfi' llf2 be Noetherian graded Rr resp. R 2-modules with d,:= dim llf. 2 for i = 1,2 and let llf aUlll> llf2 ). llf is locally Cohen-llfacaulay and equid~men~ional if and only if llfl and llf2 are locally Cohen-llfacaulay and equidimensional. Proof: By Corollary 0.4.15 and the existing relations between the local cohomology modules l!:n(llf) and the cohomology modules !P(llf) := !Jl(R, llf) l~ Ext~(m", llf) n

(d. Chapter 0, § 2, 3.) we have that llf is a locally Cohen-Macaulay module if and only 1) since if !Ji(llf) are Noetherian R-modules for all i < d l + d 2 - 2 (= dim llf Rm is an epimorphic image of a local Gorenstein ring. Further, we recall that the Segre product of two Noetherian graded modules is again a Noetherian graded module and that the Segre product of a Noetherian and an Artinian graded module is a Noetherian (and Artinian) graded module. If 1lfl and M2 are locally Cohen-Macaulay modules these remarks and our Kiinncth 2, formulas (Proposition 0.2.1O(ii)) imply that !In(llf) is Noethcrian for all n < d l + d2 i.e. ill is a locally Cohen-Macaulay module. Conversely, assume that llf is a locally Cohen-Macaulay module. Let 0 i < d l - 1. Then again by our Kiinneth formulas a(!J'(llfl ), !Jd.-l(llf2)) is a direct summand of !Ji+d.-l(llf) and hence Noetherian. But there is an integer e such that [!Jdd(llf2 )]p =4= 0 for all p e (see Lemma 4.1(ii)). Therefore there must be an integer a with [!Ji(llfl)]q = 0 for all q < a, i.e. !Ji(llf 1 ) is Noetherian. Hence llfl is a locally Cohen-Macaulay module. Exchanging llfl and llf2 we obtain that llf2 is also a locally Cohen-Macaulay module, q.e.d. We next state some corollaries and make some comments in a geometric context.

Corollary 4.11. Let V p.. and W 1 by defining them on the basis elements by

+

q

Oq[V o, VI,

••• , V q ] =

1:

;=0

where 'Vi denotes that Vi is missing. It is easily verified that Oq indeed extends to a O. The chain complex C(Ll) homomorphism Cq(d) --:.. Cq_ 1(d), and that OqOq+l = {Cq(d), Oq} is the oriented chain complex of d. Define an augmentation e: Co(d) --:.. A by e(x) = 1 for every vertex x E V. The augmented ehain complex (C(Ll), e), is the augmented oriented chain complex of d (over A). Then the qth reduced homology group of L1 with coefficients A, denoted Hq(d; A), is defined to be the qth homology group of the augmented oriented chain complex of Ll over A.

§ 1. Foundations

109

Furthermore, the reduced Euler characteristic i(L1) of L1 is defined by

E

x(L1)

(-1)q rank Hq(L1; A).

q2~1

It is independent of A and is also given by i(L1}

=

-1

+ 10 -/1 + ... ,

where Iq is the number of q-simplices in L1. If X(L1) is the ordinary Euler characteristic then X(L1) = X(L1) 1. ' If L1 4= 0, then Hq(L1; A) = 0 for q < O. If L1 = 0, then H q(0; A)~ A for q = 1, and 0 for q 4= -1. In particular1 X(0) = -1. We now wish to define the homology groups of a space X, rather than a simplicial complex L1. Let X be a topological space. Let L1q denote the standard q-dimensional ordered geometric simplex (Po, ... , pq) whose vertices Pi are the unit coordinate vectors in Rq+l, A singular q-simplex in X is a continuous map (1:

L1q -i>- X •

Let Oq(X) be the free A-module generated by all singular q-simplices. The elements of Oq are formal finite linear combinations E ca(J, where (J is a singular q-simplex a

e!:

and Co E A. Given a vertex Pi of L1 q, there is an obvious linear map L1q-l -i>- L1q which sends L1q-l to the face of L1q opposite Pi' The ith face of (J, denoted by (J(i), is defined to be the singular (q - 1)-simplex which is the composite

We now define a linear map (= A-module homomorphism) Oq: Oq

-i>-

Oq_1 by

where (J is a singular q-simplex. It is easily checked that Oq-l Oq = 0, so O(X) = {Oq(X}, Oq} is a chain complex, the singular chain complex of X (over A). Define an augmentation c:: Oo(X) -i>- A by C:«(J) = 1 for all singular O-simplices (J, The augmented chain complex C(X) is the augmented singular chain complex of X (over A). Then the qth reduced singular homology group of X with coefficients A, denoted Hq(X; A), is the qth homology group 01 the augmented 8%1Igular chain camplex of X (over A). Considering this case the reduced Euler characteri8tic i(X) of X is defined by i(X}

E

(-l)q rank Hq(X; A).

q2~1

It is independent of A. If L1 is a simplicial complex and L11 and L12 are subcomplexes of L1, then there is an exact sequence (whose definition we omit)

(with all coefficients A), called the reduced Jfayer- VUtOri8 8equence of L11 and L1 2. Similarly, if X is a topological space and Xl, X 2 are "nice" subspaces (e.g., if Xl U X 2

110

II. Hoohster-Reisner theory for monomial ideals

= (intx,ux, Xl) u (intx.ux. X 2 ), where inty Z denotes the relative interior of Z in the space Y), then we have a reduced Mayer-Vietoris sequence of Xl and X 2 exactly analogous to that of ,11 and ,12' We now come to the relationship between simplicial and singular homology: Let ,1 be a finite simplicial complex and X = [,1[. Then there is a (canonical) isomorphism for all q:

i1q(.d; A) '" iiq(X; A). F6r example, let Sd-l denote a (d l)-dimensional sphere. Then Hq(.d; A) A for q d - 1 and 0 for q =to d - 1. A simplicial complex ,1 or topological space X is acydic (over A) if its reduced homology with coefficients A vanishes in all degrees q. (Thus the null set is not acyclic, since ii_l(0; A)~ A.) Let Y be a subspace of X. Then the singular chain module Oq( Y) is a submodule of Oq(X), so we have a quotient complex O(X, Y) = O(X)/O(Y) = {Oq(X)/Oq(Y), 8q}. Define the relative homology of X modulo Y (with coefficients A) by

We next want to define reduced cohomology of simplicial complexes and spaces. The simplest way (though not the most geometric) is to dualize the corresponding chain complexes. Let 0'(,1) = 0(,1, e) be the augmented oriented chain complex of the simplicial complex ,1, over the ring A. The qth reduced ln1u.Jular cohorrwlogy group of ,1 with coefficients A is defined to be iiq(.d; A)

iiq(Hom,,(O'(.d), A)),

where Hom,,(O'(.d), A) is the cochain complex obtained by applying the functor Hom" ( ,A) to 0'(,1). Exactly analogously define iiq(X; A) and Hq(X, Y; A). Sometimes one identifies the free modules Oq(.d) and Oq(.d) Hom,,(Oq(.d), A) by identifying the basis of oriented q-chains (J of Oq(.d) with its dual basis in Oq(.d). Similarly one can identify Oq(X) with oq(X). There is a close connection between homology and cohomology of ,1 or X arising from the "universal-coefficient theorem for cohomology". We merely mention the (easy) special case that when A is a field k, there are "canonical" isomorphisms

i1q(.d; k) '" Homk(iiq(.d; k), k), i1 q(X; k)~ Homk(iiq(X;

k), k).

Thus in particular when iiq(.d; k) is finite-dimensional (e.g., when ,1 is finite), we have and similarly for X, but these isomorphisms are not canonical. We recall that a topological n-manifold (without boundary) is a Hausdorff space in which each point has an open neighborhood homeomorphic to R". An n-manifold with boundary is a Hausdorff space X in which each point has an open neighborhood {(Xl' ... , X,,) E R" I Xi O}. The boundary which is homeomorphic with R" or R,: oX of X consists of those points with no open neighborhood homeomorphic to R". It follows easily that oX is either void or an (n - I)-manifold.

i1q(.d; k)~ i1q(.d; k)

§ 1. Founda.tions

111

Suppose X is a compact connected n-manifold with boundary. Then one can show H,.(X, A) is either void or isomorphic to A. A compact connected n-manifold X with boundary is orientable (over A, if we have H,.(X, A) = A. (The usual definition of orientable is more technical but equivalent to the one given here; see also Definition 3.13 below.) For example, every compact connect n-manifold with boundary is orientable over a field of characteristic two. H a compact connected n-manifold X is orientable over A, then we have Hq(X; A) '::::: H,,-q(X; A). This is the so-called Poincare D'lMtluy Theorem. An n-dimensional p8eudomani/old withm.a boundary (resp., with boundary) is a simplicial complex A such that:

ax;

ax;

(a) Every simplex of A is the face of an n-simplex of A. (b) Every (n - I)-simplex of A is the face of exactly !wo (resp., at most two) n-simplices of A. (c) H F and F' are n-simplices of A, there is a finite sequence F F l , ... , F m = F' of n-simplices of A such that Fi and Fi+l have an (n I)-face in common for 1< i< m. The boundary 0,1 of a pseudomanifold A consists of those faces F contained in some (n - I)-simplex of A which is the face of exactly one n-simplex of A. Let A be a finite n-dimensional pseudomanifold with boundary. Then H,.(A, 0,1; A) '::::: A or O. In the former case we say that A is orientable over A; otherwise nonorientable. Let I be the unit interval [0, 1]. The 8WJpeWlWn EX of a topological space X is defined to be the quotient space of X X I in which X X 0 is identified to one point and X X 1 is identified to another point. The n-/oid 8USpeWlWn E" X is defined recursively by E"X = E(E"-lX). For any X and q we have

flq(X; A) '::::: Hq+l(EX; A). The purpose of this chapter also is to introduce a new kind of partially ordered set: Buchsbaum poset. The notion of a Cohen-Macaulay poset originated in Baclawski's thesis, see Baclawski [3]. It is now known that this concept provides some interesting connections among algebraic topology, combinatorics, commutative algebra and homological algebra. LetP be a /~nue poset; that is, a partially ordered set. We need some auxiliary concepts. A chmn of P is a totally ordered subset of P. We will usually write Xl < ... < Xn for a typical chain of P. The rank of a chain is the number of elements in it; thus r(xl < ... < xn) = n. More generally, the rank of P, written r(P), is the rank of the longest chain of P. The length of P, written l(P), is given by l(P) = r(P) - 1. The length is a more topological notion whereas the rank seems to be more combinatorial. Apparently topologists start counting at zero while combinatorialists prefer to begin at 1. We will do both. A poset is said to be ranked if every maximal chain has rank r(P). Given a poset P, we will write P for the poset obtained by adjoining a new pair of elements to P, written Ii, t such that Ii < if < t for all x E P. H we only require that Ii or i be adjoined, we will write Po or pI respectively. We use the convention that (j or i is never an element of P. The context should indicate to which poset (j or i is to be adjoined.

112

II. Hochster-Reisner theory for monomial ideals

,

A subset J ~ P will be called an order-ideal if for every x E J, Y x implies y E J. The dual definition gives the concept of an order-filter. The order-ideal generated by a subset 8 P will be noted J(8) or J p (8); while V(8) V p (8) denotes the order-filter generated by 8. The special case J(x) for x E P can also be denoted (6, x]. If P is ranked, then so is every subset J(x), and we write r(x) for r(J(x)). The function r takes values in the set Lr(P)] which by definition denotes {I, 2, ... , r(P)}. The length of an open intervall will be denoted l(x, y) instead of l( (x, y)). We will often use the Mobius function. For a poset P we write ",(P) for ",(0, i) as computed in P. For x E P we will write ",(x) or ",p(x) for ",(J(x)). Finally, for x yin P we will think of ",(x, y) as an abbreviation for "'( (x, y)). . For a finite set 8, let B(8) denote the poset of nonempty subsets of 8. A finite 8implicial complex is an order-ideal of B(8). The minimal elements are called vertices and elements in general are called simplice8. Much of what we do in the sequel may be extended routinely to simplicial complexes. As we have defined it, a simplicial complex is a special kind of poset. However, given a finite poset P, we can define the order complex of P, denoted LI(P), to be the subset of B(P) consisting of the nonempty chains of P. By this device one may view posets as a special kind of simplicial complex. We now review the correspondence between monomial ideals of the polynomial ring 8 = K[xo, ... , x n] and finite simplicial complexes. Let LIn denote the standard n-simplex; that is, the complete simplicial complex on (n+ I)-vertices which we label as xo, ... , x n • Recall that this means that LIn is the set of all subsets of {xo, Xl> "" x n}, Let K be a field and I an ideal of 8. Let V(I) be the subset of Kn+l where the elements of I vanish. If I = (Xi., "', Xi), we refer to V(I) as a coordinate hyperplane. Then we get 1-1 correspondences between: 8 1 = {subcomplexes of LIn}, 82 {ideals of 8 generated by square-free monomials}, 8 a {unions of coordinate hyperplanes in Kn+l}. We describe some of these 1-1 correspondences in detail, The correspondence 8 1 -+ 8 2 is defined by 1:~IJ.:

where 1: denotes a subcomplex of LIm and IE denotes the ideal of 8 generated by the monomials Xi• ••• Xi., io < ... < if) such that the simplex (Xi., , •• , Xi) is not in 1:. The correspondence 8 1 *- 8 2 is defined by 1:[ ~ I where I denotes a square-free monomial ideal of 8, and 1:I denotes the subcomplex of LIn consisting of all simplices (Xi., ... , Xi,) such that the monomial Xi, .' • Xi, is not in 1. For the correspondence

8 2 -+ 8 a ,

I

~

V(I)

we simply associate to I its vanishing locus in Kn+l. The correspondence 8 3 -+ 8 1 is given by H~1:H'

Here H denotes a union of coordinate hyperplanes in K"+l, and 1:ll denotes the subcomplex of LIn consisting of all simplices (Xi" ... , Xi) such that the element of K,,+l whose Xi., •• ,' Xi, coordinates are 1 and whose other coordinates are 0 is in H.

§ 1. Foundations

113

If the field K is the real field R, we may view this last correspondence in terms of ,geometric realizations. First we recall:

Definition 1.3. Let the vertices xo, ... , x" of An be identified with the canonical basis of R"+1. Then if 1: is any subcomplex of A", its geometric realization i1:1 is defined as U cx(O') C R"+1; 11:1 R,,+1 is a topological space (see McMullen-Shephard [1]). Then aEE

R the above correspondence may be viewed as 8 a --"" 8 1 given by

if K

H ___ 1,1,,1 n H = I1:H I.

We now relate these correspondences to primary decomposition. Lemma 1.4. (i)

If P c::: 8 is a 8quare-free monomial ideal of the form (Xi" •.. , Xi,), then the a880ciated Bimplwial complex 1:p c::: A" is a simplex of codimension h := n - dim 1:p ~'n An.

(ii) If I c::: 8 ~'n any 8quare-free monomial ideal, let PI n ... n Pm be an irredundant primary decompoBition of 1. Then each P j i8 of the type described in (i), and we have III

1:1

U 1:p a8 the decompOBition of 1:1 into its maximal8implWie8 . • =1

'

(iii) If I c::: 8 is any 8quare-free monomial ideal, ht I Proof: (i) is obvious from the correspondence 8 2

--""

codim(1:J, An). 8 1 above.

(ii) That each Pi is of type (i) follows from the splitting lemma. The fact that the III

intersection of PI n ... n Pm corresponds to the union ,U 1:p, follows from the cor.~l

respondences 8 2 --"" 8 a --"" 8 1, The maximality of the simplices 1:p , in 1:1 follows from the irredundancy of the primary decomposition. (iii) is a direct consequence of (i) and (ii).

Definition 1.6. If 1: is a finite simplicial complex, we define the codimension of 1: (written codim 1:) to be codim(1:, ,1"...1) where v is the number of vertices in 1:. Lemma 1.6. Let I 8 be any 8quare-free monomial ideal, and let 1:1 8impZwial complex. Then

A" be the a880ciated

ht I.

codim 1:1

Proof: This follows immediately from Lemma 1.4(iii).

The following definitions are fundamental in the sequeL Definition 1.7. If (J' is a maximal simplex in a simplicial complex we call it a facet of 1:. Definition 1.8. If a E 1: is any simplex, we define the 8tar of a to be the subcomplex Starl' a

{s

E 1: I 0' usE 1:}.

Definition 1.9. If a E 1: is any simplex, we define the h'nk of rJ to be the subcomplex Ikl'rJ = 8 Buchsbaum Rings

{8

E 1: I 0' n s

0 and rJ

U

s is in 1:) .

114,

II. Hochster-Reisner theory for monomial ideals

Lemma 1.10. Let E be a simplicial complex. Then: <X) Any Unk IkE 0' is an iterated link 01 vertices (the vertices in 0'). p) II v is a vertex 01 E, StarEv is the simplicial cone with base Ik.z;v and vertex v. 1') Any StarE 0' is an iteratUm 01 cones over IkEa (by the vertices in 0'). Lemma 1.11. Let E be a simplicial complex which is equi-dimensional 01 dimension d. rl'hen: <X) 110' is any simplex 01 E, StarEa is an equi-iUmensional complex 01 dimension d. fJ) 110' is a k-simplex 01 E, IkEa is an equi-dimensional complex 01 dimension d - k - 1. Definition 1.12. H E is a simplicial complex lct Conep(E) denote the simplicial complex given by the simplices {a

I 0' is a simplex of E}

u {(a, P) I 0' is a simplex of E} u {P}.

Note that Conep{E) contains E as a subcomplex. Conep{E) is called "the simplicial cone over E" or "the simplicial cone with base E and vertex P". The following lemmas are of basic importance. Lemma 1.13. E is equi-dimensional more than E).

~

Conep{E) is equi-dimensional (01 dimensUm one

Prool: Observe that ( ,P) induces a 1-1 correspondence {facets of E) ~+ {facets of Conep{E)} • 1-1 Since ( ,P) raises the dimension of any simplex by exactly one we conclude that one set of facets is equi-dimensional if and only if this is true for the other set. Lemma 1.14. Let T be a simplex 01 0 Conep(E). Then there are three possibilities lor IkcT: <X) II T = P, then IkcT = E. fJ) II T Ol P (i.e. T (a, P) lor some simplex 0' E E), then IkcT = IkL'a. (') II T ~ P (i.e. TEE), then IkcT Conep{lkET). Prool: <X) This is obvious from the definition of Conep(E). fJ) Ikc(a, P) = Iklkco P (recall Lemma 1.10)

= IkconepOkEo) P (by 1')) = IkEa (by <x)). (') Step 1: 0' E E and 0' n T 0 ~ (a, P) E 0 and (0', P) n T Step 2: 0' uTE E ~ (0', P) uTE O. We now present two examples of simplicial complexes. Example 1.10. X

E= 1

=

0.

§ 2. The homological Cohen.Macaulay criterion of Reisner

For each Xi =l= X o, IkrXi

= _____ and StarrX. =

115 '

A

XIX.

I I and StarrXo x.x. is the simplicial cone with base I I and vertex XO'

However, for Xi

=

X o, Ik,!;'Xo

=

I.

X,X,

I

x,x,

Example 1.16.

IkX j For Xi

= Xo or Xl>

-----

IkXi=-L

For edges e =l= XOXb For e

XoXl>

I is the simplicial cone over

lk,!;'e = • • •

..l-., q.e.d.

Stars and links are important in the study of simplicial manifolds, as we shall see in the following sections.

§ 2.

The homological Cohen-Macaulay criterion of Reisner

Let I c: A" be any finite simplicial complex. We denote by K[IJ the quotient ring S/lz. We will see that the graded K-algebra K[I] is closely related to the combinatorical and topological properties of I. Therefore we have interactions between commutative algebra and combinatorics. By combining these techniques we open the way to a deeper study of K[I] and I. First, we begin by presenting Reisner's Cohen-Macaulay criterion. ii ;(I; K) will always denote the ith reduced homology group of I with coefficients in K (see, for example, E. H. Spanier [1J, p. 168). In 1976, G. A. Reisner [1] proved the following statement: Theorem 2.1 (the Reisner Cohen-Macaulay criterion). Let I c: A" be any linire simplidal complex. Then the 10l101.MTtg conditions are equivalent: (i) K[I] is Oohen-Macaulay. (ii) K[lkEx;J is Cohen-Macaulay lor every verrex Xi E I; and iii(I; K) = 0, i < dim I. (iii) Hi(IkEO'; K) 0, i < dim IkE 0', lor all simplices 0' E I; and Hi(I, K) 0, i < dim I. Prool: Reisner [1], Theorem 1 (see also Weibel [1)). 8*

116

II. Hochster-Reisner theory for monomial ideals

If the geometric realization 11:1 of 1: (see again E. H. Spanier [1]) is a manifold (with or without boundary) the condition on the links holds automatically. This follows by using excision (see E. H. Spanier [1]). Hence we have; . Corollary 2.2. Ij 1: All is a simplicial complex suck tkat 11:1 is a topological manijoldwitk-a, tken tke jollow£ng conditions are equivalent: (i) K[1:] Us Coken-Macaulay. (ii) 11,(1:; K) = 0, i < dim 1:.

Furthermore, by proving Theorem 2.1 Reisner has also investigated the property for 1: to have Cohen-Macaulay links of its vertices. From the point of view of commutative algebra the key is the following simple lemma. Lemma 2.3. Let R be a Nnitely generated graded K-algebra witk Ro = K. Suppose R = K[xo, ... , x.] wkere xo, ... , x" are jorms oj degree 1. Tken tke jollow~ng conditions are equivalent: (i) Rp Us Coken-Macaulay jor all prime ideals P except, perkaps, tke irrelevant ma~mal ideal (xo, ... , x,,). (ii) R/(xj - 1) Us Coken-Macaulay jor i = 0, ... , n. Prooj: Assume (ii), and let P =F (xo, ... , x.) be a prime ideal. Choose Xj ~ P. It suffices to show that R", (i.e. R localized at the powers of Xi) is Cohen-Macaulay. But this follows from the well-known isomorphisms (see Grothendieck [2], Chapter II, 2.2.5) R",

(R",)o [(l/Xi),

xd,

(R,,)o~ R/(Xi -

1)

and the fact that a ring S is Cohen-Macaulay if and only if S[t, lit] is Cohen-Macaulay with t an indeterminate. . Assume (i). Then the above isomorphisms and since Xi - 1 is not a zero divisor in R If, imply (ii). "-

We will be interested in Lemma 2.3 when R K[1:] for some 1: c An. The preceding will be used in proving the following Buchsbaum criterion. Theorem 2.4 (the Reisner locally 'Cohen-Macaulay criterion). Let 1: c A" be any jinite ~mplicial complex. Then tke jollowing conditions are equivalent: (i) (K[1:J)p Us a Coken-Macaulay r£ng jor all pnme ideals P dijjerent jrom tke irrelevant ideal (xo, ... , XII) • K[1:]. (ii) For all 8£mplices (J E 1: we kave

fi .(Ikz;

(J;

K) = 0

ij

i

=F dim lkz; (J •

. (iii) K[1:] Us a (local) Buchsbaum nng. Prooj: The equivalence of (i) and (ii) is proved by G. A. Reisner [1]. The implication (iii) =} (i) is trivially true because a Buchsbaum ring is locally a Cohen-M!,-caulay ring. The converse (i) =} (iii) results immediately from Proposition 1.3.10 and the following Lemma 2.5(ii).

§ 2. The homological Cohen-Macaulay criterion of Reisner

Lemma 2.5. Let 1: c L1n be any jznite simplicial complex. Set R jar the local cohomology module !1~(R) the jollowing pr'Operties: (i) [!1~(R))n = 0 jor all n > 0 and i E Z. (ii) AS8Uming that !1~(R) is oj finite length then

117

K[1:). Then u'e have

!1~(R) = [!1~(R))o.

Proof: In order to prove these facts on local cohomology we make use of the techniques of Hochster and Roberts about the purity of the Frobenius homomorphism (in characteristic p =l= 0) and the fact that R has a presentation of relative graded F-pure type (in characteristic zero). The key is then the following assertion. (We use terminology from Hochster-Roberts [21.) .

Claim. Let K denote a perfect field of prime characteri:5tic p > O. Then K[1:] is F-pure. For an arbitrary field K, K[1:] haB a presentation oj relative graded F-pure type. Proof oj claim: First let K be a perfect field of prime characteristic p > O. We show that the Frobenius map F: R ~ R is pure. To this end we prove that F(R) is a direct summand of R as F(R)-module. This is equivalent to showing that there is an R-retraction, sayr: R ~ F(R). Here the point is that we can define r for monomials by r(x~'

x~' ... x!"

... x!")

o

{

for k; 0 mod p, O:S:; i otherwise

< n,

and extend it K -linearly to R. Since K is a perfect field we get r(R) F(R). To complete the proof we note that r is the identity on F(R) and r is an F(R)-module homomorphism. For an arbitrary field we get from Z[xo, ... , xn]/h a presentation of relative graded F-pure type, see M. Hochster and J. L. Roberts [2]. Having proven this claim we can now apply conclusions on local cohomology from Hochster-Roberts [2]. Therefore our Lemma 2.5 results immediately as a consequence from this important paper, q.e.d. Analyzing the proof of J.emma 2.5 it becomes desirable to look for more precise information. Before stating results in this direction, we make some general observations and collect some known results from Hochster-Roberts [2J which will be needed. In the following we call a homomorphism of rings R ~ S pure, if

M

~

M

S

via

m

t4

m

1

is injective for every R-module M. A ring R of prime characteristic lJ F-pure, if the Frobenius map F:R~R,

> 0 is called

rt4r P,rER,

is pure. Fe denotes F iterated e-times. We note that R is F-pure if and only if Fe: R ~ R is pure for all e ~ 1. If R is F-pure, it follows that it is reduced. A pure homomorphism of rings R ~ S induces a homomorphism of complexes j':X'

~X'

@RS

for a complex of R-modules X'. In fact, it follows by M. Hochster and J. L. Roberts [1], Proposition 6.5, that the induced homomorphisms Hi(/'): Hi(X')

~

Hi(X' @R. S)

118

II. Hochster-Reisner theory for monomial ideals

are injective for all i E Z. Assume that R -+ S is a pure homomorphism of graded rings such that R -+ S multiplies degrees by d. Let K"(~t; R) be the Koszul complex of R with respect t{) ~t (xL ..• , x~), t> 1, where (Xl>"" x,,) is a system of forms with Rad(Xl' .'O, x,,)R = m, the irrelevant maximal ideal of R. Then the maps

are injective for all i E Z and for all k E Z. Here, ~/ = (x~, ••. , 'x~) denotes the image of ~ = (Xl' ••.• xn) by the pure homomorphism,R -+ S. By taking the direct limit of the cohomology of the Koszul complexes it follows that the maps [H:n(R)Jk -+ [H:n'S(S)]kd

are injective for all k E Z and for all i E Z, where m'S denotes the image of m in S. In particular let R be a F-pure graded ring. Then the maps

are injective for all k E Z, i E Z, and e > 1, since Fe(m) has radical m. This implies the following lemma due to M. Hochster and J. L. Roberts [2]. Lemma 2.6. If R is a graded F-pure ring, then we have [H:n(R)]" = 0 for all n > 0 and i E Z. Suppose that the local cohanwlogy module H:n(R) is of finite length. Then H:n(R) = Lll:n(R)]o also follows. >

= 0 for n ~ 0, the first part of the lemma is obtained by previous considerations. If the local co~ homology module is of finite length, then we have in particular [H:n(R)]" 0 for n ~ O. Thus, the second statement follows the same way, q.e.d.

Proof: Using the fact that H:n(R) are artinian R-modules, i.e. [H:n(R)]"

In characteristic zero M. Hochster and J. L. Roberts [2], Lemma 4.7, proved a corresponding result for rings R which have presentations of certain F-pure types. For the definition and related technical results in particular the definition of "R has a presentation of relative graded F-pure type" - we refer to the fundamental paper of M. Hochster and J. L. Roberts [2]. In fact, Lemma 2.6 and the corresponding result in characteristic zero lead to a number of Buchsbaum rings.

Theorem 2.7. Let R be an equi-dimensiorwl graded k-algebra suck that Rtl is a OohenMacaulay rz'ng for all prime ideals d~fferent fr()'fl1, tke %rrele'l.!ant ideal m. Assume tkat R is F-pure re8p. has a pre8entation of relative graded F-pure type. Then it follow8 tkat R is a Buchsbaum rz'ng.

< dim R, are modules of finite length. the proof will follow. From this property we get by Lemma 2.6:

Proof: If we show that all the local cohomology modules H:n(R), 0 S i [H:n(R)] ..

=

0

for all n

0 and 0

i

< dim R.

Then Proposition 1.3.10 does imply that R is a Buchsbaum ring. Now, it is clear that the finite length of local cohomology modules H:n(R), 0 S i < dim R, is equivalent to

§ 2. The homological Cohen-Macaulay criterion of Reisner

(see also Corollary 0.4.15) , (i) R is equi-dimensional and (ii) R:p is a Cohen-Macaulay ring for all prime ideals Hence Theorem 2.7 is proved, q.e.d.

~

119

'*' m.

Next we investigate a geometrical interpretation of Theorem 2.7. To this end we state an interpretation of Lemma 2.6. Let X c:: Pk P be a projective scheme such that for a coherent sheaf (F and an integer t: (a) The canonical map M" -+ HO(X, (F(n))

'*'

is bijective for all n t, where M denotes the graded module associated to (F, and (b) Hi(X, (F(n)) = 0 for all n t and 0 < i < dim (F. Then J" is arithmetically Buchsbaum, i.e., M is a graded Buchsbaum module. Using this notion, our Theorem 2.7 says: Let R be an F-pure graded k-alfJebra resp. R has a presentation 0/ relative graded F-pure type. 1/ (X, Ox) = Proj(R) i8 a pure dt'mensional (locally) Oohen-Macaulay 8(Jheme, then X is arithmetically Buchsbaum.

'*'

Now we will examine some Buchsbaum rings which arise from the purity of the Frobenius. Example 2.8. Let k be a field of characteristic p > 0 reap. of characteristic zero, and let R be a graded k-algebra which is F-pure reap. has a presentation of relative graded F-pure type. Suppose R is a domain such that R:p is regular for every prime ideal ~ different from the irrelevant ideal m. Let S c:: R be a graded k-subalgebra which is pure in R. Then S is a Buchsbaum ring. M. Hochster and J. L. Roberts [2], § 5, showed that S is F-pure resp. has a presentation of relative graded F-pure type. Also, they showed that S:p is a Cohen-Macaulay ring for all prime ideals ~ different from the irrelevant ideal which follows from the main result of Hochster-Roberts [1], i.e., a pure subring of a regular ring with characteristic p > 0 is a Cohen-Macaulay ring. From this it follows that the ring of invariants of linearly reductive affine linear algebraic groups acting on regular rings are CohenMacaulay rings, compare Hochster-Roberts [1]. Therefore, our Theorem 2.7 shows that rings of invariants of those groups acting on certain singular rings are Buchsbaum rings. Example 2.9. Let k, R be as in the previous example. Let G be a linearly reductive affine linear algebraic group over k acting on R by preserving degrees. Then the ring of invariants S = RG is a Buchsbaum ring. We note that, S = RG is in general not a Cohen-Macaulay ring, if R is singular. For this we consider an example of M. Hochster-J. L. Roberts P]. Let

R

k[Xl' ... , x,,]/(xi

+ ... + x:).

Then R is F-pure for a perfect field k with characteristic p 1 mod n. Therefore we have a presentation of relative graded F-pure type, if k is a field of characteristic zero. Let S k[Yl' Y2]' Let G = Gl(l, k) k" to} acting on R resp. S by multiplication of

120

II. Hochster-Reisner theory for monomial ideals

a form of degree m by gttl resp. g-ttl for 9 E G. The tensor product

R @k S = k[Xl' •.• , XII' y), Y2]/(x~

+ ... + x:)

is F-pure resp.has a presentation of relative graded F-pure type. Furthermore, all the assumptions of Example 2.9 above are fulfilled. Thus (R @k S)G is a Buchsbaum ring. In fact, (R @k S)G is the Segre product of Rand S. Because R is improper in the sense of W. L. Chow [1], i.e., (see Chapter IV, Corollary 4.4 and 4.5) [M~(R)]o =F 0, d = dim R, we obtain (R @k S)G is not a Cohen-Macaulay ring, compare Chapter I, § 4. More generally, we can show that certain Segre products are Buchsbaum rings. For this we extend the investigate of Chapter I, § 4. I~t R;, j = 1, 2, be two graded k-algebras over a field k with dim RI 1. Let S denote their Segre product. Let (Xj,Ox1 ) Proj(Ri ), j = 1,2, and (W, Ow) = Proj(S). Then we have W

Xl Xk X 2

and

Ow(n) = p!Ox,(n) @kP:OX,(n),

where Pi: W -+ Xi> i = 1, 2, denote the canonical projections. Proposition 2.10. Let rObe an integer. Assume that the following conditions are f1dlfiled = 1,2:

for i

a) The canonical map

[RjJn -+HO(X j , Oxj(n)) ,

n =F r,

is bijectz·ve.

b) Hi(Xj,OX/n)) 0 for all n =F rand 0 < i < dim Xi' c) Hdl(X;, Ox/n)) = 0 for all n 0, n =F r and d j = dim Xi' Then W is arithmetically Buchsbaum. Additionally W 8atisfies conditions a), b), c). Proof: By virtue of the above remark, it is enough to show that conditions a), b), c) are

satisfied on the cohomology of W. Using the Kunneth formula H'(W,Ow(n)):::::

EB

Ha(x), Ox,(n)) @kHb(X2, Ox,(n)) ,

O-r b =8

compare Proposition 0.2.10, this follows immediately, q.e.d. Thus by analyzing the proof of Lemma 2.5 we have obtained our results 2.6-2.10 on Frobenius purity and the arithmetical Buchsbaum property. These assertions are also contained in the paper by P. Schenzel [1], 4.4. Now we return to Reisner's locally Cohen-Macaulay criterion of Theorem 2.4. By virtue of this statement it is possible to construct many Buchsbaum rings arising from simplicial complexes. Example 2.11 (Reisner's example from [1]). The fact that the Cohen-Macaulay property of K[E] depends upon K follows now immediately from Theorem 2.4. Take, for example, a triangulated manifold M, whose only nonzero homology is pure p-torsion (for some prime pl. Then K[x), ... , x~]/{u is Cohen-Macaulay if K has characteristic other than P and is not Cohen-Macaulay if char K = p. Examples of such manifolds are Lens spaces (see, e.g. Hilton-Wylie [1], p.223). For a simpler example, one can take M to be the projective plane. In particular, if we consider the minimal triangulation of the projective

§ 2., The homological Cohen.Macaulay criterion of Reisner

121

plane (Fig. 1) then In this case K[.E] is not Cohen-Macaulay if char K = 2 and it is Cohen-Macaulay if char K =1= 2. It will also follow from Corollary 2.12 below that K[.E] is not CohenMacaulay but Buchsbaum for char K 2. We also could have taken LI. to be a finite triangulation of the ~al projective n-space P~. We then get for the reduced simplicial

Fig. 1

homology of LI" with coefficients in an abelian group G (see, e.g., Hilton-Wylie [1], 3.9.4) : H.(LI . G ~ { G/(2) G if i is odd, , ", )T 2(G) . 1'f' ~lseven.

where T2(G) we get for 0

Ig E G such that 2g




Definition 3.3. ,A simplicial complex 1: will be called a comhinatarial manifold-with-8, or simply a manifoU, if: a) 1: is equi-dimensional, and b) Sing 1: = 0. Definition 3.4. A comJnnatorial sphere (resp. comlnnatorial disc) will mean a combinatorial manifold-with-81: such that 11:1 is homeomorphic to a sphere (resp. a disc). We will write c-sphere, c-disc, for short notation. Lemma 3.5. Let 1: be an equi-dimensional simplicial complex. Then the ,/ollowzng conditions are equivalent: (i) 1: is a manifoU. (ii) For all 81,'mplices (1 E 1:, lkE(1 is a sphere or disc. (iii) Far all simplices (1 E 1:, IkE(1 is a c-sphere or c-disc. (iv) Far all 81,"mplice8 (1 E 1:, StarE(1 is a c-disc. Proof: (il ~ (ii) by definition of manifolds. (ii) ~ (iii): The implication (~) is trivial. The implication (=?) results from the fact that the link of any simplex in 1: is an iteration of links of vertices (see Lemma 1.10). (iii) ~ (iv): The implication (~) is easy to prove, since IkL'(1 is just IkstarEu(1, and StarL'(1 is a combinatorial manifold by hypothesis. To prove the implication (=?) recall that StarE (1 is just an iteration of simplicial cones over IkE(1. Thus it suffices to show that if 1: is a c-sphere or c-disc, then Conep(1:) is a c-disc. But this follows from the link formulas of Lemma 1.14, q.e.d.

Remark 3.6. According to our definitions, an equi-dill1ensional simplicial complex is a combinatorial manifold if and only if Sing 1: 0, i.e. the link of every simplex is homeomorphic to a sphere or disc. This is weaker than the usual definition, which would have every link PL-homeomorphic to the standard simplicial sphere or disc (it is enough to require this for links of vertices). Our definition is a hybrid of the topological and PL approaches which makes it easier to detect the singular simplices. . Proposition 3.7. Let 1: be a simplicial complex. Then if 1: is a manifold, 11:1 is a topological manifoU-u;ith-8. Proof: Let P be any point of 11:1. We have to show that P has an open neighborhood U in 11:1 homeomorphic to an open subset of the closed half-space Hll R" where d is the dimension of 1:. Now, P lies in the interior of some simplex (1 of 1:. Note that we may perform a modified barycentric subdivision of 1: (with P as the barycenter of !(1!) to

=

§ 3. The topological Cohen.Macaulay criterion of Schwartau

125

make P into a vertex in a new triangulation E' of lEI. Under such a barycentric subdivision we have IStarE 0'1· ~ IStarE.PI. Since E is by hypothesis a manifold, it follows from Lemma 3.5(iv) that IStarE,PI is homeomorphic to a disc (of dimension d). Thus the required neighborhood U above is provided by IStarE.PI, q.e.d. Now we want to analyze the singularities of cones and of stars. Proposition 3.8. Let E be an equi-dimensional simplicial complex. Then we have: (i) (s~'ngularities of cones). There are three p(xmoilities for the simplicial cone Conep(E): 1) a) E is a c-8phere or c-dz'sc and Conep(E) is a c-disc. b) E is a manifold other than a c-sphere or c-disc and Sing Conep(E) {Pl. 2) Sing E 9= 0 and Sing Cnep(E) = Conep(Sing E) 9= 0. (ii) (singularities of stars). Let 0' be any szmplex of E. Then for any vertex v E 0', 0' E Sing E zJ and only if 0' E Sing StarEv. Proof: (i) This proof depends on the link formulas tx, {3, y of Lemma 1.14. (ii) Note that v EO'=? 0' E StarEv, thus the statement is meaningful. The proof follows by showing that ( *)

IkE 0' = IkStarL" 0' •

To show this, note that in any complex, the link of a simplex 0' is obtained by finding all simplices of the complex containing 0' and "pulling off" 0'. But since v E 0', {simplices of E containing O'} . {simplices of StarEv containing O'} and (*) is proven. We may now apply (*) to show that Sing E, Sing StarE v have the same facets; hence we have an equality of complexes Sing E Sing StarEv, q.e.d. Next we need to make some observations and collect known results on quasimanifolds. Definition 3.9. A simplicial complex E is a quasi-manifold if (i) E is equi-dimensional, and (ii) Sing E contains no codimension 1 simplex of E. Remark 3.10. This definition means that the link of every codimension 1 simplex is a O-sphere or O-disc; i.e. every codimension 1 simplex is contained in at most two facets. This constitutes only part of the usual definition of "pseudo-manifold" found in the literature. Definition 3.11. A simplicial complex E is called a regular non-quaai-mamYold (RNQM) if (i) E is equi-dimensional, and (ii) Sing E 9= 0 and is a union of codimension 1 simplices of E. Examples 3.12. (i) Any manifold is a quasi-manifold. (ii) The complex E of Example 1.15 is not a manifold, but is a quasi-manifold. (iii) The complex E of Example 1.16 is not a quasi-manifold, but is It regular non-qultsimanifold. (iv) Let E be an equi-dimensional simplicial complex. If lEI is a topological manifoldwith-o, then E is a quasi-manifold.

126

II. Hochster-Reisner theory for monomial ideall!

Proof: Suppose that E is not a quasi-manifold. But since E is equi-dimensional, 80 there must exist a simplex u E Sing E of codimension 1 in E. Let P be any point of lui. Then any meighborhood of P in iStarl:ul is a neighborhood of P in lEI. Therefore if we show that IStarl:ul is not a topological manifold at P then lEI is not a topological manifold at P, and the proof is finished. But u is contained in 3 facets of E, therefore IStarl:ul is a "fan". Separations theorems from topology guarantee that a fan cannot be a topological manifold. This does prove (iv), q.e.d.

We still need more facts on orientable simplicial complexes. Definition 3.13. Let u be a simplex in a simplicial complex E. Then u becomes an oriented simplex if we choose an arbitrary fixed ordering of the vertices. The equivalence class of even permutations of this fixed ordering is called the positively oriented szmplex +u; the equivalence class of odd permutations is the negatively oriented simplex -u. Notice that an orientation of u induces through the simplicial boundary operator an orientation on each boundary-face of u. Definition 3.14. Let E be a quasi-manifold. Then we say E is coherently oriented if the facets of E are oriented in such a way that any two facets meeting in a codimension 1 simplex induce opposite boundary orientations on that simplex. Definition 3.15. Let E be a quasi-manifold. Then we say that E is orientable if it is possible to orient E coherently; otherwise we say E is non--orientable. Remark 3.16. Note that if E is an orientable quasi-manifold, and if E' manifold of the same dimension, then E' must be orientable.

E is a quasi-

We now extend the definition of orientability to arbitrary simplicial complexes. Definition 3.17. Let E be a simplicial complex. Then we say that E is orientable if: a) E is equi-dimensional of dimension d. b) Every d-dimensional quasi-manifold E' c E is orientable. Examples 3.18. (i) Any triangulation of a cylinder is an orientable quasi-manifold of dimension 2. (ii) Any triangulation of a Mobius band is a non-orientable quasi-manifold of dimension 2. (iii) The quasi-manifold of Example 3.12(ii) is orientable. (iv) The RNQM of Example 3.12(iii) is orientable. Lemma 3.19. Let E' c E be equi-dimen8ionalsz'mplicial complexes of the same dimen81,·on. Then: E is orientable ~ E' is orientable. Proof: Immediate from the definition of orientability.

Corollary 3.20. Let E be a simplicial complex, anit v a vertex of E. Then: E is orientable ~ Starl:v is orientable.

Lemma 3.21. Let E be a szmplicial complex. Then: E is orientable

{?

Conep(E) is orientable.

§ 3. The topological Cohen-Macaulay criterion of Schwartau

127

Proof: As 'in the proof of Lemma 1.13, we need the 1-1 correspondence:

{facets of E) + (;~) • {facets of Conep(E)) . This correspondence preserves equi-dimensionality, and preserves coherent in a natural way; thus the Lemma 3.21 is proven. Corollary 3.22. Let l' be a

8~mplic£al

orie~tations

complex, and v a vertex of E. Then:

l' 'is orientable :::} IkI v 'is or£entable. Proof: IkIv is the base of the simplicial cone StarIv. We now apply Corollary 3.20 and

Lemma 3.21. Corollary 3.23. Let l' be a s£mplicial complex and then IkI

(1

(1

any ~mplex of E. If l' 'is orientable

is orientable.

Proof: As in Lemma 1.10, any link IkI (1 is just an iterated link of vertices. Now apply

Corollary 3.22, q.e.d. It is not too difficult to show that equi-dimensional simplicial complexes of codimension 0 or 1 are all c-spheres or c-discs. Therefore it was Schwartau's idea to investigate simplicial complexes of codimension 2. All his results depend on the following decomposition theorem:

Theorem 3.24 (Schwartau [1], Theorem 165). Let d > 1, l' an equ£-d£mensional d-dimensional ~mplicial complex of codt'mens~on 2. Then there exist subcomplexes S, St, G = S n St of l' such that: (i) S =!= 0, St =!= 0, and l' S u St. (ii) S 'is a c-sphere or c-d'i8c of d~menstOn d. (iii) St 'is an equi-dimensional simplicial cone of d£mens~on d and of cod£mension < 2. (iv) G is conta1ned in the base of the cone St. (v) G is a nonvoid union of (d-1)- and (d

2)-~mplices.

By considering orientable simplicial complexes of codimension 2 Schwartau obtained the following result (see Schwartau [1], Theorem 167): Theorem 3.26. Let l' be an orientable quasi-manifold, either:

s~mplicial

complex of

cod~mension

2. Then if l' 'is a

a) E £8 a manifold, or b) Sing E has a facet of codtmension 2 in E.

The key in analyzing the singularities of codimension 2 complexes is the following interesting lemma proved by Schwartau [1], Lemma 166: Lemma 3.26. Let X, Y be two-dimensional simplicial ocmplexes, each of which 'is a manifold or an RNQ~!, joined along two disjoint line segments. Then if cod~m (X u Y) 2, X u Y must be a M 6bius band. We can now sharpen Theorem 3.25 and we get the topological Cohen-Macaulay criterion of Schwartau [1], Theorems 172, 173, 174 and 175.

128

II. Hoohster-Reisner theory for monomial ideals

Theorem 3.27 (Schwartau's topological Cohen-Macaulay criterion). Let .E be an equidimensional orientable 8implicial complex of codimension 2 and d~"mension d 2. Then the following conditions are equ~valent: (i) E is Oohen-Macaulay. (ii) Sing E is equi-dimensional of dimension d - 1 or i8 0. (iii) E i8 a manifold. (iv) E is a regular non-quasi-manifold. This is Schwartau's beautiful Cohen-Macaulay criterion. Example 3.28. The complex of Example 1.15 is not Cohen.Macaulay, but the complex of Example 1.16 is Cohen-Macaulay. This is now a direct consequence of Theorem 3.27. In the following observations we show how to put Schwarlau's topological CohenMacaulay criterion into practice. Let E be an equi-dimensional simplicial complex of codimension 2 and dimension d. Then by definition E is a subcomplex of L1 d+2, the standard (d + 2)-simplex. It follows that each facet of E contains all the vertices of L1 d+2 save two. Thus we may denote each facet of E by two parameters, namely by the vertices it does not contain. If we now draw vertices to indicate the vertices of L1 d+2 , each facet of E may be represented by an edge connecting two vertices. This is the socalled associated graph of E. Example 3.29. Consider the ideal I (xo, Xl) () (X2' xa) in S = K[xo, ... , xa]. Then I is a square-free monomial ideal of height 2. Therefore the Reisner complex (see § 1) has codimension 2. We have a decomposition of Er into its facets given by Er

E(31•. :&,)

U

E(31,,31.)'

By the definition of Reisner complexes, of S except Xo,

Xl'

is the simplex given by all the variables o 1 Therefore in the graph for E[ this simplex is represented as - ; E(:&••z,)

3

similary, the simplex

-o

1

3

2

E(z •• z,)

2

is represented as _ . The total graph for

E[

is thus:

Notice that in Pi the ideal J defines two skew lines; by coincidence the graph is a simplicial model of this fact (also note that the graph of E[ in this case happens to coincide with E[ itself). Remark 3.30. In case that a monomial ideal I of S is not a square· free monomial ideal, we must take the polarization of J before obtaining the Reisner complex (see Definition 1.1). Now suppose dim E 2 and T is a simplex of codimension 2 in E. Since IkET is equi-dimensional of dimension 1 and since codim E 2 it follows that codim IkET 2 (linking reduces vertices by at least as much as dimension). Thus IkET must be a i-dimensional subcomplex of .1 3 , the standard 3-simplex. It follows that ikE T is either a 1-sphere, or a 1-disc, or

t I, or contains a sUbcomplex of the form --1-.

§ 3. The topological Cohen.M.acaulay criterion of Schwartau

129

In the last case (T, v) is a singular simplex of E containing T. Thus if T is a facet of Sing E this cannot occur. Alternately lk£T could also not be a sphere or dies. Thus we must have Ik£T =

a

c

b

d

I !.

Conversely, if T has such a link, T must be a facet of Sing E. For T is clearly in Sing E, so we only need to check that any codimension 1 simplex of E containing T is non-singular. But the link in E of any such simplex is simply the link of a vertex in Ik.!,'T. This is always a O-disc and we are done. Thus we have shown:

T ,is a facet of Sing E of codimension 2 in 1: § lk£ T

=

I I. a

c

b

d

Notice that T has such a link if and only if (T, a, b), (T, e, d) are facets of E and (T, a, c), (T, a, d), (T, b, c), (T, b, d) are not. Thus T is an isolated codimension 2 singularity of E if and only if the edges ab, cd appear in the associated graph of E and the edges ae, ad, be, bd do not. Notice that the four missing edges are the only possible edges which could connect the disjoint edges ab, cd to each other. Conversely, suppose there exist disjoint edges ab, cd in the graph which are not connected to each other by any other edge of the graph. The facets of E represented by ab, cd must then intersect along a codimension 2 simplex T of E (given by il d+2 - {a, b, e, d} ), and lk T must be precisely

a

c

b

d

I !; hence l' is a facet of Sing E of codimension 2

in E. Therefore_we have shown that there is a 1-1 correspondence: pairs of disjoint edges in the } {Facets of Sing of codimension 2 in E} ~-+ graph of E not connecteq. to • { each other by any other edge Therefore Sing E has no facets of codimension 2 in E if and only if given any two edges of the associated graph, either they are connected to each other or else there exists a third edge connected to both of them. Let E be an equi-dimensional simplicial complex of codimension 2, and write G for the associated graph of E. Consider the dual graph G*: each edge of G becomes a vertex in G*, and two vertices of G* are connected with an edge if and only if the two corresponding edges of G meet at a vertex. Recall that the diameter of a graph is the largest number of edges needed to connect any two given vertices of the graph. In this context, we hav~ proven:

Theorem 3.31. Let E be an eqm'-dimensional simplicial complex of codimens~on a880eiated graph, and G* the dual graph of G. Then:

2, G the

assume) ( dim £';;:: 1

1. Sing E

conta~"ns

no facet8 of codimension 1 in E {::::

g

G eontainAJ no triangle8,

assume)

( dim£;;::2

2. Sing E contain8 no facet8 of codimension 2 in 1: {::::=g G* na8 diameter 9 Buchsbaum Rings

2.

130

II. Hochster-Reisner theory for monomial ideals

Corollary 3.32. Let I be an equi-dimensional orz'entable simplwi.al complex 0/ codimenstOn 2.; G the a8soci.ated graph, and G* the dual graph. Then: A) I is Cohen-Macaulay ~ G* has di.ameter 2. B) I Us a c-sphere or a c-disc ~ G* has di.ameter 2 and G contains no tri.angles. Proof: If dim I = 0 then I is always Cohen-Macaulay (see Example 2.14(ii)), and the result follows. If dim I 1, I must be a subcomplex of Ll 3 • I is Cohen-Macaulay if and only if I is connected (see Example 2.14(iii)), and again the result follows directly. Thus we may assume dim I 2, whence both parts of Theorem 3.31 apply. If codim I < 2, the corollary is a trivial consequence of Theorem 3.31 by use of our observations after Corollary 3.23. Thus we may assume codim I = 2. But then we finish the proof as follows: A) :::}: Theorem 3.31 and Theorem 3.27. A) ~: Theorem 3.31, Theorem 3.27 and an application of the following fact which improves Theorem 3.25: Let I be an equi-dimensional orientable simplicial complex of codimension 2. Th~n either I is a manifold or else Sing I is a non-empty union of codimension 1 and codimension 2 simplices of I. B) :::}: Theorem 3.31. . B) ~: Theorem 3.31 and the just mentioned fact. We also note that if I is a manifold then I is a c-sphere or c-disc, q.e.d.

Now let 1 c S = K[xo, ... , x n ] be a monomial ideal of height 2; we relate the above theory to the Cohen-Macaulay property for S/l. First note that Sll cannot be CohenMacaulay unless 1 is equi-dimensional and unmixed; that is, every associated prime of 1 has height 2. It follows that the Reisner complex I associated to 1 is equi-dimensional and of codimension :;; 2. In fact by these results, each facet of I is given by a Reisner complex I(Xa,Xb) c LlN for (xu. Xb) an associated prime of Polar 1 c SIN} = k[xo, "', XN]' By the definition of Reisner complexes, X a , Xb are the two vertices of LlN not contained in the facet I(x•. x.); hence this facet correspond to the edge :-----1 in the graph G which we associate to I. In short, G (or G*) may be obtained directly from the primary decomposition of Polar 1. Thus we now refer to G* as "the graph or the primary decomposition of Polar 1". In addition, we now call 1 orienlable or a qua8't"-mani/old, etc., if the Reisner complex I of 1 has these properties. It follows immediately from Corollary 3.32A); CoroUary 3.33. Let 1 c S be an orientable monomi.al Ukal 0/ hei,ght 2, and let G* be the graph 0/ the primary decomposition 0/ the pol{lrization 0/ 1. Then: S/l is Coken-Macaulay zj and only tj (i) I i8 equi-dimensional and unmixed, and (ii) G* has di.ameter 2.

Example 3.34. Let 1 c S be the square-free monomial ideal (Xo, Xl) n (Xl' X2) n ... r (Xk-l, Xk)' It is not difficult to see that 1 is an orientable quasi-manifold. Thus Corollary 3.33 applies. The conclusion is: S/l is Cohen-Macaulay if and only if k 3.

§ 3. The topological Cohen-Macaulay criterion of /3chwartau

131

We now apply Schwartau's theory to answer the following question: Consider the ideal

Which exponents a, b, c, d 0) define arithmetically Cohen-Macaulay curves in P~. Examining the dual graph G* of the Reisner complex of I we get the following general result: Example 3.35. Let G* be the graph associated to Polar 1. Then G* has diameter if and only if:

+c

b

Case 2. a, b, c > 0, d = O;a

+c

Case 1. a, b, c, d

>

0: a

+ d + e, for e b

+

2

-1,0, 1.

1.

Case 3a. a, b > 0, c = dO: always. Case 4. a

>

>

0, b d = 0: never. 0, b = c = dO; always.

Case 3b. a, c

Proof: Left to the reader (see also Schwartau [11, Theorem 186).

Now we recall the original motivation for this section; Consider the following ideals in S: and

We will see from liaison addition in Chapter III that S/1 is always Cohen-Macaulay, and S/I' is always Buchsbaum. Example 3.35 provides also' an explanation for this phenomenon independent of liaison addition. By Example 3.35 the rings Sj]' are not Cohen-Macaulay since the 'associated Reisner complex E has isolated codimension 2 singularities (i.e. Sing E has facets of codimension 2 in E). In addition it follows that the Reisner complex for the ideal]' has precisely n isolated codimension 2 singularities. In order to show that S/1' is a Buchsbaum ring with invariant n we have to use another method. Applying Schwartau's liaison addition from Chapter III we will prove that the curves in P~ defined by l' have liaison invariant 11" (up to shift); that is, 8jl' is always Buchsbaum. Furthermore, by constructing the minimal graded free resolution of the ideal from Example 3.35 we can sharpen the result of Example 3.35 (see Schwartau [1], Chap. 3) ;

!

Proposition 3.36. The ideal (xo, xll a n (Xl> x 2)b Il (X2' xa)C n (xa, xo)t! defines an arithmetically Cohen-Macaulay curve in Case 1. a, b, c, d Case 3a.

Pk if and only if:

+ d + e, for e = 0; a + c < b + 1.

> 0, d = a, b > 0, c = d = a, c > 0, b d

Case 2. a, b, c Case 3b.

> 0; a + c

b

0: alwaY8. 0; never.

Ca8e 4. a> 0, b = c ='d = 0; always. 9*

-1,0,1.

'132

II. Hochster-Reisner theory for monomial ideals

§ 4.

Further applications to algebraic topology and combinatorics

In this paragraph we will investigate properties of Buchsbaum complexes of simplicial complexes. For example, following P. Schenzel [1], we will examine a term which can be interpreted as measuring the error when we no longer have the Cohen-Macaulay case of simplicial complexes. We note that we actually do not need Schenzel's approach for these applications to combinatorics (see, for example, Corollary 4.7' and Lemma 4.14'). The first topic of this pltragraph will 15e the characterization of Buchsbaum modules using dualizing complexes. For this we will follow an idea initiated by R. Kiehl [1]. P. Schenzel [1] obtained some generalizations of Kiehl's approach by improving the locally Cohen-Macaulay criterion of Reisner of Theorem 2.4. We will first of all review the relevant commutative algebra. Let M': ... -'>- Mk -'>- Mk+1 -'>- ... be a complex of A-modules, By ,8M', resp, "M', we denote the truncated complex ,., -'>-

jJfk

lV8 -

1 -'>-

MH2)

-'>-

-'>- , •• -'>-

Im(2If$-

0

resp.

o If r


-

lVr+1

..•

-'>- ~Jfk -'>- .,'

s we have Hk(,BM') ,

~

=

{OHk(M')

for k for 'I'

or k ~ s, k < s, r


(iii). To this end we make use of the following:

134

II. 'Hochster-Reisner theory for monomial ideals

Proposition 4.3. Let

r: K" -+ D

be a homomorphi8m of complexes of A-modules su,ch that

(a) K' is a complex of Jc..vector spaces, and

(b) HV): H'(K") -+ H'(D), i E Z, is a surjective homomorphism, Then the complex. L' is quaIJi-isomorphic to a complex of k-vector spaces,

Bi.

Proof: We denote by Bk., the image of the homomorphism Kl-l -+ Ki resp. V-I -+ V, and we denote by Zk" Z~, the kernel of the homomorphism Ki -+ Ki+l resp, V-+ V+l. Then we have the following commutative diagram with exact rows

)

Hi~K')----):'()

where the homomorphisms Bk. -+ BL Z~, -+ ZL and Hi(K') -+ Hi(£") are induced by r. Because K" is a complex of k-vector spaces, the canonical homomorphism Zk -+ Hi(K') splits. Furthermore, Hi(j'): Hi(K') -+ Hi(D) is a surjective homomorphism of k-vector spaces, i.e., it splits. That is, we get a homomorphism Hi(L') -+ £i. In fact, it is a homomorphism of the cohomology complex H'(L') of L' to L', which induces isomorphisms on the cohomology modules. This proves Proposition 4.3, Next we prove (i) =? (iii) of Theorem 4.1. By Theorem 1.2.15, for a Buchsbaum module M, the canonical homomorphism of co~plexes TtlK'(mj M) -+ TdK"'(m; M)

induces surjective homomorphisms on the cohomology modules. Because K'(m; M) is a complex of k-vector spaces it follows that rtlKOO(m; M) is quasi-isomorphic to a complex Erm(M) in the derived of k-vector spaces, see Proposition 4.3. Because KOO(m; M) category, we have that T-dE HomA(M, I A) is quasi-isomorphic to a complex of k-v~ctor spaces by virtue of the Local Duality Theorem, Theorem 0.3.4. For the proof of Theorem 4.1 it remains to show (iv) =? (i). To this end we consider the Koszul complex K'(q; M) of M with respect to an arbitrary parameter ideal q (Xl' .. ,' Xd) of M. By the definition we have K"(qj M) = K'(q; A) ®AM,

where K'(q; A) is a complex of finitely generated free A-modules such that the cohomology modules Hi(qj M) of K'(q; M) are A-modules of finite length, compare Chapter 0, § 1 for properties of Koszul complexes, By virtue of our Lemma 4.2 it follows from (iv) that TdK'(q; M} is quasi-isomorphic to a complex of k-vector spaces. In pa,rticular we have mHd-1(q; M) = 0

§ 4. Further applications

for an arbitrary parameter ideal q of M. We set q' following short exact sequence 0.-+ Hd-2(q'; M)@AAjxdA

.-+

H·H(q; M)

.-+

(Xl' ..• ,

Xd-l)'

135

Then we have the

q'M :Xdjq' Jl .-+ 0, ~

compare Chapter 0, Lemma 1.6. Therefore we get m· (q'M :Xd) Buchsbaum module by virtue of Proposition 1.1.10.

q'111', i.e., .Jl is a

The following corollaries indicate, in our opinion, how substantial Theorem 4.1 is. First, we recall that an ideal a of a regular local ring R is called perfect, if Rja is a CohenMacaulay ring. That is equivalent to the vanishing of Hi(E Hom(Rja, R») ~ Ext~(R/a, R)

for all i

n. For Buchsbaum rings a corresponding result is valid.

dim R - dim Rja

Corollary 4.4. The larol ring RIa is a Buchsbaum ring ?/ and only i/ TnE Hom(Rja, R) is to a complex 0/ k-vector spaces, where k denotes the residue field 0/ R.

qu.a~i-ilJomorphic

The proof follows from Theorem 4.1 because E Hom(Rja, R) is up to a shift isomorphic to the (normalized) dualizing complex of Ria. If M is a Buchsbaum module, then it follows that T-d

HomA(ilf, fA)::::::: Hom".(C·(M), k»)

by the Local Duality Theorem and Matlis duality. Corollary 4.5. Let M be a Buchsbaum module. Let F' be a bounded complex 0/ finitely generated free A-module8 8uch that Fi 0 lor i < 0 and F" @Ak has trivial boundary map8. A8sume that Hi(F' @M), i E Z, are modules o//inite length. Then we get

m· Hi(F'@M)

=

0

and i

diIDt Hi(F' @M)

= 1: rank FH dim". Hfn(M)

i

lor 0

< dim M.

Proof: First we notice that m· Hi(F'@M) 0, i < dim M, follows from Lemma 4.2. Then the quasi-isomorphism (*) given in the proof of Lemma 4.2 implies Hi(F'@M)

= H'(F'

C'Ull») ,

i


••• , Xr ; M) ~ 1m d'-I.

Having this one obtains the implication (i) ~ (iv) as follows: First, choose a EJj M/(xi" .• ,' Xi) , M)' -basis Yl> ••• , y, of m (see our Definition 1.1.7 and Pro-

( l-s;,j,< .. · ... , x r- 1) • ill: m)

n (Yl'

..• , Ys) . M

S (Xl> •.• , Xr-l) . M which by applying Lemma 1.1.14, equals 1m dr2 ; that is, a = dT - 2(a'). Hence we obtain for all i < r that a di - 1 (a'). Then

dH ((-l)'x ra'+b)

(-l)'xra

+ di-1(b)

0;

that is, (-1)' xra'

+ b E Ker di - I

n (Yl> ... ,

1m di - 2 •

y" x r )· Li-l

Thus we see that b = 1).-1 xral + di - 2(b') with b' E £1- 2 ; that is, (a, b) = di - 1 (a', b') i I E 1m d - • This proves the claim and therefore also CDrollary 4.7'. Next we are interested in the Buchsbaum property of the canonical module of a Buchsbaum ring. We recall the definition of the canonical module.

Definition 4.8. A Noetherian A-module KA is called a canonical module ?f A if KA@AA

HomA(Hi!t(A}, E),

as A-modules, where E

d = dim A,

E(k) denotes the injective hull of the residue field k of A.

The notion of the canonical (or dualizing) module was introduced by Herzog and Kunz in [1]. It is uniquely determinated if it exists. In case A possesses a dualizing complex lA' A has a canonical module. More precisely, KA '::: H-d{lA) '

d = dim A.

'This follows from the Local Duality Theorem, compare § 3, Chapter O.

Theorem 4.9. Let A denote a d-dimenswnal Buchsbaum ring which has a canonical module K A. Then KA is a Buchsbaum module with H:n(KA) "-' HomA(Ili!t-i .1(A), E)

lor 2

i


0 we have (aN

+ anN):N m =

aN:Nm

+ all-1(aN +

aN):Nm).

Passing to M := NjaN we may assume without loss of generality that a

= O. Then

hence anM:m

=

(aIlM:m) n (a ll - 1M

=

O:m

=

+ O:m)

+ all-1(aIlM:m):an-l) O:m + afl-I(aM + O:m):m)

+ (all - M n (aIlM:m)) O:m + afl-1(aIJM:afl-1):m)

O:m

1

O:m

a,,-l(aM:m) ,

q.e.d. Now we sketch Suzuki's proof of Theorem 4.9 for modules: The canonical module KM of a Buchsbaum module M is also a Buchsbaum module. We set JJ'(.) = Hom...{H!n(.), E(k)). We may assume that A = A and we procede by induction on d dim M. Let d > 3 and additionally that depth M > 0 since H'fn(M) H~(MjH:h(M)). Let al> ... , ad be any system of parameters for K M • We have an exact sequence

o ~ K l /aKM ~ K{M/aM) ~ Dd-l(M) ~ 0, l,[

where a

(aI' .•• , ad)'

140

II. Hochster-Reisner theory for monomial ideals

Consider the long exact sequence of Koszul homology modules with respect to a' = {a2, ••• , ad},

EI(a'; K M,) -+ EI(a'; V) -+ KM/(a) KM -+ KM'/(a') K M, -+ V -+ 0

where M' = M/aM and V

D'H(M). If we have that the mapping

EI(a'; n) : EI(a'; K,w) -+ E 1(a'; V)

is a zero map, then the equality

holds. On the other hand, we have

and therefore, by the induction assumption, we can conclude that the difference lA(KM/(a) KM) - eo(a; K M)

does not depend on the chOIce of the system of parameters at> ... , ad for M; which is the definition of the Buchsbaumness of K M • Consider the direct system {Ei(a~, ... , a~; M), «1>~.n+l} with the limit E~(M) (see Chapter 0, § 1), where Ed-i(a; M) ~ Ei(HomA(K.(a; A), M))~ Ei(a; M).

Let an {a~, ... , a~} and a'n {a~, ... , a:}. A commutative diagram is induced by the exact sequence (see Lemma 0.1.5) 0 -+ aM -+ M ~ MjaM -+ O. Ed-I(a", M) - - - - - - - - - - -..... Ed-I(a"; M/)

II!

II!

E 1 (a";M)

1\b:'

H'f;;I(M)

Eo(a''';M')EBEj(a''';M')

1IP:'.

E'f;;I(M')

d

and

1: a/i

o.

(#)

j=2

ltsuffices to show that for any j 2, ... , d, Ii· A = 0. Let z E E'fn-I(M). Then there exists (u, v) E ZI(a"; M) c: Ko(a'fI, M) EB K1(a'fl; M) such that «1>l:c(lu, vI) = z. Note first that the cycle condition implies that a!u E (a~, ... , a~) M; hence uE UM(a;, ... ,a~).

We claim that h·«1>l:c,·E I (a",n) (Iu, vi) the homology class of a cycle c. Let Iii,

vi

(##)

O. Here we use the notation lei for

(lu, Ivl) := HI(a"; n) (Iu, vi) E Eo(a'''; M')

EI(a'''; M').

§ 4. Further applications

141

It is not too difficult to see that (/>~'(Iu,

vi) =

(I(az, .•• , ad)

ul, 1(1)

E Ho(a'fI+l; M/)

EB Hl(a'fl+l; M ' ).

Therefore it suffices to show that

with lui E Ho(a;+I, ... , a~+l; M/aM) M/(a, a~+l, ... , a~t1) M. By Goto's Lemma, from (it it) we obtain the following expression: 'U

= .E

ai~luI

IM~l(l(az, , •. , ad) a~-lihi) If I 9= {2, .. " d}, there exists i

For I

~

O.

I, hence

= {2, ... , d} ,

Ii' (/>~,;1(I(a2' ... , ad)fI-l alaII)

Ii' (/>;'~1(I(a2' ... , ad)" uII)

= a/I' (/>;';1(I(a2' ... , ai' , .., ad)" aj-lull) , by the cycle condition (it),

= -Lad;' (/>~,;I(I(a2' ... , aj, ... , ad)" aj-luII) i+i -_

"I i ' 'PM' """+l{I(a2' •• ,' Ujai, A • "', ad )fI ain-lai11+1UI I) •

-,,;;.,

i+i

Since lai +lUI I = 0

in

M/(a~d,

... , a~+l) M',

we conclude in this case also that

It· (/>~~1(I(a2' .. ,' ad),,-l alaII)

0

as required, q.e.d. Our Theorem 4,9, proved by P. Schenzel in [1], is an extension of the main result of Kiehl's paper [1] to Buchsbaum rings. Using the criterion of Theorem 1.2.10, Kiehl showed that KA is a Buchsbaum module, if i_ttIA is isomorphic t.o a complex of kvector spaces, If A is a two-dimensional local ring admitting a canonical module K A , then K,d is a Cohen-Macaulay module. Therefore, the converse of Theorem 4.9 is not true in general, even if we assume that A has small dimension or that KA .is a CohenMacaulay module.

142

II. Hochster-Reisner theory for monomial ideals

Theorem 4.10. Let A denote a local riWJ haviWJ the canonical module K A • Assu1ne A saf:i8fies 8 2 , i.e.,

~that

depth All > min(2, dim All) for all p E Spec A. If KA is a B'IJJ)hsbaum module, then A is a B'IJJ)hsbaum riWJ. Proof: First without loss of generality we can assume A A, i.e., A possesses the dualizing complex I;. by virtue of the Cohen Structure Theorem. For dim A < 2 there is nothing left to prove. Therefore, we can assume dim A > 2. By Proposition 0.3.6, we have

Supp KA

= Spec A.

It follows from Theorem 4.9 that H~{KA)'-"'" HomA(H~-i+l(A), E)

for 2

i


- Erm(K'(f; R)) ->- Hom(lJ 'Hom(K(f; R), KR[d]), E) ->- 0

by virtue of the Local Duality Theorem. Since (Hom(E Hom(K(;r; R), KR[dl),

for all i


- Hr-l(f._l; R) ®RRfx;R ->- H'(;r8-l> x!; R). If s d and r < d, we saw that m annihilates the left modules for all i2. 1. By Nakayama's 1.emma it follows that m· H,-l(f._l; R) = 0, i.e., for 8 - 1 = d - 1 and, r < d the cohomology modules are of finite length. By an induction argument it follows that H'(f.; R), 0 r < s, are modules of finite length, in fact k-vector spaces. Re.peating the above arguments we have, as in the case s = d, that r K' (f.; R) is isomorphic to a complex of k-vector spaces. Thus, the formula for the cohomology modules can be derived here also in a similar manner, q.e.d.

Lemma 4.14 provides a key to applications in combinatorics by use of homological algebra in derived categories. We want to describe an elementary approach to these applications by giving a new and simple proof of 1.emma 4.14. Also our Lemma ,4.14' improves Lemma 4.14. Lemma 4.14'. a) Let M be a Buchsbaum module 0/ dimension d and let {Xl' •.. , Xd} be a system 0/ parameters 0/ M. Then we have (i)

H~(M/(Xh ... , x;) 1\f)

4> (EEl H~i(M)) for all i and j with i + j < d. I=(}

10 Buchsbaum Rings

({)

146

II. Hochster·Reisner theory for monomial ideals

(ii) H'(x I , b) Let R

••• ,

=

EB H:;I(M)) ( ({)

Xi; M) '"

EB R;

lor all i and j with i < j.

be a Noetherian graded ring with Ro

=

K a lield, m

R i • Let

i:2:0

M be a Noetherian graded R-module 01 dimension d. Let {Xl> ••• , xa} be a By8tem 01 homogeneOUB parameter8 01 M with ti := degree 01 Xi lor i = 1, ... , d. Then we have (i') !1:n(M/(XI' •.. , Xi)'

M)

lor all i and j with i

+j

(ii') IP(xv ... , xi; M) '"

(

lor all i and j with i


... , Xi-I) ·.J.ll ) EB Hm i+ l( M /(Xl' •.• , X,-2) . M) Hm

and an easy induction on j proves the result (i). (ii) l'his result is clear for j 1 since HO(XI; M)

If j

>

O:MXl~ H~(.lll).

1 we consider the following exact sequence (E') (see Lemma 0.1.6)

0.....;.- Hl(Xi; Hi-I(XV ... , xj-I; M)).....;.- Hi(Xl' ... , Xj; .J.ll)

(E')

.....;.-}]O{Xi; }]i(X1' ... , xj-I; M)) .....;.- O.

If i


,." Xj-I) • M)

and we therefore have: Hi-I(X I , "" Xi; M) "" Hi- 2 (xl> "" xj-I; M)

Hr;.,(M/(xl> .," XI-I)' .lll").

(i) and an easy induction on j prove (ii), b) The exact sequence (E) of part a) has the following form: 0->- M/(xl> "" xj-I) , M : m( - t i ).!4 M/(x 1 , ->-

M /(Xl> "" Xi) • M

->-

".,

xj-I) . M

0,

Therefore again we obtain (i/) by induction on 1. For (ii') we point out that the exact sequence (E/) has the following form:

o ->-IP(xi; IP-l(xl> ""

Xj-I; M)) (tj) ->-IP(xI , , .., Xi; M)

->- !fO(Xj; lli(xI' "" xj-I; M)) ->- 0,

Furthermore we have: llO( Xj; llH(x v .'" Xi-I; M)) ((XV"" Xi-I) • M : m/(x1,

""

XI-I) , M)(tl

+ .,' + tj- 1 )

+ .. , + tj-I)'

~ llr;.,(M/(Xl> "., Xj_l) , M) (tl

Hence our result (ii') follows by the same arguments as above, q.e,d,

Remark 4.15. It follows from Lemma 2.6 and Lemma 4,14 that ll~(R) ~ [Hi(;!i; R)]o.

0

i

< d.

For this it is not necessary to assume that if {Xl> ., "Xd} is a system of parameters consisting of forms of the same degree. Following the reasoning of the proof it is seen immediately that it is enough to assume for if = {Xl' , •• , xn} to be any set of forms m. Or put differently, it is not necessary contained in m and such that ROO;!iR to take a direct limit in order t{) comlJUte the local cohomology modules via Koszul complexes for those rings, This was proved by M. Hochster and J, L. Roberts in [2], Theorem 1.1(b), in case of an P-pure graded k-algebra over a perfect field k of prime characteristic p and in Hochster-Roberts [2], Theorem 4.8(b), in case R has a presentation of perfect graded P-pure type. Hence, in the case of the ground field of characteristic zero our result leads to a slight improvement of Hochster-Roberts [2], Theorem 4.8(b). For our considerations in the following t{)pics Lemma 4.14' has an important application with respect to quotients of certain ideals. It allows us to give an explicit description of certain Hilbert functions. 10*

148

II. Hochster-Reisner theory for monomial ideals

Corollary 4.16. Let R denote as belore a graded Ie-algebra. Let 2: = {Xl' ... , Xd} be a system 01 parameters consisting ollorms 01 degree t. Then, lor 1:C;; s:C;; d there are isomorphisms s ~-l

((Xl' .'"

where ri

= (s

~

XS-I)

R : Xs )/(xv , .. , XS-l) R '"'-'

1£"( -t't) ,

1) dimk[H~(R)]o'

Prool: For the cohomology modules of K08zul complexes there are the following short exact sequences 0----+ Hs-2(2:s-t; R) ----+ Hs-l(:fs; R) ----+ (2:s-tR : xs/2:s-tR) ((s -

1)

t)

0,

where it was used that llS-2(2:,H; R) is annihilated by m. Since this short exact sequence is a sequence of Ie-vector spaces it splits. By virtue of Lemma 4.14' our statement now follows, q.e.d. In point of fact, the graded rings considered in I..emma 4.14' are Buchsbaum rings by virtue of Theorem 2.7. ('A)ITesponding results hold for an arbitrary graded Buchsbaum ring without assuming the purity condition. However, for two reasons we restrict ourselves to this special case. Purity implies that the non-vanishing cohomology of T d!1rm(R) is concentrated in at most the zE,lro'th graded piece. This is no longer true for an arbitrary graded Buchsbaum ring. For our purposes here it is enough to consider pure rings for which the statements can be formulated in an easier manner. Let .1 denote a finite simplicial complex with the vertex set {Xl>"" x n}. Let /; be the number of i-dimensional faces of .1. Thus 10 nand 1-1 = I, since .1 has the unique I)-face 0. The vector I (/-1,/0' ... , Id-t), d dim .1 1, is called the I-vector of .1. Since 1e[.1] is a graded algebra, we can associate to it its Hilbert lunction H(m, k[.1]), that is H(m, 1e[.1]) = dimk[Ie[.1]jm, mE Z,

+

where [k[.1]jm denotes the k-vector space of all homogeneous forms of degree m in k[.1]. For m large, H(m, k[.1]) coincides with a polynomial, the Hilbert polynomial. R. P. Stanley [3], Proposition 3.1 showed that the I-vector describes the Hilbert function exactly.

Proposition 4.17. For the Hilbert lunction H(m, k[.1]) we have H(m, k[.1])

where

(~!)

=

=i:i:>

0 lor i

+

(m i

1 and

1),

C=~) =

Prool: I,et Xi denote the image of

Xi

m 2:

0,

1.

by the canonieal projection

k[Xh ... , xn] ----+1e[.1).

x:' .....

A k-basis of [k[.1]jm consists of all monomials x = X: such that deg x all = m and Supp (x) E .1, where the support Supp(x) is defined by

+ ... +

Supp(x)

=

{x;

i

a;

> O}.

8

at

§ 4. Further applications

If

(J

E .1 has exactly i

+

1 elements, then the number of monomials of degree m

+

2:: 0

(m . 1). Hence, the propo~ition is

whose support is contained in (J coincides with proved, q.e.d. In particular by Proposition 4.17 dim k[L1) = dim .1

149

~

1.

Next we will prove an estimate of the number of i-faces of certain simplicial complexes. To this end we define integers hi by (1 - T)d

E H(m, k[L1]) Tm = 11,0 + hIT + ... + hdTd. m2'O

It is easily seen that the degree of the polynomial on the left does not exceed d. The vector h = (11,0' • '" hd) is called the h-vector of .1. In fact, the form!!>] power series E H(m, k[L1]) Tm is the Poincare series of the graded k-algebra k[L1). By the Theorem m2'O

of Hilbert-Serre there is a polynomial/(T, R) such that F(T, R)

I(T, R)j(1

T)d

for the Poincare series F(T, R) of a d-dimensional graded k-algebra R. We recall that H(m, ) and F(T, ) are additive functions on short exact sequences. For our particular situation we remark: knowing the I-vector of .1 is equivalent to knowing its h-vector. Proposition 4.18. For the h-vector and I-vector 61 a linite simplicial complex the 10Uowing relations hold:

i~ (-1)0-' (~ -:) Ii-!

hv lor v

0, 1, ... , d, d

= dim .1

and

IH

ito (~ _:) hi

+ 1.

The prool follows by an easy calculation. We omit it. Now we prove one of the main results of our applications in algebraic topology and combinatorics. Theorem 4.1D. Let .1 denote a simplicia}, BuchBbaum complex with n vertices and we denotp d = dim .1 1. For the I-vector and the h-vector 01 .1 there are the lollowing bounds

+

IV-I

d) (:) (v

(V E.

,,-2

i~-l

1,

1) dimk H,(L1; k)

+1

and

h. (n - d+v v 1) where 0

v

(-I)"

(d) .=1:. V

v- 2

(-1)i dim k

Hi(L1;k),

1

d and k denotes an arbitrary lield.

Prool: By the previous Proposition 4.18 we have an exact expression of the Hilbert function or even of the Poincare series of k[L1].

150

II. Hochster-Reisner theory for monomial ideals

'Vhat now follows is another way to calculate the Poincare series by techniques from commutative algebra. To this end we consider the associated graded k-algebra R k[,1], where k denotes the fixed field. Without loss of generality we can assume k an infinite field. Otherwise we extend k to the field of rational functions k(t) in one {x}' ... , Xd) of variable. Then there exists a homogeneous system of parameters:li R consisting of forms of degree 1. Let x E R be homogeneous of degree one. Then .there is the following exact sequence of graded R-modules R(1) --+(R/xR) (1) --+0.

Because F'(T, ) is additive, it follows that (1 - T) F'(T, R) = F'(T, RlxR) - TF'(T, OR:X).

Iterating this argument d-times we obtain a-} (1 - T)d F'(T, R)

where Q;, 0 S i d

F'(T, RI;rR)

T(l - T)i F'(T, Q;),

1, denotes the quotients

((x}' ... , Xd_H)R: Xd-;/(X ll •.• , Xa-H)) R.

By virtue of Corollary 4.16 we get F'(T, Q;)

a-i-}

= ,1:

(d

1=0

It now follows from simple calculations that (1- T)d F'(T, R)

=

F(T,R/;JdR)

-1: (d)v (.EI(-1)v-i-Idimk[l!~(R)]0) T". ~=}

'=0

In particular, the last equality shows that F'(T, Rj;JdR)

flo

+ flIT + ... + flaTd

does not depend on the system of parameters ;r (consisting of forms of degree 1). We define ((Jo, flI' ... , fla)

fI

the fI-vector of the simplicial Buchsbaum complex ,1. By the definition of the h-vector it follows that

h. = g. where 0

v

(~) v

.:i (_1)V-i-l dimk[l!~(R)]o, 1

.=0

d. Using the second formula of Proposition 4.18 we get

.1: (d)v '=0

v-I

(V-l) . . dimk[l!:n(R)]o, ~

v 0, 1, ... , d, by some simple calculations. In point of fact, the fI-vector of ,1 is the Hilbert function of the O-dimensional graded k-algebra R/;JdR, i.e. fl.

dimk[R/;JdR]v,

0

... , ed and ho, hi> ... , hd such that Z(P,m)

d+l =.EeH

(m).

1=1

'/,

and

E Z(P, m) Tm

(1 - T)dH

hoT

+ hlTt + ... + hdTd+l,

m: