Combinatorics and Commutative Algebra [1st ed.] 978-0-8176-3112-3;978-1-4899-6752-7

Table of contents :
Front Matter ....Pages i-viii
Background (Richard P. Stanley)....Pages 1-29
Nonnegative Integral Solutions to Linear Equations (Richard P. Stanley)....Pages 30-61
The Face Ring of a Simplicial Complex (Richard P. Stanley)....Pages 62-85
Back Matter ....Pages 86-94

Citation preview

Progress in Mathematics Vol. 41 Edited by J. Coates and S. Helgason

Springer Science+Business Media, LLC

Richard P. Stanley

Combinatorics and Commutative Algebra

Springer Science+Business Media, LLC 1983

Author: Richard P. Stanley Mathematics Department, 2-375 Massachusetts Institute of Technology Cambridge, MA 02139

Library of Congress Cataloging in PubUcation Data Stanley, Richard P., 1944Combinatorics and commutative algebra. (Progress in mathematics ; v. 41) Bibliography: p. 1. Commutative algebra. 2. Combinatorial analysis. I. Title. 11. Series: Progress in mathematics ; 41. QA251.3.S72 1983 512'.24 83-17915

CIP-Kurztitelaufnahme der Deutschen Bibliothek StanIey, Ricbard P.: Combinatorics and commutative algebra/ Richard P. Stanley. - Boston; Basel; Stuttgart : Birkhäuser, 1983. (Progress in mathematics ; 41) NE:GT ISBN 978-0-8176-3112-3 ISBN 978-1-4899-6752-7 (eBook) DOI 10.1007/978-1-4899-6752-7 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior permission of the copyright owner.

© Springer Science+Business Media New York 1983 Originally published by Birkhäuser Boston, Inc in1983. Softcover reprint of the hardcover 1st edition 1983 ABCDEFGHIJ

TABlE OF CONTENTS Preface Notation Chapter 0

vii viii BACKGROUND § §

§

Chapter I

10 2. 3 0

7 22

NONNEGATIVE INTEGRAL SOLUTIONS TO LINEAR EQUATIONS l. 2. 3.

4. 5. 6.

7. 8. §

9.

§1O. § ll. § 12. § 13.

Chapter II

Combinatorics Commutative algebra and homological algebra Topo 1ogy

Integer stochastic matrices (magie squares) Graded algebras and modules Elementary aspects of JIJ -sol utions to linear equations Integer stochastic matrices again Dimension, depth, and Cohen-Macaulay modules local cOhomology Local cohomology of the modules Mffi'>',0: Reci proc ity Reciprocity for integer stochastic matrices Rational points in integral polytopes Free resolutions Duality and canonical modules A final look at linear equations

30 31

33 37 39 43 45

50

51

52 54

56 60

THE FACE RING OF f\ SH1PLICIAL COMPLEX 1. 2. 3. 4. 5. 6. 7. 8.

Elementary properties of the face ring f-vectors and h-vectors of complexes and multicomplexes Cohen-Macaulay complexes and the Upper Bound Conjecture Homological properties of face rings Gorenstein face rings Gorenstein Hilbert functions Canonical modules of face rings Buchsbaum complexes

62 64 68 70 74 77 80 84

86

References

v

PREFACE

These notes are based on aseries of eight lectures given at the University of Stockholm during April and May, 1981.

They were intended

to give an overview of two topics from "combinatorial commutative algebra", viz., (1) solutions to 1 inear equations in nonnegative . integers (wh ich is equivalent to the theory of invariants of a torus acting 1 inearly on a polynomial ring), and (2) the face ring of a simpl i ci al compl ex.

In order to give a broad perspect ive many detai ls

and special ized topics have been regretfully omitted.

In general,

proofs have been provided only for those results which were obscure or inaccessible in the 1 iterature at the time of the lectures. The original lectures presupposed considerable background in commutative algebra, homological algebra, and algebraic topology.

In order to broaden the

accessibi 1 ity of these notes, Chapter 0 has been prepared with the kind assistance of Karen Col 1 ins.

This chapter provides most of the back-

ground information in algebra, combinatorics, and topology needed to read the subsequent chapters. I wish to express my gratitude to the University of Stockholm, in particular to Jan-Erik Roos, for the kind invitation to visit in conjunction with the year devoted to algebraic geometry and commutative algebra at the Institut Mittag-Leffler.

I am also grateful for the

many insightful cümments and suggestions made by persons attending the lectures, including Anders Björner, Ralf Fröberg, Christer Lech, and Jan-Erik Roos.

Special appreciation goes to Anders Björner für the

time-consuming and relatively thankless task of writing up these lecture notes.

Finally I wish to thank Maura A. McNiff and Ruby Aguirre

for their excel lent preparation of the manuscript.

Richard Stanley Cambridge, Massachusetts May, 1983

vii

NOTATION

c:

eomplex nUlllbers

IN

nonnegative integers

IP

pos i t i ve i nte'ge rs

~

rational numbers

R

real numbers

~

integers

IR+

nonnegat ive real numbers

[n]

for

N-matrix

a matrix whose entries belong to the set

N[x]

polynomials in

N[[x]]

nEIN,

the set

x

{], 2, ... ,n}

whose eoeffieients belong to the set

formal power series in set

N N

x whose eoeffieients belong to the

N

# S

eardinalitv of the finite set

1·1

eardinal ity or geometrie realization, aeeording to eontext

T e: S

T

is a sub set of

S

T e: S

T

is a subset of

5

cx > 0

for a veetor

0:

and

S

T # 5

= (0: 1 , ••• , O:n) E IR n ,

this means

0:.

1

for all k*

nonzero elements of the field

kE

veetor spaee over

9!

symbol for isomorph i sm

Rj

symbol for homeomorph i sm

$,

11

k wi th basis

E

direct sum (of veetor s paces or modules)

im f

image feH)

ker f

kerne 1 of

vol P

vol urne

6 ..

the Kroneeker delta

IJ

k

of the homomorphism

f: M.... N

f: M.... N

(= Lebesgue measure) of the set (= 1 if i=j,

viii

and

pe: IR n

= 0

if i # j)

> 0

CHAPTER 0 BACKGROUND

§Io

Combinatoricso The purpose of this introduction is to provide the reader with the

relevant background from combinatorics, algebra, and topology for understanding of the texL

In general the reader may prefer to begin

with Chapter land refer back to this chapter only when necessary.

We

assurne the reader is famil iar with standard (first-year graduate) material but has

no special ized knowledge of combinatorics, commuta-

tive algebra, homological algebra, or algebraic topology. We begin with a discussion of rational power series in one variLet F(x) = L f(n)x n E ([[x]] be a formal power

able [St 5 , §IV]o

n~O

series with complex coefficientso i oeo,

F(x)Q(x) = P (x)

ity we may assurne

10 I

in the ring

Q(O)

THEOREM.

= 10

t[[x]]o

Define

f: tJ.... (

are eq u i val en t :

and

deg F(x)

= deg

= p(x)/Q(x), 1055

of general-

p(x) - deg Q(x).

QTxT ,

=

1 + 0lx +., ,+ 0dx d

degree less than

( i i)

Without

p(x)

(i)

Q(x)

is rational if there F(x)

1 2 " " , 0d be a fixed sequence of complex 0d # 00 The following conditions on a function

d

where

F(x)

for which

° ,°

Let

numbers,

~

We say

p(x), Q(x) E ([x]

exist polynomials

For all

and

d,

n

~

0 ,

1

p(x)

is a polynomial in

x of

2

o. ( i i i)

For all

n;" 0 •

k

n

where and

(l-y.x)

I

Sketch of proof.

d.

I

i=l is a polynomial in n

P. (n)

(1)

are distinct,

I

of degree less than

d.

I

Q(x) = 1 + alx + ... + adx d

Fix

Define four

complex vector spaces as follows: such that (i) holds}

Vl = {f:IN .... t {f: IN ....

V3

{f: N .... t

such that (i i i) holds}

V4

{f: IN .... It

such that

k

2: G. (x) (l-Y.x)

i=l

such that (i i) holds}

V2

I

I[

-d.

I

for some polynomials di ,

It is easily seen that

where

dim Vi = d .J

~

V2 , V4

~

We next cons i der rat i onal funct ions

I

1.2

fl:E f -

Ef c: tl

I

have

5

4.

One readily

V2 = V3 (= V4 )

Vl

F(x) =

p(x)/Q(x)

D

with

deg F(x) ;" 0 .

PROPOSITION. n f(n)x = p(x)/Q(x)

n;"O finite set

d.

5 j

for

Hence

Vl , V4 ~ V3

deg P ;" deg Q, i. e. ,

and

Yi

(i i i) }.

the same meaning as in

Vl

Gi (x)

I

of degree less than

shows

f(n)x n

2:

n;"O

Let where

f: IN .... [ P,Q

and suppose that

E [[x].

Then there is a unique

(the exceptional set of

[1, = [ - {O} such that the function

f)

and a unique function

g:lN .... [

defined by

3 f (n) ,

if

n ~ Ef

{

9 (n)

f(n) + f 1 (n),

if

nE Ef ,

n I g(n)x = R(x)/Q(x), where R E [[x] and deg R < deg Q. n2!O Moreover, assuming Ef 1

{x) special cases.

and discussed some

This algorithm was subsequently used by MacMahon [MM,

Sections VIII-X] to investigate a wide variety of combinatorial

6

problems.

In particular, in [MM, Section VIII, Ch. VII] he computes

the number H3 (r) of 3> M 8 N ... 0 •

Def i ne

Tor nA(11,N) The

n

ker(d

n

8 1 )/im(Cl

n+

l!! 1) .

Tor~(M,N)

does not depend, up to isomorphism, on n the choice of projective resolution of M. Moreover, both TorA(M,-) and

A-module

H (P 8 N)

Tor~(-,N)

are covariant functors.

A basic property of

Tor

Tor~(11,N) ';; M 8 A N. TorA(M,N) ';;: TorA(N,M). n n

Note that

is the isomorphism

17 2.28

DEFINITION.

If

M and

N are A-modu1es, then

denotes the set of all A-modu 1e homomorph i sms HomA(M,N)

f: 11 .... N.

Hom A(t1,N) The set

has the structure of an A-modu1e via

x(f(u)) ,

(xf) (u)

fE HomA(M,N), u E M. I f M and N are free A-modu1es of ranks m and n, then one can identify in an obvious way (after for

x E A,

choosing bases for matricesover

A,

basis u 1 "'" um' defined by u~(u.) J

I

and

r~

He also set then

Hom A(t1,N) If

M'~=HomA(M,A),

with

mXn

isfreewith

11

is free with dual basis

U

,';

l'

•••

,

u ,',

m

= 6 .. IJ

Next we note that if

N) elements of

is a contravariant functor.

HomA(-,N)

Name1y,

f: X .... Y is a homomorphism of A-modu1es, then define f"': HomA(Y,N) .... HomA(X,N)

as fo11ows: 2.29

given

g: Y .... N and

DEFINITION.

With

u E X,

let

P, M, N as in

(f"'g) (u)

gf( u) •

Def, 2.27, we have a

cochain complex: Cl

HomA(P,N): ..•

+

*

a;'

Hom(P n+ 1 ,N) ~ Hom(pn,N) ~

, •• -

°

H (C) ';; { q

~O(C) • A, q

0.

Similarly of course we can define augmented cochain complex, reduced cochain complex, and reduced cohomology groups. that the monomorphism

E:A

~

The only difference is

Co will not in general split, since

need not be an injective A-module. Of course, if A is a field then A is injective and-thus HO(C) ~ HO(C) • k •

A k,

21 2.38

DEFINITION.

C = {C ,d}

Let

q

chain (or cochain) complexes over

C 9 C'

{D ,S}

=

n

n

A.

q

and

C'

= {C'q ,a'} be two q

The tensor product

is the chain (or cochain) complex defined by

(C

D

n

e

C')

il

n

i+j=n

(C.

e

I

C~) J

:3.C. 8 C~ + (_1)i C. 8 d~C~

sn (c i ""C J'.)

I

The reader should check that

J

I

sn_lsn

J J

I

= 0

(or

snsn_l

= 0).

For example, in Chapter 1.6 there is considered over a ring s 8 (0 ... R ... R ... 0) • ~Jhen s = 2 th i s becomes Yi i=l

R

a

complex

o ... R 8 R .... (R 8 R ) Yl Now

R8M';;;R

2.39

and

Hence we obtain

DEFINITION. let

For specified basis

(R 8 R ) ... R eR ... 0 • Y2 Yl Y2

$

e.

I

Ae.

I

Let M be an A-module and xl ,"', x r e A . be a free A-module of rank one with a

Let

K(x. ) I

K (x.) = 0 q I

denote the chain complex satisfying:

if

q cF 0, I

xx.

I

This is denoted

x.

K(x.): 0 .... Ae. ~ A .... 0 . I

If

I

M is an A-module, then we have a complex

22

a

K(x.) I

Define the Koszul complex xl"'"

xr

x. M: 0 ~ Me. ~ M ~ 0 I

K(X l ,···, x r ' 11)

wi th respect to

by

K(x

l'

...

x

,

r'

H)

I f we pu t e i ... i = u 1 a ... €I ur' where u.I = e.I for {i ..• i} land q u.=l for other i, then Kq (x l' ... , x ,tl) i E I' , q I n is a free A-module with basis {ei'" i I I ,;; i l 2'

now eome to the relationship between simplieial and singular

homology. THEOREt1.

3.6

Let

I'>

be a finite simpl ieial eomplex and

X

11'>1.

Then there is a (eanonieal) isomorphism

Hq(I'>·A) '

~

Hq(X·A) "

for all q,

3.7 PROPOSITION.

Let

Sd-l

denote a

(d-l)-dimensional sphere.

Then

Hq (I'> ;A) 3.8 X

DEFINITION.

is aeyel ie (over

A)

vanishes in all degrees

H-1 (QJ; A)

3.9

A, q

d - 1

0, q

I d - 1

{

A simplieial eomplex I'> or topologieal spaee if its redueed homology with eoeffieients A q.

(Thus the null set is not aeyel ie, sinee

~ A.)

DEFINITION.

eha i n modu 1e eomplex

_

C (y)

C(X,Y)

homology of

q

Let

Y be a subspaee of

isa submodu 1e of

C (X), q

= C(X)/C(Y) = {C q (X)/C q (Y),

X modulo

Define the relative

d}

H (C(X, V})

Then the singular

so we have a quot i ent

-

Y (with eoeffieients

q

X.

q

Al

.

by

27 \Je next want to define reduced and spaces.

cohomology of simpl icial complexes

The simplest way (though not the most geometrie) is to

dualize the corresponding chain complexes. DEFINITION.

3.10

Let

C'(L'I)

=

C(L'I,O:)

chain complex of the simplicial complex

be the augmented oriented

L'I,

q-th reduced singular cohomology group of

L'I

over the ring

A.

with coefficients

The A is

defined to be

where

HomA(C'(L'I),A)

functor and

Hq(X,Y;A). C (L'I)

C (X) q

is the cochain complex obtained by applying the to

C' (L'I).

q

Exactly analogously define

Sometimes one identifies the free modules

= HomA(Cq(L'I) ,A)

Cq(L'I) of

HomA(-,A)

C (L'I)

and

q

by identifying the basis of oriented q-chains

with its dual basis in

with

Hq(X;A)

Cq(L'I).

0

Similarly one can identify

Cq(X).

There is a close connection between homology and cohomology of or

L'I

X arising from the " un iversal-coefficient theorem for cohomology."

We merely mention the (easy) special case that when

A is a field

k,

there are "canonical" isomorphisms q

~

Hq(L'I;k) ---=.... Homk(R (L\;k) , k) Rq(X;k) ~ Homk(Rq(X;k), k) • Thus in particular when is finite), we have

R (L\;k) q

R (L\;k) q

';;t

is finite-dimensional (e.g., when

Rq(L\;k)

and similarly for

X,

L\

but

these isomorphisms are not canonical. 3.11

DEFINITION.

A topological n-manifold (without boundary) is

a Hausdorff space in which each point has an open neighborhood IR n An n-manifold with boundary is a Hausdorff

homeomorphic to space

X in which each point has an open neighborhood which is homeomorphic with IR n or IR: = {(xl'···' x n) E IR n I xi 2 O}. boundary

3X

of

homeomorphic to

The

X consists of those points with no open neighborhood It follows easi ly that dX is either void or IR n

28

an

(n-l)-manifold. Suppose

X is a compact connected

Then one can show 3.12

H (X,aX;A) n

DEFINITION.

n-manifold with boundary.

is either void or isomorphic to

A compact connected n-manifold

A.

X with

boundary is orientable

(over A) if H (X,3X;A) = A . (rhe usual n definition of orientable is more technical but equivalent to the one

given here.) 3.13

PROPOSITION.

Every compact connected n-manifold with

boundary is orientable over a field of characteristic two. 3.14

POINCARE DUALITY THEOREM.

X is orientable over

3.15

A,

DEFINITION.

q

An n-dimensional pseudomanifold without

boundary (resp., with boundary) (a)

Every simplex of

(b)

Every

6

Fand

If

is a simpl icial complex

F'

6

are n-simpl ices of

F contained in some one n-simplex of 3.16

of a pseudomanifold

06

6,

there is a finite

Cn-J}-simplex of

6

suchthat

F. I

6 consists of those faces which is the face of exactly

6

PROPOSITION and DEFINITION.

sional pseudomanifold with boundary.

O.

6

6.

F=F 1 ,F2 , .. ·,Fm F' ofn-simplicesof 6 $ i < m Fi + 1 have an (n-l)-face in common for

The boundary

such that:

is the face of exactly two (resp.,

sequence and

6

is the face of an n-simplex of

(n-J}-simplex of

at most two) n-simplices of (c)

If a compact connected n-manifold

H (X;A) ~ Hn-q(X;A).

then

In the former case we say that

Let

6

be a finite n-dimen-

Then either 6

H (6,d6;A) n

is orientable over

~

A or

A', other-

wise nonorientable. 3. 17 suspension

DE F I NIT I ON . EX

quotient space of and

Xx

Le t

be the unit interval

of a topological space Xx I

in which

[0,1].

The

X is defined to be the

Xx 0

is identified to another point.

is defined recursively by EnX = E(En-1X).

is identified to one point The n-fold suspension

EnX

29

3.18

PROPOSITION.

For any

X

and

q,

H (X;A) ';; R 1 (LX;A) • q

q+

CHAPTER I NONNEGATIVE INTEGRAL SOLUTIONS TO LINEAR EQUATIONS

§l.

Integer stochastic matrices (magic sguares)

The first topic will concern the problem of solving linear equations in nonnegative integers. In particular, we will consider the following problem which goes back to MacMahon. Let Hn(r) := number of n

x

n

~-matrices

having line sums r,

where a line is a row or column, and an ~-matrix is a matrix whose entries belong to ~. Such a matrix is called an integer stochastic matrix or magic sguare. Keeping r fixed, one finds that Hn(O) = 1, Hn(l) = n!, and Anand, Dumir and Gupta [A-D-GJ showed that

L

H (2)x n

ex/2

--,;n:.!....-_

n~O (n!)2

I~

See also Stanley [St s ' Ex. 6.11J. Keeping n fixed, one finds that Hl(r) = 1, H2(r) = r + 1, and MacMahon [MM, Sect. 407J showed that

Guided by this evidence Anand, Dumir and Gupta [A-D-GJ formulated the following 1. 1 CONJ ECTURE. (i)

H (r)

E

IJlfrl

Fi x n

~

1.

Then

31 deg H = {n_l)2

(ii)

n

Hn ( -n+ 1) = 0,

H{-n-r) n

(_1)n-1 Hn {r) •

This conjecture can be shown equivalent to: ,

d

= n2

- 3n + 2

0,1, ... ,d .

The following additional conjectures can be made: hi

(i v)

0 ,

~

\'Je will verify conjectures (i) to (iv). Conjecture (v) is still open. The solution will appear as a special case of solving linear diophantine equations. This will be done in a ring-theoretic setting, and \~e will nO\ 1, e.g. letting R = k[x,y,z]/(xy-z2), deg x = (2,0), deg y = (0,2), and deg z = (1,1), then {x,y} and {z} are maximal homogeneous R-sequences. In terms of the Hilbert series we get the following characterization: 81 ,8 2 , ... ,8 r E H(R+) is an M-sequence if and only if

F(r1,A)

F(M/(8 1rH'" +8 r M),A) r deg e. 11

i=l

(1 _

A

')

5.7 DEFIrlITION. (i) If m = 1, let depth M := length of longest homogeneous M-sequence. (ii) If m > 1, specialize the grading to all-grading in any way and define depth /·1 as in (i). (It can be shown that this definition is independent of the specialization.) It is clear that depth /·1 ~. dim M. The case of equality, i .e., when some hsop is regular, is of particular importance. 5.8 DEFHIITIOil. 5.9 THEOREM.

M is Cohen-Macaulay if depth 1'1 Let /·1 have an hsop.

Then M is

=

dim N. C-~1

~

every hsop is regular

~

M is a finitely-generated and free k[e]-module for some (equivalently, every) hsop 8 = (8 1 ,8 2 "" ,8 d).

5.10 THEOREM. Let Mbe C-M, with an hsop Let nl,n2, ... ,nt E H(M). Then

9

= (9 1 ,8 2""

,9 d)·

42 M =

M lli k[e] -111,112"" t

a choice of e's and

I1'S

t

I" i=l

F(r1,,,)

d TI

j=l

(1-"

'l1t is a k-basis for r'1/e~1.

For such

it follows that deg 11.

'

deg e.

J)

Returning to our ring R~ once more, we can now state the following theorem, which will be proved later. 5.11

THEOREM (Hochster [H l ]).

5.12

COROLLARY.

I

r;:,O

Hn(r)"r

R~

=

is Cohen-Macaulay.

~-)-2- , PI,,) (l_,,)(n-l) +1

E

lN[,,]

The corollary follows since permutation matrices have degree one. It is an open problem to compute PI,,) or even P(l) in a simple way. In particular, can P(l) be computed more quickly than PI,,)? 5.13 THEOREf1. Let dirn R~ = d. There exist free commutative monoids Gl ,G 2 , ... ,G t ~ E~, all of rank d, and also 111,112"" ,l1 t E E~ ,

such that E.

'"

t

=

~ i =1

(11. + G,')' where ~ denotes disjoint union. ' t 11.

In terms of t:,e ri ng t:,i s theorem says that R. =

'"

I I x ' k[G.].

1=T

'

This is analogous to the C-M property, but differs in that the Gi's change. The proof is combinatorial, and uses the shellability of convex polytopes (due to Bruggesser and f1ani). The proof is sketched in [St ll , §5]. 5.14 EXAMPLE.

For the equation xl

+

x2 - x3 - x4

o we

Here xl x3 corresponds to the solution (1,0,1,0) as usual, and the geometry of the cone of solutions after triangulation is

get

43

Geometrically,

\~e

have taken all integer points in cone A, and

"pushed off" cone B from its intersection with A by translation by

(0,1,0,1).

§6.

Local cohomology ~ie

now turn to the proof that RIP i s Cohen-/1acaul ay, and more

generally to the question of deciding depth 11",

""Cl

use the tool of local cOhomology,

~Ihic:l

always, all rings Rand modules

are graded.

I~

•

For this we shall

will first be reviewed.

As

Local cohomology will be

defined with respect to the irrelevant ideal R+. Let LR (r1) +

= L(M) = {u

E

/1IR~U =

then L(f) : L(r1) + L(N) by restriction. a left-exact additive functor, so

R\.

6.1

DEFINITION.

Hi(M)

\~e

Hi (M) R+

° for some n

>

O}.

If

f:11+N,

It is easy to check that L is

can take the right-derived functors

= R\R

+

(M).

Some of the fundamental properties of local cohomology Hi(M) will nO~1

be stated.

In particular, the reader unfamil iar or unenamored with

homological algebra can adopt the of Hi(M).

follo~Jing

theorem as the definition

In the following R denotes R localized at Yi (i.e., with Yi

respect to the multiplicative set generated by y,.), and M Yi 6.2

THEOREM.

M®R R Yi

44

where Yl'Y2"" 'Y s The complex

H(R+) and Rad(Yl , ... 'Ys) = R+.

E

s

Q! (0 .... R .... R .... 0) ® ~1 is denoted K(y'" ,r1). i=l Yi cohomology is depth sensitive in the following sense.

6.3 TlIEORH1. He(M) t 0, Hd(M) t

Local

HiUn = 0 unless e = depth r·1,; i ,; dirn N = d; and

o.

Theorem 6.2 imposes on Hi (rl) in a natural way the structure of a 7l m-graded module: Hi (1) = 1L Hi(~1) . Since Hi (1) is known to be Cl E Zln Cl artinian, it follows that Hi(M)Cl = 0 for Cl »0. usually not finitely-generated. Define

6.4 THEOREM.

F(M,A)", =

d

I

i=e

.

However, Hi(M) is

.

(_1)lF(H1(M),A)

In this formula F(M,A) signifies that F(r.1,A) is to be expanded as a Laurent series around "'. F~r instance, for R = k[x], Hl(R) = x-lk[x- l ] 1

A- l

n

1

and F(R,A) = T=i = - -------1 = - I A = -F(H (R),A). Let deg F(R,A) l-A n< 0 denote the degree as a rational function, i.e., degree of numerator minus degree of denominator. 6.5 COROLLARY. then deg F(R,A) < o.

If m =

and Hi(R) Cl = 0 for all Cl

>

0 and all i

'

Proof. Write (uniquely) F(R,A) = G(R,A) + L(R,A) where deg G(R,A) < 0 and L(R,A) E 7l[A]. The expansion G(R,A) has only negative exponents, while L(R,A) = L(R,A). Hence if Hi(~) = 0 for Cl > 0 and all '" Cl i, then by the previous theorem F(R,A) = G(R,A). 0 The condition deg F(R,A) < 0 is equivalent to saying that the Hilbert function H(R,n) has no "exceptional" values. E.g., if R is generated by Rl then H(R,n) is a polynomial for all n ? O.

45

§7.

Local cohomology of the modules

M~

'>',a

.

Let be an r x n ?Z-matri x and a E?Zn as before, and reca 11 the definitions of E, E,a' R and r',a' Let E denote the group generated by E in ?Zn, and let E,a := E + E,a' the coset of E in ?Zn containing E . The nonnegative real solutions to S = 0 form a ,a + n convex polyhedral cone C = {S E (IR) Is = O}. Any cross-section of C' i.e., bounded intersection with a hyperplane meeting the relative interior of C' is a convex polytope, and different cross-sections are combinatorially equivalent. If F is a face of the cross-section polytope P, define supp F := supp S for any S E F - aF, where as before sUPP(Sl""'Sn) = {ilsi > O}. He now try to compute the local cohomology of R by considering the complex K(,t"',r1), r'1 = r',~ . Recall the set of completely funda"',a mental solutions CF = {ol ,0 2 "" ,os}, consisting of the integer points nearest the origin on the extreme rays of C' Thus, CF is in one-to-one correspondence with the vertices of the polytope P' Set Yi = x SEE",

o·

1

""a

o+

ER. I'Je know by Theorem 3.7 that Rad(Yl""'Ys) = R+. consider the part of K(l.'" ,~1) of degree S: Mß +

11 lM ) i

Yi S

+

11

i such that y > 0, then for ß E E : r ß = (J ß < 0 (strict inequality in all coordinates). Hence, for et = 0 there is the follo~/ing "reciprocity" result (when E n lP n ~ (J): R is spanned by nonnegative solutions to 8 = 0, and Hd(R) is spanned by strictly negative solutions to ß = O. In general, Hd(M,et} ,; k{xßIß structure given by

=1

y+ß

x

xy • xß

0

,

E

E,et ' r ß

if r Y+ß

= (J},

with the R-rnodule

= (J

otherwise ,

7.6 COROLLARY. (a) (Hochster [H l ]) R is Cohen-Macaulay, and (b) for any specialization of R to an ~~grading, deg F(R,y) < O.

48

Proof. (a) He must sho~1 that if ß E E

: M.... EA(k) has degree

57

MVV = M, the A+-adic completion of M.

Fact 2.

12.2 EXAMPLE. Let A = k[x,y], R = A/(xy), deg x = (1,0) and deg y = (0,1). A homogeneous k-basis for RV consists of those ~ : R + k[x-l,y-l] such that ~(l) = 1 or ~(l) = x- n , n > 0, or ~(l) = y-n, n > 0, since we cannot have negative exponents of both x and y appear in the image of the element 1. Thus,

F(RV,A)

=

L

1 +

l

n>O F(R,A)

L

1 +

2

(A n + A n)

,and

(A~ + A~)

n>O The functors Ext and v are related to local cohomology by the following remarkable result [Ha, §6]. 12.3 LOCAL DUALITY THEOREM.

Extl(M,A)v

Hs-i(M)

Let M be a Cohen-Macaulay module of dimension d \·lith a minimal free resolution

.t

Let g(M) = coker = At / im .t = Ext:-d(M,A). Equivalently, g(M) is the unique finitely-generated R-module whose completion ~(M) g(M) SR R is isomorphie to HdU~)v. Then

° (4) is an exact sequence, because C-M-ness ensures that Hi(M) t °only for o-

.*

AÜ....J........ Ai

----+ ...

~*

---.!- At

°

----+ g(M)

~

i = d, hence by local duality Extl(M,A) t only for i = s - d = hdAM t. In fact, (4) is a minimal free resolution of g(M). g(M) is

=

58

ca 11 ed the canoni ca 1 modul e of M, and it can be shovtn di rectly that as an R-module n(M) is independent of A. It is seen from (4) that the Betti numbers of nU']) are the reverse of those of 11:

n(M) has a natural llm_grading such that F(n(M),A) = (-l)dF(M,l) as rational functions. In the following table \~e record the vJay that the Hilbert series varies with the fundamental modules associated with the Cohen-Ilacaulay module 11. The subscripts 0 and "" signify expansion of a rational function around the origin and infinity respectively. Module

Hil bert series

11

F(rl,A)O

= L hell Cl

F(tl

,tt

=

L

hClA- Cl

Cl

(_l)dF(tl,t)O =

L huA u Cl

(-l)dF{M,A)""

= L ~uA-u a

Define the soele of a module M by soc M : = {u follows from noetherianness that'dim k soc M< "".

E

MIR+u = O},

It

12.4 THEORH1. Let 11 be a Coilen-f':1acaulay module of dimension d over A = k[xl, ... ,x s ]' Then the fo1101'ling numbers are equal:

(b)

the minimum number of generators of n(rl) (as an A-module or an R-module)

59

Proof. The equivalence of (a), (b) and (c) follO\·/s ties mentioned earlier, such as il(M) = Hd(M)v, etc. (c) straightfon~ard use of tile long exact sequence for local The number just characterized is called the ~ of 12.5 THEOREM. ing are equivalent:

Let R = AII be Cohen-Macaulay.

(a)

type R = 1 ,

(b)

il(R)

~

from properequals (d) by cohomology. 0 M.

Then the follow-

R (up to a shift in grading) .

A Cohen-~lacaulay ring of type one is said to be Gorenstein. Thus a minimal free resolution of a Gorenstein ring is "self-dual". In particular, A Ss- d-1. (R)

12.6 THEOREM.

If R is Gorenstein then for some a

E

ZZm,

F(R,tJ = (_l)d"aF(R,,,) . Proof. F(R,t)O = (_l)dF(il(R),A) = (_l)d"aF(R,,,) If m = 1 and R is Gorenstein with Hilbert series F(R,,,) =

0

hO + hl " + ... + ht"t d

y.

TI (1 - " 1) i=l then by the previous theorem hi = ht _i , i = O,l, ... ,t, and a = t - zYi = deg F = max {jIHd(R)j t O}. Also, if a 2 0 and each Yi = 1 then a is the last value ~Ihere the Hilbert function and the Hilbert polynomial disagree. The converse to the preceding theorem is false. For instance, the ring k[x,y]/(x 3 ,xy,i), deg x = deg y = 1, is Cohen-~lacaulay, artinian and F(R,,,) = ,,2 F(R,tJ, but is not Gorenstein. For a reduced counterexample one can take k[x,y,z,~/]/(xyz,xw,yvl). In the positive direction the following can be said [St 6 , Thm. 4.4].

60 12.7 THEOREM. If R is a Cohen-r1aeaulay domain, then R is Gorenstein .. F(R,A) = (-l)dAaF(R,t) for some a E ~m.

Let us nO~1 reVie\1 a few more faets about eanonieal modules of Cohen-rlaeaul ay rings. The bas i e referenee i s [H-K]. 12.8 THEOREM. >l(R) is isomorphie to an ideal of R -- Rp is Gorenstein for every minimal prime p (e.g., if R is a domain).

If m = 1 we ean obtain an isomorphism >l(R) ; I as graded modules, up to a shift in grading. This is in general false for m > 1. Take R = k[x,y,z]/(xy,xz,yz), deg x = (1,0,0), deg y = (0,1,0) and deg z = (0,0,1). Then R is a C-f'l ring, the loealization at every minimal prime is a field, and >l(R) ; (x-y, x-z), but there is no way of realizing >l(R) as a homogeneous ideal. However, if m > 1 and R is a domain one ean realize >l(R) ; I as graded modules up to a shift in grading. 12.9 THEOREM. lf >l(R) ; I then R/I is Gorenstein and either R or dirn R/I = dirn R - 1. 12.10 THEOREM. Let 6 1 "" ,6 d be an hsop for M and S k[8 1 , ... ,8 d]. Then a(M) ; Homs(M,S).

The isomorphism here is as R-modules. There is a standard 't/ay of making HomS(M,S) into an R-module: if x E R, ~ E HomS(M,S) and u E M, define (x~)(u) = ~(xu).

§13.

A final look at linear equations

We shall now return for the last time to the rings R~ of linear diophantine equations. Reeall that ~ is an r x n ll-matrix of maximal rank, E~ = {ß ElNnl~ß = O} and R~ = kE~, the monoid algebra of E~ over k. The following discussion could be extended to the modules M~,a' but for simplicity \~e eonsider only R~ \~hich is ahlays Cohentlacaulay. IIssume there exists ß E E1> such that ß > O. Recall that Hd (R~) = k{x ßI ~ß = 0, ß < O} and that in general >l(M) = Hd (M) v . A

13.1

COROLLARY. >l(R~) - k{xßIß

E

E~, ß

>

O}.

61

Thus, Q(R~) is isomorphie to an ideal in R~. Sinee R~ is a domain \'ie knOl'i Q(R~) ean in fact be real ized as a graded ideal, and the above eorollary identifies this ideal. 13.2 COROLLJ\RY. R~ is Gorenstein ..... 3 unique minimal ß (i.e. , if y > 0, Y E E~, then y - ß ;:0, 0). E~ 13.3 COROLLARY.

Rq, is Gorenstein if (1,1, ... ,1)

E

>

0 in

Eq,'

The last result has a niee equivalent formulation in terms of invariant theory: if T ~ SLn(k) is a torus aeting on R = k[x l , ... ,x n], then RT is Gorenstein. In this eonneetion we ~/Ould 1ike to mention the follOl·!ing eonjeeture of Hochster, Stanley and others: If G ~ SLn(k) is 1inearly reduetive, then RG is Gorenstein. This is kno~m to be true for finite groups (Hatanabe [WatJ), tori (just shovm) and sef11isif11ple groups (Hochster and Roberts [H-R]). Also, RG is known to be Cohenrlaeaulay for any linearly reduetive G ~ GLn(k) (Hochster and Roberts [H-R]) . Finally, eonsider again the algebra of magie squares. Let E~ be the set of n x n ~-matriees having equal line SUf11S. [ 1;.

11':]

E

Eq,' henee R~ is Gorenstein, henee Hn(r)

(-1)n-1Hn(-n-r).

Conversely, if the last equality is proved eombinatorially, \~hieh ean be done, then F(Rq"A) = (-1)dAaF(R~,t) , whieh by Theorem 12.7 implies that Rq, is Gorenstein sinee Rq, is a domain. The same arguments go through also for symmetrie magie squares.

CHAPTER 11 THE FACE RING OF A SIMPLICIAL COrWLEX

§1.

E1ementary properties of the face ring

Let I::. be a finite simp1 icia1 comr1ex on t:,e vertex set V = {xl , ... ,x n}· Recall that t:,is means that I::. is a collection of subsets of V such that F ~ GEI::. - F E I::. and {xi} E I::. for all xi E V. The elements of I::. are ca11ed faces. If F E 1::., then define dirn F := IFI - 1 and dirn I::. := max (dirn F). Let d = dirn I::. + 1 Given any fie1d k we now FE I::. define the face ring (or Stan1ey-Reisner ring) k[l::.] of the comp1ex 1::.. 1.1

DEFINITION.

I = [x. x .... x. I i 1 u '1'2 'r A

= k[x 1 , ... ,x n]/II::.' where

k[l::.]
O}. a l a2 an Clearly, all monomials u = xl x2 ... xn such that supp u E n form a k-basis for k[n]. By counting such monomials u according to their support F E n we arrive at the following expression for the Hilbert series of the fine grading:

64 F (k[fI] ,A)

Now replace all Ai by A to obtain Theorem 1.4.

§2.

f-vectors and h-vectors of complexes and multicomplexes

What can be said in general about f-vectors of simplicial complexes? There is the following characterization given independently by Kruskal and Katona in response to a conjecture by Schützenberger (see [G-K] for references). Given two integers t,i > 0 write

A unique such expansion exists. Define (ni)

R-(i)

(ni -1)

+

;+1

i

+ ••• + (n j ) j+1 •

2.1 THEORHl (Kruskal, Katona). A vector (fO,fl, ••• ,fd_l ) is the f-vector of some (d-l)-dimensional simplicial complex fI .. 0


d we also write h(r) = (hO, ... ,h d). A sequence (h O,11 1 , ... ) which is the h-vector of some non-void multicomplex r will be called an M-vector. Recall the definition of ~(i), and define in analogy with the earlier notation

2.2 THEOREM (essentially r1acau1ay [rlac]).

fl-vector - 11 0 = 1 and 0 " hi+ 1 "l1 i ' i ~ 1.

(11 0 ,h 1 , ... ) is an

Just as in the case of simplicial complexes, list all monomials of degree i in reverse lexicographic order. E.g., for i = 3:

Given h = (hO,h l , ... ) ~Jith hO = 1, let r h = i~O {first hi monomials of degree i in above order}. To prove Theorem 2.2, one then verifies that the following are equivalent: (i) h is an M-vector (ii) r, is a multicomplex, 1 - k. For n ( ') s 0$ j $ S define a matrix A J , whose rows are indexed by M. and columns by rl s ., by the rule A(j} = a(uv}. Let h. = rank A(j): Then -J uv J (hO,hl,···,h s ) is a Gorenstein sequence (over k) with hl $ n, and all such Gorenstein sequences arise in this way.

80

§7.

Canonical modules of face rings

The next topic will be canonical modules of Cohen-~lacaulay face rings. Recall (Theorem 12.8 of Chapter I) that n(R) is isomorphie to an ideal I sR" R is generically Gorenstein (i.e .• Rp is Gorenstein for all minimal primes p). Now. a face ring k[ö] is generically a field. This suggests the following general problem: Imbed n(k[ö]) as an ideal I of k[ö]. and describe k[ö]/I. Recall that if R is a graded Cohen-rlacaulay algebra of dimension d then F(n(R).") = (_l)dF(R.}) (up to a shift in the grading of n(R». The right-hand side can be explicitly computed for any R = k[ö]. 7.1 THEOREM. Let ö be any (d-l)-dimensional simplicial complex and give k[ö] the fine grading. Then (_l)dF(k[Ö].}) =

I FEÖ

Let a = (al •...• a n)

E

~n and F = {xi!ai

>

O}

E

ö.

The coefficient of

(-l)!F!+!G'!=(_l)d+!F!-l -(lk "a is (_l)d E (_l)!G! = (_l)d E GEÖ G'Elk F x G.=F

n.O

81

7.2 COROLLARY. void). Then

Let

I~I

be a

~anifold

(_l)dF(k[~],t) = (_l)d-lx(~) +

with boundary (possibly

L FE~-a~

Ft