Journal of Algebra 244, 706᎐736Ž. 2001 doi:10.1006rjabr.2001.8932, available online at http:rrwww.idealibrary.com on
Local Cohomology of Stanley᎐Reisner Rings with Supports in General Monomial Ideals
Victor Reiner, Volkmar Welker, and Kohji Yanagawa
View metadata, citation and similar papers at core.ac.uk brought to you by CORE School of Mathematics, Uni¨ersity of Minnesota, Minneapolis, Minnesota 55455 E-mail: [email protected] provided by Elsevier - Publisher Connector
Fachbereich Mathematik und Informatik, Philipps-Uni¨ersitat¨ Marburg, 35032 Marburg, Germany E-mail: [email protected]
Department of Mathematics, Graduate School of Science, Osaka Uni¨ersity, Toyonaka, Osaka 560, Japan E-mail: [email protected]
Communicated by Craig Huneke
Received January 2, 2001
i wx⌬ ᎐ We study the local cohomology modules HkI Ž . of the Stanley Reisner ring wx ⌺ wx k ⌬ of a simplicial complex ⌬ with support in the ideal I⌺ ; k ⌬ corresponding to a subcomplex ⌺ ; ⌬. We give a combinatorial topological formula for the multigraded Hilbert series, and in the case where the ambient complex is Goren- stein, compare this with a second combinatorial formula that generalizes results of Mustata and Terai. The agreement between these two formulae is seen to be a disguised form of Alexander duality. Other results include a comparison of the local cohomology with certain Ext modules, results about when it is concentrated in a single homological degree, and combinatorial topological interpretations of some vanishing theorems. ᮊ 2001 Academic Press Key Words: Stanley᎐Reisner rings; local cohomology modules; Alexander duality; Lichtenbaum᎐Hartshorne vanishing theorem; Gorenstein complex.
1. INTRODUCTION
i [ i r j The local cohomology module HRI Ž. limª Ext RŽ R I , R. of a Noetherian commutative ring R with support in aŽ. nonmaximal ideal I is still a very mysterious object. Recently there have been several instances where more explicit information on local cohomology modules was ob-
706 0021-8693r01 $35.00 Copyright ᮊ 2001 by Academic Press All rights of reproduction in any form reserved. LOCAL COHOMOLOGY OF STANLEY-REISNER RINGS 707 tained in special cases. One such case is a combinatorial approach to the local cohomology of affine semigroup ringsŽ. e.g., polynomial rings and at their monomial idealsŽ seewx 12, 14, 16, 18, 20, 21. . In this paper we construct an analogous theory for a Stanley᎐Reisner ring kwx⌬ using combinatorial topology. s wx ⌬ : Let S kx1,..., xn be a polynomial ring over a field k. Let 2Ä1,...,n4 be a simplicial complex and let ⌺ be a subcomplex of ⌬. Then the Stanley᎐Reisner ideal I⌬ of ⌬ is contained in the Stanley᎐Reisner ideal wx I⌺⌺of ⌺. We denote the image of I in k ⌬ s SrI⌬by J. We study the i wx⌬ ޚ n local cohomology module HkJ Ž ., making use of its natural -grading. Hochster gave a famous formula for the ޚ n-graded Hilbert function of i wx wx wx Hkᒊ Ž ⌬ ., and Terai 18 and Mustat¸ˇa 16 gave a similar formula for i Ž. HSI⌺ . Our result, Theorem 3.2, is a generalization of both formulas. wx i Ž. wx⌬ Most results of Mustat¸ˇa 16 for HSI⌺ remain true for Gorenstein k . In Theorem 4.9, we prove that i wx⌬ r wx⌬ : i wx⌬ r w2x wx⌬ : иии : i wx⌬ Ext kw⌬ x Ž.k J, k Ext kw⌬ x Ž.k J , k HkJ Ž., and these inclusions are well regulated if we consider the ޚ n-grading, where J w jx is the jth ‘‘Frobenius power’’ of J.IfŽ kwx⌬ is not Gorenstein, a somewhat weaker result holds.. A general commutative ring R and an ideal I usually do not have such a simple property. There is one big difference betweenwx 16 and our case. In wx 16 , it is i r i proved that Ext SJŽ.S J, S ‘‘determines’’ HSŽ..Inourcase, i wx⌬ r w2x wx⌬ i wx⌬ i wx⌬ r wx⌬ Ext kw⌬ xŽk J , k . determines HkJkŽ ., but Ext w⌬ xŽk J, k . i cannot. The reason is that HSJ Ž.is a straight moduleŽ after suitable i wx⌬ degree shifting. , but HkJ Ž . is only quasi-straight; see Section 2. The paper is structured as follows. In Section 2 we prove some basic i wx⌬ properties of HkJ Ž .Ž. In Section 3 we give a topological formula Theo- ޚ niwx⌬ rem 3.2. for the -graded Hilbert function of HkJ Ž .. As an applica- tion, we give a topological proof and interpretation of the Lichtenbaum᎐ Hartshorne vanishing theorem for Stanley᎐Reisner ringsŽ. Theorem 3.5 . In wx⌬ ᎐ i Section 4 we assume that k is Cohen Macaulay and study HJkŽ.w⌬ x wx⌬ for the canonical module kw⌬ x of k , which is in some sense much i wx⌬ easier to treat than HkJ Ž .. In Section 5 we use some of the results from i Section 4 to give a topological formulaŽ. Theorem 5.1 for HJkŽ.w⌬ x .Of wx⌬ wx⌬ course, when k is Gorenstein, so that k and kw⌬ x are isomorphic up to a shift in grading, this formula is equivalent to that of Section 3. But the equivalence is not trivial and is derived from a variant of Alexander duality Ž.Lemma 6.8 . The final section is an appendix in which we collect the tools and the terminology from combinatorial topology used in Sections 3 and 5. i wx⌬ We hope that our results will motivate further study on HkJ Ž . and provide good examples for the general theory of local cohomology modules. 708 REINER, WELKER, AND YANAGAWA
2. BASIC PROPERTIES
s wx Let S kx1,..., xn be a polynomial ring over a field k. Consider an ގn s [ s [ a a s n ai -grading S a g ގ nnSa a g ގ kx , where x Ł is1 xi is the s monomial with exponent vector a Ž.a1,...,an . We denote the graded ᒊ maximal ideal Ž.x1,..., xn by . ޚ n s [ Let M be a -graded S-module; that is, M a g ޚ n Ma as a k-vector : g ޚ nng ގ space and SMbaM aqb for all a and b . Then MŽ.a denotes s ޚ n the shifted module with MŽ.a baM qb. We denote the category of - graded S-modules and their ޚ n-graded S-homomorphisms by *Mod. Here ª ޚ n : g ޚ n we say that f: M N is -graded if fMŽ.aaN for all a .We denote the category ofŽ. usual S-modules and their S-homomorphisms by Mod.
In this paper, Hom SŽ.M, N always means Hom ModŽ.M, N , even if M and N belong to a subcategory of Mod.If M, N g *Mod and M is finitely i generated, then Hom SSŽ.ŽM, N more generally, Ext Ž..M, N has a natural ޚ n s wx -grading with Hom SŽ.M, N a Hom*ModŽM, N Ža .. Žsee 8. . Similarly, if i J is a monomial ideal, then a local cohomology module HMJ Ž.for M in *Mod is also ޚ n-graded. For a g ޚ n, define the following support subsets of wxn [ Ä41, 2, . . . , n :
[ N ) suppq Ž.a Ä4i ai 0 [ N - suppy Ž.a Ä4i ai 0 [ N / suppŽ.a Ä4i ai 0.
We extend this terminology to Laurent monomials x a in the obvious way; a that is, suppŽ x .Ž.[ supp a , and similarly for suppqy , supp . We use a q Ä4i q g ޚ n g ޚ n to denote a eiifor the ith unit vector e . When a satisfies s s ai 0, 1 for all i, we sometimes identify a with F suppŽ.a . For example, we write MF for Ma. Žwx. ޚ n s [ DEFINITION 2.1 19 . A -graded S-module M a g ޚ n Ma is called square-free if the following conditions are satisfied: ގn s / ގn Ž.a M is finitely generated and -gradedŽ i.e., Ma 0ifa .. 2 ¬ g Ž.b The multiplication map Ma y xyi MaqÄi4 is bijective for g wx s g ގn / all i n and all a Ž.a1,...,aniwith a 0.
EXAMPLE 2.2. A free module SŽ.yF is square-free for all F : wxn .In ޚ n s y particular, S itself and the -graded canonical module S SŽ.1 are square-free. Here y1 syŽ.1,...,y1. Let ⌬ : 2w nx be a simplicial complexŽ i.e., if F g ⌬ and G ; F, then G g ⌬.. The Stanley᎐Reisner ideal of ⌬ is the square-free monomial ideal LOCAL COHOMOLOGY OF STANLEY-REISNER RINGS 709
F I⌬ [ Ž x N F f ⌬. of S. Any square-free monomial ideal is I⌬ for some ⌬. wx We say that k ⌬ s SrI⌬ is the Stanley᎐Reisner ring of ⌬. Stanley᎐Reisner ideals and rings are always square-free modules.Ž Seewx 19. 20 for more information.. Let ⌺ : ⌬ : 2w nx be two simplicial complexes. Then we have Stanley᎐Reisner ideals I⌬⌺: I . We always assume that ⌺ is not the void simplicial complex л; that is, we assume that I⌺ / S, while we allow the case ⌺ s Ä4л Ž.i.e., I⌺ s ᒊ . Throughout this paper, we denote the image wx of I⌺⌺in k ⌬ by J. Sometimes we also denote I itself by J. For a wx k ⌬ -module M, the ideal I⌬ lies in the kernel of the action of S on the i Ž. local cohomology module HMI⌺ of M regarded as an S-module. Thus i Ž. wx⌬ HMI⌺ can be regarded as a k -module, which is known to be isomor- i wx⌬ phic to the local cohomology module HMJ Ž.of M as a k -module. Therefore, the convention that we have chosen is harmless. ޚ n s [ DEFINITION 2.3. A -graded S-module M a g ޚ n Ma is called straight if the following two conditions are satisfied: - ϱ g ޚ n Ž.a dim k Ma for all a . 2 ¬ g Ž.b The multiplication map Ma y xyi MaqÄi4 is bijective for g wx g ޚ n / all i n and all a with ai 0. : wx EXAMPLE 2.4. For a subset F n , let PF denote the monomial prime N f r r ideal Ž.xiFi F of S. The injective hull ESŽ.P of S PFin the category *Mod is a straight module. In fact,
k if suppq Ž.a : F Ž.2.1 ESŽ.rP s F a ½ 0 otherwise,
r 2 ¬ b g r and the multiplication map ESŽ.PF a y xy ESŽ.PF aqb is always wx r surjectiveŽ see 14, Example 1.1.Ž. . Note, that ES PF is not injective in s ᒊ ⅷ Mod unless PF . Nevertheless,Ž. 2.1 is still useful. For example, if E ޚ ni is an injective resolution of a -graded module M in *Mod, then HMJ Ž. i ⌫ ⅷ ⌫ for a monomial ideal J is isomorphic to H ŽŽJJE .., where is the ⌫ s g N m s 4 endofunctor on *Mod defined by J Ž.M Ä x M Jx 0 for m 0.4 wx i s In 20 , Yanagawa proved that a local cohomology module HJSŽ. i y HSJ Ž.Ž1 .is a straight module for any monomial ideal J. But in general, i wx⌬ HkJ Ž . is not straight even after a degree shift. Thus we must introduce a new concept. ޚ n s [ DEFINITION 2.5. A -graded S-module M a g ޚ n Ma is called quasi-straight if the following two conditions are satisfied: - ϱ g ޚ n Ž.a dim k Ma for all a . 710 REINER, WELKER, AND YANAGAWA
2 ¬ g Ž.b The multiplication map Ma y xyi MaqÄi4 is bijective for g wx g ޚ n / y all i n and all a with ai 0, 1.
EXAMPLE 2.6. Square-free modules and straight modules are always r quasi-straight. The injective module ESŽ.Ž.PF a is quasi-straight if and s f r only if aiF1, 0 for all i F.If ESŽ.Ž.P a is quasi-straight, then it can r : wx l s л be written as ESŽ.Ž.PGF for some G n with F G . Note that w r xws : = r x ESŽ.Ž.PGF a k if suppqya F and supp a G, and ESŽ.Ž.PGF a s 0 otherwise. s Let M be a quasi-straight module. Then we have dim k Ma dim k Mˆa g ޚ n g ޚ n for all a , where the ith component aˆˆi of a is defined by ¡ G 1ifai 1 s~ s aˆi 0ifai 0 ¢y Fy 1ifai 1.
Hence, to know the Hilbert function of a quasi-straight module M,itis g ޚ n gy enough to know the dimensions dim k Ma for a with ai Ä41, 0, 1 . A quasi-straight module M is finitely generated as an S-module if and only if it is ގn-graded. Of course, M is square-free in this case. We denote the full subcategory of *Mod consisting of quasi-straight modules by qStr. One can easily check that this is an abelian subcategory of *Mod closed under extensions and direct summands. For further study of qStr, the concept of the incidence algebra of a finite poset is very useful. For the reader’s convenience, here we recall the basic properties of the incidence algebra. Let P be a finite poset. The incidence algebra A s IPŽ., k of P over k N g F is the k-vector space with basis Ä4ex, y x, y P, x y and multiplication s ␦ defined k-linearly by eex, yz, wy, zxe , wxx. We write e for e , x. Then A is a s finite-dimensional associative k-algebra with 1 Ý x g Pxe . Note that ee xy s ␦ x, yxe . s [ If M is a right A-module, then M x g P Mex as a k-vector space. We ª write Mxxfor Me .If f: M N is an A-linear map of right A-modules, : : s / then fMŽ.xxN . Note that Mexx, yyM and Me xy, z 0 for y x. For each x g P, we can construct an injective object ExŽ.in the category mod A of finitely generated right A-modules. Let ExŽ.be a N F k-vector space with basis Ä4e y y x . Then we can regard ExŽ.as a right A-module by
e if y s z and w F x e и e s w yz, w ½ 0 otherwise. w xws F x s Note that ExŽ.yyke if y x, and ExŽ.y0 otherwise. LOCAL COHOMOLOGY OF STANLEY-REISNER RINGS 711
¨ PROPOSITION 2.7. The category mod A has enough projecti es and enough injecti¨es. An indecomposable injecti¨e object is isomorphic to EŽ. x for some x g P, and any injecti¨e object is a finite direct sum of the copies of EŽ. x for ¨arious x. Proof. Since A can be regarded as an algebra associated with a qui¨er with relation Žseewx 1. , the assertions follows fromwx 1, Proposition 1.8 and the arguments inwx 1, pp. 61᎐62 . ޚ n $ F We partially order by setting Ž.Ž.a1,...,an ᎏ b1,...,bniiif a b for all 1 F i F n. Denote by 3w nx the subposet of ޚ n w nx [ g ޚ n Ny F F 3 Ä4a 1 ai 1 for all i . w x w x LEMMA 2.8. Let A [ IŽ3,n k. be the incidence algebra of the poset 3 n ¨ ¨ ( o er k. Then there is an equi alence of categories mod A qStr. g s [ ( Proof. For N mod A, let M a g ޚ n Maaabe a k-vector with M Nˆ for each a g ޚ n. Then M has an S-module structure such that the 2 ¬ b g 2 ¬ multiplication map Maay x y M qbaais induced by Nˆˆy e , Žaqb. и g wx y NŽaqb.. By an argument similar to the proof of 21, Theorem 3.2 , we 2 ¬ can see that M is quasi-straight and the correspondence mod A N M g ( qStr gives a category equivalence mod A qStr. Note that inwx 21 , the author used the term ‘‘sheaf on a poset’’ instead of the equivalent concept of ‘‘module over the incidence algebra of the poset.’’
PROPOSITION 2.9. The category qStr has enough projecti¨es and enough injecti¨es. An indecomposable injecti¨e object in qStr is isomorphic to r : wx l s л ESŽ.Ž.PF G for some F, G n with F G . Proof. Let A s IŽ3,w nx k. be the incidence algebra of 3w nx over k. By the ª functor mod A qStr constructed in Lemma 2.8, the injective object EŽ.a , g w nx r s s a 3,inmod AFis sent to ESŽ.Ž.PG, where F suppqŽ.a and G suppyŽ.a . Because of the explicit description of the indecomposable injectives in qStr, we know that an injective object in qStr is also injective in *Mod. Hence we deduce the following.
PROPOSITION 2.10. Let M be a quasi-straight module and let E ⅷ be a minimal injecti¨e resolution of M in *Mod. Then each E i is quasi-straight. Proof. Take a minimal injective resolution of M in qStr. Then this is isomorphic to E ⅷ.
THEOREM 2.11. Let J be a monomial ideal of S. If M is quasi-straight, i G then so is the local cohomology module HJ Ž. M for all i 0. 712 REINER, WELKER, AND YANAGAWA
Proof. Let E ⅷ be a minimal injective resolution of M in *Mod.By Proposition 2.10, each E i, i G 0, is a direct sum of finitely many copies of r : wx l s л quasi-straight modules ESŽ.Ž.PGF for F, G n with F G . Note that ESŽ.Ž.rPG if P = J Žequivalently, F g ⌺. ⌫ Ž.ESŽ.Ž.rPGs FF JF ½ 0 otherwise. G ⌫ i i Thus, for each i 0, J Ž E . is a direct summand of E and a direct sum of r g ⌺ ⌫ i the finitely many copies of ESŽ.Ž.PGFJwith F . In particular, ŽE . i s is quasi-straight again. Since qStr is an abelian category, HMJ Ž. i ⌫ ⅷ H ŽŽJ E ..is quasi-straight. ⌫ s 0 Remark 2.12. If M is quasi-straight, then so is JJŽ.M HMŽ.. ⌫ y i y Hence JJŽ.is an endofunctor on qStr, and H Ž.is its right-derived functorŽ. on qStr . Since a Stanley᎐Reisner ring is square-free, and hence also quasi- straight, we have the following. wx COROLLARY 2.13. Let k ⌬ be a Stanley᎐Reisner ring, and let J = Ibe⌬ i wx⌬ another monomial ideal of S. Then HJ Ž k . is quasi-straight. In particular, i wx⌬ ( i wx⌬ HkJJŽ.aaHk Ž.ˆ. i wx The following is a generalization of the well-known fact that if Hkᒊ Ž ⌬ . is finitely generatedŽ. or, equivalently, if it is of finite length , then i wx ᒊ Hkᒊ Ž ⌬ . s 0. i COROLLARY 2.14. In the situation of Theorem 2.11, if HJ Ž. M is finitely i s generated as an S-module, then JHJ Ž. M 0. i Proof. Since HMJ Ž.is quasi-straight and finitely generated, it is si s 4 square-free. On the other hand, JHJ Ž. M 0 for some s 0, since i i i HMJJJŽ.is finitely generated. Since HMŽ.is square-free, we have JHŽ. M s 0. i wx⌬ Moreover, by Theorem 4.5 in Section 4, if HkJ Ž . is finitely generated i wx⌬ as an S-module, then HkJ Ž . is isomorphic to a submodule of i wx⌬ r wx⌬ Ext kw⌬ xŽk J, k .. Remark 2.15. What we have called a quasi-straight module is essen- tially the same as a 2-determined module in Millerwx 14 , where 2 s Ž.2,...,2 g ގn. More precisely, M g *Mod is quasi-straight if and only if MŽ.y1 is 2-determined. Millerwx 14 also defined a-determined modules for each a g ގn. For readers familiar withwx 14 , it is not hard to check that our methods can be used to show that the category of a-determined modules has enough injectives, and that injectives in this category are also injective LOCAL COHOMOLOGY OF STANLEY-REISNER RINGS 713 in *Mod. Thus, for any monomial ideal J, the local cohomology module of i HMJ Ž.of an a-determined module M is a-determined again. Inwx 14 , Miller also defined positi¨ely a-determined modules for each a g ގn. A positively a-determined module is always finitely generated, and a positively 1-determined module is the same thing as what we call a square-free module. If M is positively a-determined, then the shifted y q i module MŽ.1 is Ža 1 .-determined. Thus HMJ Ž.of a positively a- determined module M is Ž.a q 1 -determined Ž after degree shifting by y1..
3. A TOPOLOGICAL FORMULA FOR THE HILBERT FUNCTION
In this section we give aŽ. combinatorial topological formula for the ޚ n i wx⌬ -graded Hilbert function of the local cohomology module HkJ Ž ., following Hochster. We use the fact that it may be computed via a Cechˇ complex of localizations of kwx⌬ whose boundary map is similar to the simplicial boundary map. w nx As before, ⌬ : 2 is a simplicial complex, ⌺ a subcomplex, and I⌬⌺: I wx are their associated Stanley᎐Reisner ideals, with the image of I⌺ in k ⌬ wx⌬ denoted by J. Let m12, m ,...,m denote the images within k of the square-free monomials corresponding to the faces of ⌬ y ⌺ which are minimal with respect to inclusion, so that J is generated by m12, m ,...,m. g wx⌬ wx⌬ wx⌬ For f k , let k f denote the localization of k at the multiplica- n tive set Ä f 4nG 0. ⅷ i wx⌬ The Cechˇˇ complex C computing HkJ Ž . is the complex ª wx⌬ ª [[ wx⌬ ª wx⌬ 0 k k mmttk mt F F 11F - F 2 1 t111 t t2
ª иии ª wx⌬ иии ª k m1 m 0, where for ˇi s [ wx⌬ C k Ł t g Ttm T:wx , < PROPOSITION 3.1. wwk ⌬ x x is at most one-dimensional as a k-¨ector Ł t g Ttm a space and is non¨anishing exactly when both : Ž.i suppy Ža .supp ŽŁ t g Ttm . and j g ⌬ Ž.ii supp ŽŁ t g Ttm .suppq Ž.a . The nonvanishing condition in the proposition has a nice rephrasing, once we have introduced a little terminology. Regard the map s 2w xwª 2 nx ¬s T supp Ł mt ž/tgT as an order-preserving map of Boolean algebras. Then if we regard any simplicial complex on vertex set wxn , such as ⌬, as an order ideal in 2w nx , its inverse image sy1 Ž.⌬ is an order ideal in 2w x and hence a simplicial complex on wx . We also use the notions of stars, deletions, and links of faces in a simplicial complex; seeŽ 6.1 . . Setting Fqqyy[ supp Ž.a , F [ supp Ž.a , Proposition 3.1 may be rephrased as saying that wwk ⌬ x x is one- Ł t g Ttm a y1 dimensional exactly when T is a face of s ŽŽ..star⌬ Fq which does not lie y1 ŽŽ.. in the subcomplex s delstar⌬ Ž Fq. Fy . Consequently, one can check that there is an isomorphism of cochain complexes of finite-dimensional k-vec- y ⅷ tor spaces, up to a shift by 1, between the Cechˇˇ complex Ca and the Ž.augmented simplicial relative cochain complex with coefficients in k for the pair y1 y1 s star⌬ Fq , s del Fy . ž/Ž.Ž. Ž.star⌬ Ž Fq.Ž. This yields the following theorem. THEOREM 3.2. Let ⌺ : ⌬ be simplicial complexes and let a g ޚ, Fqs suppqyya and F s supp a. Then i iy1 y1 y1 3.1 Hkwx⌬ ( Hs˜ star⌬ Fq , s del Fy ; k , Ž. J Ž.a ž/Ž.Ž. Ž.star⌬ Ž Fq.Ž. where H˜ ⅷ Ž.y, y; k denotes simplicial relati¨e reduced cohomology. Equi¨alently, i wx⌬ ( iy1 5555y ⌺ Ž.3.2 HkJ Ž.a H˜ Ž star⌬Ž.Fq , 55y 5⌺5 delstar⌬ Ž Fq.Ž.Fy ; k. , where H˜ ⅷ Ž.y, y; k denotes singular relati¨e reduced cohomology and 55⌬ denotes the geometric realization of a simplicial complex ⌬. LOCAL COHOMOLOGY OF STANLEY-REISNER RINGS 715 Proof. The first equation is immediate from the preceding discussion of the Cechˇ complex. The second is equivalent by Proposition 6.5. Readers concerned with effective computation might be slight disturbed by the use of singular cohomology in Eq.Ž. 3.2 . However, one has also the following equivalent formulation in terms of simplicial cohomologyŽ see Proposition 6.3. : i wx⌬ ( ˜iy1 y ⌺ Ž.3.3 HkJ Ž.a H žSdŽ. star⌬Ž.Fq , Sd del Fy y ⌺ ; k , Ž.star⌬ Ž Fq.Ž. / where SdŽ.⌬ y ⌺ means the subcomplex of the barycentric subdivision Sd ⌬ induced on the vertices which are barycenters of faces not in ⌺. Equivalently, it is the order complex of the poset of faces in ⌬ y ⌺.Ž The order complex of the face poset of ⌬ itself is the cone over SdŽ.⌬ with the vertex л. But since we assume that ⌺ / л, the poset ⌬ y ⌺ does not contain л.. 5 555 If star⌬Ž.Fq y ⌺ s л and i s 0, then the right side of Eq.Ž. 3.2 is a little bit confusing, but its ‘‘correct’’ meaning is given by Eq.Ž. 3.3 . Regard- 0 wx⌬ / s ⌬ s ⌺ less, it is clear that HkJ Ž .Ž.0 if and only if J 0 i.e., . In this 0 wx⌬ s wx⌬ case, HkJ Ž . k . w i wx⌬ x / COROLLARY 3.3. With the foregoing notation, if HJ Ž k . a 0, then Fqqyg ⌺ and F j F g ⌬. f ⌺ ŽŽ..y ⌺ Ž Ž. Proof. If Fq , then both Sd star⌬ Fq and Sd delstar⌬ Ž Fq. Fy y ⌺. are cones over the barycenter of Fq. Thus the assertion follows from Ž. j f ⌬ Ž.s Ž. Eq. 3.3 . If FqyF , then delstar⌬ Ž Fq. Fy star⌬ Fq . Hence the assertion follows from Eq.Ž. 3.1 . i wx wx Hochster’s formula on Hkᒊ Ž ⌬ .Ž17, II, Theorem 4.1. easily follows from Theorem 3.2. Since ⌺ s Ä4л in this case, we may assume that n a gyގ by Corollary 3.3. Then star⌬Ž.Fq s star⌬ Ž.л s ⌬, and we have i wx⌬ ( iy1 555⌬ 5 HkJ Ž.a H˜ Ž., del⌬Ž.Fy ; k iy1 ( H˜ Ž.⌬, del⌬Ž.Fy ; k iy< F COROLLARY 3.4. Assume that there are at most fi¨e minimal faces of ⌬ not lying in ⌺, that is, F 5. Then the dimensions of the graded components ⅷ wx⌬ HkJ Ž .a are independent of the field k. Proof. Pairs of simplicial complexes on at most five vertices have torsion-free cohomology; one does not obtain torsion until one reaches the minimal triangulation of the real projective plane on six vertices. The statement then follows from Eq.Ž. 3.1 and the universal coefficient theo- rem in relative cohomology. i wx⌬ ) s wx⌬ The fact that HkJ Ž . vanishes for i d dim k corresponds to i wx⌬ the fact that Eq.Ž. 3.3 expresses each component HkJ Ž .a in terms of relative cohomology for simplicial complexes of dimension at most d y 1. On the other hand, it is interesting to see what further information the Lichtenbaum᎐Hartshorne vanishing theoremŽ. LHVT inwx 10, Theorem 3.1 for kwx⌬ tells us about the topology of simplicial complexes. THEOREM 3.5Ž. LHVT for Stanley᎐Reisner rings . Let ⌺ : ⌬ be simpli- s wx⌬ s d wx⌬ s cial complexes, with d dim k and J I⌺. Then HJ Ž k . 0 if and only if e¨eryŽ. d y 1-face of ⌬ contains a ¨ertex of ⌺. Although short proofs of the LHVT are known even in the general case, we give a proof of this special case to better understand its topological meaning. Proof. For the forward implication, we show the contrapositive. As- sume that some Ž.d y 1 -face F contains no vertex of ⌺. Then we claim that the multidegree a having Fqq[ supp a s л, F yy[ supp a s F satis- d wx⌬ ( d wx⌬ fies HkJ Ž .a k, and consequently HkJ Ž . does not vanish: d wx⌬ ( dy1 55⌬ y 555⌺ 5y 55⌺ HkJ Ž.a H˜ Ž., del⌬Ž.F ; k ( HF˜dy1 Ž.555, Ѩ F 5; k ( HF˜dy1 Ž.555r Ѩ F 5; k ( H˜dy1 Ž.ޓdy1 ; k ( k, LOCAL COHOMOLOGY OF STANLEY-REISNER RINGS 717 where the first isomorphism is Theorem 3.2, the second isomorphism is excisionŽ we excise away 55⌬ y 55⌺ y 55F and use the fact that F con- tains no vertex of ⌺., and the third isomorphism is Proposition 6.1. For the reverse implication, assume that every Ž.d y 1 -face of the Ž.d y 1 -dimensional complex ⌬ contains a vertex of the subcomplex ⌺. We must show that this implies the combinatorial topological assertion that ˜dy1 5 5y 555⌺ 5y 55⌺ s H Ž.star⌬Ž.Fq , delstar⌬ Ž Fq.Ž.Fy ; k 0 for every pair of disjoint faces Fqy, F of ⌬ with Fq@ F yg ⌬. However, since star⌬Ž.Fq is again a complex of dimension at most d y 1, just as ⌬ was, we can replace star⌬Ž.Fq by ⌬ Žor, equivalently, we may assume that Fqs л.. Setting D [ d y 1, we can then rephrase the combinatorial topological assertion that we must prove as follows: Let ⌬ be a simplicial complex of dimension at most D and let ⌺ be a subcomplex containing at least one vertex from each D-face of ⌬. Then for any face F of ⌬, D 55 555 5 55 H˜ Ž.⌬ y ⌺ , del⌬Ž.F y ⌺ ; k s 0. We actually show that these hypotheses imply that the pair 55 555 5 55 Ž.⌬ y ⌺ , del⌬Ž.F y ⌺ is relatively homotopy equivalent to a pair of subcomplexes of Sd ⌬ of dimension at most D y 1. To do this, we start with the fact from Proposition 6.3 that the foregoing pair is relatively homotopy equivalent to Ž.SdŽ.⌬ y ⌺ , SdŽ. del⌬ Ž.F y ⌺ . The latter is a pair of simplicial complexes of dimension at most D, which we denote by Ž.X, A for short. We proceed to remove all D-faces of X Žor of both X and A. by repeating the following procedure: Whenever we find a D-face of X Ž.Ž.or both X and A that has some D y 1 -face contained in no other D-face of X Žand hence also in no other D-face of A.Ž., we remove both and from X or from both X and A . Each such removal is either an elementary collapse Žseew 2,Ž. 11.1x. on X which affects no simplices of A, or a simultaneous elementary collapse on X and A.In either case, it does not change the relative homotopy type of the pair Ž.X, A . To see that this process will remove all of the D-faces in X Ž.and A , assume the contrary. That is, assume that at some stage after performing 718 REINER, WELKER, AND YANAGAWA some of these collapses, we are left with a pair Ž.XЈ, AЈ having some D-faces in XЈ, but that every such D-face has all of its Ž.D y 1 -faces lying in some other D-face, so that no further collapses are possible. Let be one of the remaining D-faces in XЈ, and let G be the unique D-face of ⌬ in which it lies, so that the barycenter bG of G is one of the vertices of . y Then lk X ЈŽ.bG is a subcomplex of the simplicial Ž.D 1 -sphere ( Ѩ lk Sd ⌬Ž.bG ŽSd G ., having these two properties: Ž.iItis ŽD y 1 . -dimensional, by virtue of the existence of . Ž.Ž.ii A D y 2 -face always lies in exactly two Ž.D y 1 -faces. Ž To y j Ј see this, consider the Ž.D 1 -face bG of X and use the assumption on XЈ.. s These two properties force lk X ЈŽ.bG lk Sd ⌬ Ž.bG . But this contradicts our assumption that G contains at least one vertex of ⌺, so that XЈ : X is missing at least some of the D-faces of Sd G. For the remainder of this section, we regard J, not S, as an ideal of kwx⌬ . Recall that grade J is defined as the length of the longest k wx⌬ - sequence contained in J Žseewx 4, Section 1.2. . It is well knownw 4, Proposition 1.2.14x that grade J F ht J with equality whenever kwx⌬ is Cohen᎐MacaulayŽ seewx 4, Corollary 2.1.4. . It is also knownwx 9, Theorem 3.8 that s N i wx⌬ / grade J minÄ4i HkJ Ž.0. We are interested in the following problem. ⌬ ⌺ i wx⌬ s Problem 3.6. Characterize pairs Ž., such that HkJ Ž . 0 for all i / grade J. If ⌬ s 2Ä1,...,n4 Ž i.e., kwx⌬ s S.Ž., then a pair ⌬, ⌺ satisfies the condition of Problem 3.6 if and only if kwx⌺ is Cohen᎐Macaulay. Similarly, if ⌺ s Ä4л Ž.i.e., J s ᒊ , then the condition of the problem is satisfied if and only if kwx⌬ is Cohen᎐Macaulay. But in the general case, the situation is more delicate. Set dim kwx⌬ s d.If ⌺ / Ä4л Ži.e., J / ᒊ or, equivalently, d ) ht J G grade J .Ž., then all d y 1 -faces of ⌬ must contain a vertex of ⌺ to satisfy the condition of Problem 3.6, by Theorem 3.5. Assume that kwx⌬ is Cohen᎐Macaulay and that dim ⌺ s 0ŽŽ i.e., dim kwx⌬ rJ ..s 1 . Then LOCAL COHOMOLOGY OF STANLEY-REISNER RINGS 719 s y i wx⌬ s - y grade J d 1, and hence HkJ Ž . 0 for all i d 1. So in this case, the condition of Problem 3.6 is satisfied if and only if every facet of ⌬ contains a vertex of ⌺. We return to this problem in Section 5. 4. LOCAL COHOMOLOGY MODULES OF THE CANONICAL MODULE The goal of this section is to understand, in the case where kwx⌬ is ᎐ i Cohen Macaulay, how the local cohomology HJkŽ.w⌬ x is filtered by the modules i wx⌬ r w jx G Ext kw⌬ x Ž.k J , kw⌬ x for j 1, where J w jx denotes the jth Frobenius power of the ideal J Žsee Definition 4.4.Ž . The main result is Theorem 4.9, which shows in particular Corollary i i wx⌬ r w2x 4.10. that HJkŽ.w⌬ x is determined by Ext kw⌬ xŽk J , kw⌬ x.. ޚ n wx⌬ Denote the category of -graded k -modules by *Mod kw⌬ x, so that s *Mod kw⌬ x is a full subcategory of *Mod S *Mod. We review some facts about this category which may be found inwx 8, Chapters 1 and 2 . g ⌬ = If F , then PF I⌬. In this case we regard PF as aŽ. prime ideal of wx⌬ s r k S I⌬. The category *Mod kw⌬ x has enough injectives, and an inde- wx⌬ r composable injective object in *Mod kw⌬ x is of the form EkŽ PF .Ža . for some F g ⌬ and a g ޚ n, where wx⌬ r s wx⌬ r EkŽ.ŽPFSHom k , ESŽ.P F . s g r N s g Ä4y ESŽ.PF zy 0 for all z I⌬ wx⌬ r s r is the injective envelope of k PFFS P in *Mod kw⌬ x.Ž Throughout wx⌬ r this paper, we use the convention that EkŽ PF . means the injective wx⌬ r envelope of k PFkin *Mod w⌬ x, not in *Mod S.. We can easily compute wx⌬ r the structure of EkŽ PF .. LEMMA 4.1. We ha¨e kifsuppqyŽ.a : F and supp Ž.a j F g ⌬ EkŽ.wx⌬ rP s F a ½ 0 otherwise, wx⌬ r 2 ¬ b g wx⌬ r and the multiplication map EŽ k PF .a y xy EkŽ PisF .aqb always surjecti¨e. Proof. Since I⌬ is square-free, wx⌬ r s g r N G s f ⌬ EkŽ.PFFÄ4y ESŽ.P xy 0 for all G . 720 REINER, WELKER, AND YANAGAWA / g w r xwx: G / q For 0 y ESŽ.PF a and G n , xy 0 if and only if suppqŽa G.Ž: F. These conditions are also equivalent to G : suppy a.j F. Thus wx suppyŽ.a j F is the largest subset of n whose monomial does not annihilate y. Now the assertion follows from an easy computation. Let M be a ޚ n-graded S-module. For any multidegree a g ޚ n, define [ [ M%ᎏ abM , b%ᎏ a syŽ.y a graded submodule of M. When a j,..., j , we simply denote M%ᎏ a n s [ n ގ by M%yᎏᎏj. In particular, we say that M% 0aa g ގ M is the -graded part of M.If M is quasi-straight, then its ގn-graded part is square-free. LEMMA 4.2. If M is quasi-straight, then it is a *essential extensionŽ see w4, x. G Section 3.6 of any of its submodules M%yᎏ j for j 1; that is, for any homogeneous element 0 / y g M, there is some z g S such that 0 / zy g M%yᎏ j. / g s N - y Proof. For a homogeneous element 0 y Ma , set T Ä4i ai 1 y y [ 1 a i g / g and z Ł ig Tix S. Since M is quasi-straight, we have 0 zy : M%yᎏᎏ1 M%yj. LEMMA 4.3. Let j G 1 be an integer. If M is a quasi-straight module, then [ N M%yᎏ j satisfies the following conditions: - ϱ g ޚ n Ž.a dim k Na for all a . 2 ¬ g ¨ Ž.b The multiplication map Na y xyi NaqÄi4 is bijecti e for all g wx %y / y i n and all a ᎏ j with ai 0, 1. ¨ ޚ n s Con ersely, if a -graded S-module N with N N%yᎏ j satisfies conditions Ž. Ž. ( a and b , then there is a quasi-straight module M with M%yᎏ j N. Such an M is unique up to isomorphism. Proof. Straightforward. DEFINITION 4.4. As in the previous sections, J is the image of a wx G i monomial ideal I⌺⌬Ž.= I of S in k ⌬ . We denote the ideal ŽŽ x . N G f ⌺. of kwx⌬ by J wixw. More precisely, J ix is the natural image ŽŽ x G .i N G f ⌺. q wx wixw1x I⌬ in k ⌬ . We call J the ith Frobenius power of J. Note that J s J. THEOREM 4.5. Suppose that the ޚ n-graded kwx⌬ -module M is quasi- straight as an S-module. For all i, j G 0 and yj syŽ.j,...,yj , we ha¨e iiw jq1x HM s Ext w⌬ x kwx⌬ rJ , M . JkŽ.%yᎏ j Ž.%yᎏ j LOCAL COHOMOLOGY OF STANLEY-REISNER RINGS 721 In particular, iiw jq1x Hkwx⌬ s Ext w⌬ x kwx⌬ rJ , k wx⌬ . JkŽ.%yᎏ j Ž.%yᎏ j To prove the theorem, we need the following lemma. n LEMMA 4.6. Let a g ޚ with a %yᎏ j syŽ.j,...,yj . For a monomial s g ⌬ prime ideal P PF with F , w jq1x Hom w⌬ x kwx⌬ rJ , Ek wx⌬ rP k Ž.Ž.%ᎏ a Ekwx⌬ rPifJ% : P s Ž.ᎏ a ½ 0 otherwise. In other words, w jq1x Hom w⌬ x kwx⌬ rJ , Ek wx⌬ rP s ⌫ Ekwx⌬ rP % . k Ž.Ž.%ᎏ a J Ž. Ž.ᎏ a Proof. For all 0 / y g EkŽ wx⌬ rP.Ž, we have ann y.: P.If J P, then w jq1x wx⌬ r w jq1x wx⌬ r s J P. Thus Hom kw⌬ xŽk J , EkŽ P.. 0 in this case. On g w wx⌬ r x = w jq1x the other hand, for all y EkŽ P. a , we have annŽ.y P by Lemma 4.1. If J : P, then we have J w jq1x : Pw jq1x. Hence J w jq1x y s 0 and g w Ž wx⌬ r w jq1x Ž wx⌬ r ..x y Hom kw⌬ x%k J , Ek P ᎏ a. LEMMA 4.7. Let M be a ޚ n-graded kwx⌬ -module whose ގn-graded part is square-free as an S-module, and let E ⅷ be a minimal injecti¨e resolution of M i G in *Mod kw⌬ x. Then each E , i 0, is a direct sum of the copies of wx⌬ r g ⌬ g ގn EkŽ PF .Ža . for some F and a . Proof. Let MЈ be the ގn-graded part of M. Since MЈ is square-free, Ј Ј the injective envelope EMSSŽ.of M in *Mod is a direct sum of the r wx copies of unshifted ESŽ.PF by an argument in Section 3 of 21 . Since the Ј Ј wx⌬ Ј injective envelope EMkw⌬ xŽ.of M in *Mod kw⌬ x is Hom SSŽk , EMŽ..,it wx⌬ r Ј is a direct sum of unshifted EkŽ PFk.Ž.. By the injectivity of EMw⌬ x , ޚ n ª Ј Ј ¨ Ј we have a -graded map f: M EMkw⌬ xŽ.extending M EMkw⌬ xŽ.. By definition, the ގn-graded part of MЉ [ kerŽ.f is 0. Hence the ގn- Љ Љ graded part of the injective envelope EMkw⌬ xŽ.is 0, and EMkw⌬ x Ž.is a r g ގn _ direct sum of the copies of ESŽ.Ž.PF a for a Ä40 . There is a ޚ n ª Љ Љ ¨ Љ -graded map M EMkw⌬ xŽ.which extends M EMkw⌬ xŽ.. Since 0 s Ј [ Љ E EMkw⌬ xŽ.is a direct summand of EMkw⌬ xŽ.Ž.EMkw⌬ x ,itisa r g ގn direct sum of the copies of ESŽ.Ž.PF a for a . 1 ⅷ r The next term E of E is the injective envelope of EMkw⌬ xŽ.M. Since ގn the -graded parts of M and EMkw⌬ xŽ.are square-free, that of r EMkw⌬ xŽ.M is also square-free. Hence we can apply the foregoing argu- 722 REINER, WELKER, AND YANAGAWA r i G ment to EMkw⌬ xŽ.M. Similarly, we can prove the assertion for E , i 2, by induction on i. Proof of Theorem 4.5. Let E ⅷ be a minimal injective resolution of M in ⅷ wx⌬ r g ގn *Mod kw⌬ x. Then E consists of EkŽ PF .Ža . for a by Lemma 4.7. ޚ n Ž. s Ž. For a -graded module N, we have N a %yᎏᎏj N% ayj a . Hence w jq1x ⅷⅷ Hom w⌬ x kwx⌬ rJ , E s ⌫ E %y k Ž.%yᎏ j J Ž.ᎏ j by Lemma 4.6. The ith cohomology of the complex of the left side is w i Ž wx⌬ r w jq1x .x isomorphic to Ext kw⌬ x%yk J , M ᎏ j, and that of the right side is w iŽ.x isomorphic to HMJ %yᎏ j. w i wx⌬ r Remark 4.8.Ž. 1 In the situation of Theorem 4.5, Ext kw⌬ xŽk w jq1x .x iŽ. G iŽ. J , M %yᎏ j ‘‘determines’’ HMJJfor each j 1, since HMis quasi- iŽ. w i Ž wx⌬ r w jq1x .x straight. HMJkis a *essential extension of Ext w⌬ x%yk J , M ᎏ j G w i Ž wx⌬ r w2x .x for all j 1. Of course, Ext kw⌬ x%yk J , M ᎏ 1 is the smallest of these w i Ž wx⌬ r .x modules. In Example 4.13, we see that Ext kw⌬ x%k J, M ᎏ 0 is not i sufficient to recover HMJ Ž.. Ž.2 In general, we have i w jq1x i w jq1x Ext w⌬ x kwx⌬ rJ , k wx⌬ / Ext w⌬ x kwx⌬ rJ , k wx⌬ . k Ž.Ž.%yᎏ j k For example, assume that kwx⌬ is not Gorenstein and that J s ᒊ. Then i wx⌬ / G [ wx⌬ i wx⌬ s Ext kw⌬ xŽk, k . 0 for all i d dim k . But Hkᒊ Ž . 0 for all ) ގniwx⌬ ) i d. Hence the -graded part of Ext kw⌬ xŽk, k . is 0 for i d. [ nyd wx⌬ s wx⌬ We say that kw⌬ x Ext SSŽk , . with d dim k is the canoni- cal module of kwx⌬ . The following is a generalization ofw 16, Theorem 1.1x . THEOREM 4.9. If kwx⌬ is Cohen᎐Macaulay, then iiw jq1x H w⌬ x ( Ext w⌬ x kwx⌬ rJ , w⌬ x JkŽ.%yᎏ j k Ž.k for all j G 0. Proof. For convenience, we say that a ޚ n-graded S-module M is s j-graded if M%yᎏ j M. By Theorem 4.5, it suffices to prove that i wx⌬ r w jq1x s wx⌬ Ext kw⌬ xŽk J , kw⌬ x. is j-graded. Set d dim k . It is well known that i wx⌬ r w jq1x ( nydqi wx⌬ r w jq1x Ext kw⌬ x Ž.k J , kw⌬ x Ext SSŽ.k J , as ޚ n-graded S-modules. In fact, both of these are isomorphic to the dyi wx w jq1x graded Matlis dual of Hkᒊ Ž ⌬ rJ .. Hence it suffices to show that i Ž wx⌬ r w jq1x . ޚ n Ext SSk J , is j-graded. Let Fⅷ be a -graded minimal free resolution of kwx⌬ rJ w jq1x over S. By an argument using Taylor’s resolu- LOCAL COHOMOLOGY OF STANLEY-REISNER RINGS 723 Ž.y $ q tion, we see that Fⅷ is a direct sum of free modules S a with a ᎏ j 1. ⅷ [ Ž. s Ž.y ⅷ Set G Hom S Fⅷ , SS. Since S 1 , G consists of free modules ⅷ SŽ.yb with b %yᎏ j. In particular, G is a complex of j-graded modules. w q x i ⅷ s i wx⌬ r j 1 Thus HGŽ.Ext SS Žk J , . is j-graded. The following is clear from Theorem 4.9. COROLLARY 4.10. If kwx⌬ is Cohen᎐Macaulay, then we can construct i i wx⌬ r w2x HJkŽ.w⌬ x from its submodule Ext kw⌬ xŽk J , kw⌬ x. by Lemma 4.3. In wx⌬ i wx⌬ particular, if k is Gorenstein, then HJ Ž k . is determined by a submodule i wx⌬ r w2x wx⌬ Ext kw⌬ xŽk J , k .. Remark 4.11. Assume that kwx⌬ is Cohen᎐Macaulay. From Theorem 4.9, we see that depthŽkwx⌬ rJ w2x.Žs depth kwx⌬ rJ wix.Žand Ass kwx⌬ rJ w2x. s AssŽkwx⌬ rJ wix. for all i G 2. The former statement also follows from an argument using lcm-latticesŽ seewx 7. . In fact, for all i G 2, the lcm-lattice wix w2x of I⌺⌺q I⌬⌬is isomorphic to that of I q I . wx For a square-free monomial ideal I⌬ ; S, Lyubeznik 13 proved a beautiful formula on the cohomological dimension of S at I⌬, which states that [ N i / s y r Ž.4.1 cd ŽS, I⌬ .maxÄ4i HI⌬ Ž.S 0 dim S depthŽ.S I⌬ . We can extend this result to a monomial ideal J of a Gorenstein Stanley᎐Reisner ring kwx⌬ , using the second Frobenius power J w2x. COROLLARY 4.12. Let the notation be as before. Assume that kwx⌬ is Gorenstein. We ha¨e cdŽ.kwx⌬ , J s dim k wx⌬ y depthŽ.k wx⌬ rJ w2x . i wx⌬ / i wx⌬ r Proof. By Corollary 4.10, HkJkŽ . 0 if and only if Ext w⌬ xŽk w2x wx⌬ / wx⌬ i wx⌬ r w2x wx⌬ J , k . 0. Since k is Gorenstein, Ext kw⌬ xŽk J , k . is isomor- dyi wx w2x phic to the Matlis dual of Hkᒊ Ž ⌬ rJ . up to degree shifting, where s wx⌬ i wx⌬ / dyi wx⌬ r w2x / d dim k . Hence HkJ Ž . 0 if and only if Hkᒊ Ž J . 0. i wx w2xwwx 2x Since maxÄi N Hkᒊ Ž ⌬ rJ .4 s depthŽk ⌬ rJ ., we are done. If ⌬ s 2w nx Ž i.e., kwx⌬ s S.Ž., then Corollary 4.12 and Eq. 4.1 imply that wix depthŽ.SrI⌬⌬s depth ŽSrI . for all i G 1. This also follows from the fact wix that the lcm-lattice of I⌬⌬is isomorphic to that of I for all i G 1. wx EXAMPLE 4.13. Set S s kx, y, z, w , I⌬ s Ž.xz, yw , and J s Ž.z, w . That is, ⌬ consists of four edges of a quadrilateral, and ⌺ consists of three edges of ⌬. Then kwx⌬ is Gorenstein, and dim k wx⌬ y depthŽkwx⌬ rJ . s 0. But an easy calculationŽ. e.g., via Theorem 3.2 shows that w 1 wx⌬ x / i wx⌬ s G HkJ Ž . Ž1, 0, 0, y1. 0, although HkJ Ž . 0 for all i 2. Thus cdŽkwx⌬ , J .Ž.s 1, so the direct analogue of 4.1 does not hold for J ; kwx⌬ . 724 REINER, WELKER, AND YANAGAWA On the other hand, depth Žkwx⌬ rJ w2x. s 1, and in fact, J w2x ; kwx⌬ has an g 1 wx⌬ embedded prime Ž.Žx, w, z . This also means that Ž.x, w, z Ass ŽHkJ Ž .. by Corollary 4.14. Hence dim kwx⌬ y depthŽkwx⌬ rJ w2x. s 1, in agreement with Corollary 4.12. To state the next result, we introduce the following notation. For a kwx⌬ -module M and an integer i, set Assi Ž.M s Ä4P g Ass Ž.M N dimŽ.kwx⌬ rP s dim k wx⌬ y i . COROLLARY 4.14. Assume that kwx⌬ is Cohen᎐Macaulay and dim kwx⌬ s g i wx⌬ r F y d. If P AssŽ HJkŽ..Žw⌬ x , then dim k P. d i and ii s iwx⌬ r w2x Ass Ž.HJkŽ.w⌬ x Ass Ž.k J = N ⌺ < To prove this result, we need the following lemma. LEMMA 4.15Ž seewx 6, Theorem 1.1. . Let M be a finitely generated wx⌬ g i wx⌬ r F y k -module. If P Ass Ext kw⌬ xŽ.M, kw⌬ x , then dim Žk P. d i, and ¨ i s i i G we ha e Ass Ž.M AssŽ Ext kw⌬ xŽ..M, kw⌬ x for all i 0. Proof. The proof ofwx 6, Theorem 1.1 also works here. i wx⌬ r w2x : i Proof of Corollary 4.14. Since Ext kw⌬ xŽk J , kw⌬ x. HJkŽ.w⌬ x is i s an *essential extension by Theorem 4.9, we have AssŽ HJkŽ..w⌬ x i wx⌬ r w2x AssŽ Ext kw⌬ xŽk J , kw⌬ x... Hence the first statement and the equality in the displayed equality follow from Lemma 4.15. The last inclusion holds wx⌬ r w2x = wx⌬ r s wx⌺ s N since AssŽk J .ŽAss k J .ŽAss k . Ä PF F is a facet of ⌺4. i w Remark 4.16. Even HSJ Ž.can have an embedded primeŽ see 16, Example 1x. . The inclusion in Corollary 4.14 can be strict, as Example 4.13 shows. 5. A TOPOLOGICAL FORMULA FOR THE HILBERT FUNCTION IN THE GORENSTEIN CASE In Section 3 we gave a combinatorial formula for the ޚ n-graded Hilbert i wx⌬ ⌬ ⌬ ᎐ function of HkJ Ž . for general .If is Cohen Macaulay, then we can i give a formulaŽ. Theorem 5.1 for the Hilbert function of HJkŽ.w⌬ x for the wx⌬ ⌬ canonical module kw⌬ x of k . Hence, if is Gorenstein, then we get a i wx⌬ new formulaŽ. Corollary 5.2 on HkJ Ž .. LOCAL COHOMOLOGY OF STANLEY-REISNER RINGS 725 THEOREM 5.1. Assume that kwx⌬ is Cohen᎐Macaulay and has dimension d, and that ⌺ : ⌬ is a subcomplex of ⌬. As before, let J denote the image of wx n Iink⌺⌬⌬ s SrI . Then for any multidegree a in ޚ , setting Fq[ suppqya, F [ supp ya, we ha¨e H i ( H˜ star F l ⌺, del F ; k JkŽ.w⌬ x a dyiy1Ž.⌬Ž.y star⌬ Ž Fy.l ⌺ Ž.q ( ˜ Hdyiy< Fq ⅷ ª 0 ª 1 ª иии ª d ª Dkw⌬ x :0 Dkw⌬ x Dkw⌬ x Dkw⌬ x 0, i s wx⌬ r Dkw⌬ x [ EkŽ.PF , Fg⌬ < n If a g ގ Ž.i.e., Fys л , then Theorem 5.1 simplifies because star⌬Ž.Fy l ⌺ s ⌺. Consequently, we obtain for a g ގn with F [ supp a, 5.1 H i ( H˜ lk F ; k . Ž. JkŽ.w⌬ x a dyiy< F If a g ގn, then we have ii dyi H ( Ext kwx⌬ rJ, ( Hkwx⌬ rJ y JkŽ.w⌬ x aakw⌬ x Ž.kw⌬ x ᒊ Ž.a by Theorem 4.9 and local dualityŽwx 4, Theorem 3.5.8.Ž. . Thus 5.1 also follows from a well-known formula of HochsterŽwx 17, Theorem 4.1. . Let ⌬ be a Gorenstein complexŽ. over k . Set ⌬Ј [ core Ž.⌬ ,a k- homology sphere. Let V be the vertex set of ⌬Ј, so that if W [ wxn y V, ⌬ s W )⌬Ј s wx⌬ y wx then 2 and kw⌬ x k Ž.ŽW see 17, 4. . COROLLARY 5.2. Suppose that kwx⌬ is Gorenstein with dim kwx⌬ s d, and ⌺ a subcomplex of ⌬. Keeping the same notation as before, and defining for a g ޚ n, F [ Ž.Fqyqyj W y F s F j Ž.W y F we ha¨e i wx⌬ ( ˜ l ⌺ Ž.5.2 HkJdŽ.a H yiy1Ž.star⌬Ž.Fy , delstar⌬ Ž Fy.l ⌺ Ž.F ; k ( ˜ Hdyiy< F EXAMPLE 5.3. Consider the case kwx⌬ s S Ži.e., ⌬ is the full simplex w nx 2.Ž . In this case, star⌬ Fy.l ⌺ is always ⌺, and we have i ( HSJnŽ.a H˜ yiy< F THEOREM 5.4. Suppose that ⌬ is a Cohen᎐Macaulay complex of dimen- sion d y 1 and ⌺ is anŽ. s y 1-dimensional subcomplex of ⌬. Then i s / y ¨ ⌬ HJkŽ.w⌬ x 0 for all i d s if and only if for e ery face F of , the subcomplex star⌬Ž.F l ⌺ is a Cohen᎐Macaulay complex of dimension s y 1. Proof. The ‘‘if’’ part is immediate from Theorem 5.1. We prove the ‘‘only if’’ part. Set ⌺ЈŽ.F [ star⌬ Ž.F l ⌺ for F g ⌬. It suffices to show that s / : wx Ž.5.3 H˜iy wx⌬ i wx⌬ s COROLLARY 5.5. Suppose that k is Gorenstein. Then HJ Ž k . 0 for all i / grade JiŽ..e., the condition of Problem 3.6 is satisfied if and only if for e¨ery face F g ⌬ the subcomplex star⌬Ž.F l ⌺ is Cohen᎐Macaulay and has the same dimension as ⌺. Remark 5.6.Ž. 1 Assume that a pair Ž⌬, ⌺ .satisfies the condition of Theorem 5.4. Then ⌺ itself is Cohen᎐Macaulay, since star⌬Ž.л l ⌺ s ⌺. Moreover, for any facet F of ⌬, Ä4G g ⌺ N G : F s star⌬Ž.F l ⌺ is a Cohen᎐Macaulay complex of dimension s y 1. In particular, any facet of ⌬ contains a facet of ⌺. l ⌺ wx⌬ Ž.2 Here we give a ring-theoretic meaning of star⌬ ŽF . . Let k x F be the localization of kwx⌬ at the multiplicatively closed set generated by F ¸ wx⌬ ª wx⌬ the monomial x . Then the kernel of the composition S k k x F s q is Istar⌬⌬⌬Ž F .. Thus Istar Ž F .l ⌺ Istar Ž F . I⌺. EXAMPLE 5.7. Checking the combinatorial condition of Theorem 5.4 is somewhat complicated, but the following examples satisfy the condition: Ž.1 Let ⌺ be the s-skeleton of a Cohen᎐Macaulay complex ⌬.In this case, star⌬Ž.F l ⌺ is the s-skeleton of the Cohen᎐Macaulay complex F star⌬⌬Ž.F s 2 )lk Ž.F . Hence it is Cohen᎐Macaulay. Ž.2 Let ⌬ be a Ž.d y 1 -dimensional balanced Cohen᎐Macaulay com- ; wx < s g wx⌬ jiÝ x k . Ž.i sj wx w⌬x By 17, Proposition 4.3 , 1,..., d is a system of parameters of k .If s s 2 wx⌬ Ј [ N f s N Ž.i j, then xijx iin k . Set J Ž.j j T . Since J ŽŽ.xi i f T ., we have J > JЈ > J w2x and hence J s 'JЈ . Therefore, J ; kwx⌬ is a y i wx⌬ s set-theoretic complete intersection of codimension d s, and HkJ Ž . i wx⌬ s ) y wx⌬ HkJЈŽ .Ž0 for all i d s. This is true even if k is not Cohen᎐Macaulay.. On the other hand, since kwx⌬ is Cohen᎐Macaulay, i wx⌬ s - y s HkJ Ž .Ž.0 for all i d s grade J . By the same reasoning, we also i s / y have HJkŽ.w⌬ x 0 for i d s. So Remark 5.6Ž. 1 gives another proof ofwx 17, III, Theorem 4.5 , which states that ⌺ is Cohen᎐Macaulay in this case. 728 REINER, WELKER, AND YANAGAWA Naturally, since Theorem 3.2 and Corollary 5.2 both express the same local cohomology in topological terms when kwx⌬ is Gorenstein, there must be some topological explanation for their equivalence. For example, consider the case where ⌬ s 2w nx and kwx⌬ s S, as in Example 5.3. Terai’s formulaŽ. and our Corollary 5.2 state that HSi ( H˜ ⌺; k , In⌺Ž.Ž.y1,...,y1 yiy1Ž. while Theorem 3.2 gives HSii( H˜ y1 55⌬ y 555⌺ , Ѩ⌬ 5y 55⌺ ; k , I⌺Ž.Ž.y1,...,y1 Ž . where for ⌬ s 2w nxw we have Ѩ⌬ s 2 nx y wxn . For simplicity, assume that w nx 55 55 I⌺ / 0Ž i.e., ⌺ / 2. . Then ⌬ y ⌺ is contractibleŽ since it is star- shaped with respect to the barycenter of the maximum face wxn of ⌬., and hence H˜iy1Ž55⌬ y 55⌺ ; k. s 0 for all i. So the long exact sequence for cohomology shows that H˜iy1 Ž.55⌬ y 555⌺ , Ѩ⌬ 5y 55⌺ ; k ( H˜iy2 Ž.5Ѩ⌬ 5y 55⌺ ; k . Thus Alexander duality on the Ž.n y 2 -sphere 55Ѩ⌬ connects Theorem 3.2 and Terai’s formula. In the general case, the situation is much more complicated. But, as we now explain, the relation between Theorem 3.2 and Corollary 5.2 is based on the ‘‘disguised’’ version of Alexander dualityŽ. Lemma 6.8 , giving further confirmation for the observationŽ see, e.g.,wx 14. that in a combina- torial setting, Matlis duality corresponds to Alexander duality. Recall that Corollary 5.2 says that i wx⌬ ( ˜ HkJdŽ.a H yiy< F On the other hand, Theorem 3.2 says that i iy1 y1 y1 Hkwx⌬ ( Hs˜ star⌬ Fq , s del Fy ; k J Ž.a ž/Ž.Ž. Ž.star⌬ Ž Fq.Ž. iy1 y1 y1 ( Hs˜ star⌬ Fqyj F , s del j Fy ; k , ž/Ž.Ž.Ž.star⌬ Ž FqyF .Ž. where the first isomorphism is our original statement and the second isomorphism is Proposition 6.6. LOCAL COHOMOLOGY OF STANLEY-REISNER RINGS 729 Thus we need to show that when ⌬ is Gorenstein, and with the foregoing notation, ˜ Hdyiy< F iy1 y1 y1 ( Hs˜ star⌬ Fqyj F , s del j Fy ; k . ž/Ž.Ž.Ž.star⌬ Ž FqyF .Ž. Note that all complexes in this last equation are subcomplexes of F Fy star⌬Ž.Fqyj F s 2 )2 )lk ⌬ЈŽ. Ž.Fqyj F y W . Note also that the last complex in the foregoing join is a link of a face in the homology sphere ⌬Ј s coreŽ.⌬ , and hence a homology sphere itself. Using Eq.Ž. 6.3 , we can replace star⌬Ž.Fqyj F with ⌬, and star⌬Ž.Fqyj F l ⌺ with ⌺, Fywith G Žsince we may assume that Fqyj F g ⌬ by Corollary 3.3, we can keep the assumption ⌺ / л., and then we need only prove the following combina- torial topological statement: If ⌬ s 2 F )2G )ޓ for some homology sphere ޓ and ⌺ is a subcomplex of ⌬, then ( iy1 y1 ⌬ y1 H˜˜dyiy< F iy1 y1 ⌬ y1 Hs˜ Ž.⌺⌺⌬Ž., s Ž.del Ž.G ; k iy1 y1 G y1 ( ˜ )ޓ G HsŽ.lk ⌺⌺F Ž.2 , slk F Ž.del 2 ) ޓ Ž.G ; k . Now we can further replace 2G )ޓ with ⌬ and lk ⌺ F with ⌺, and then we must prove the following: 730 REINER, WELKER, AND YANAGAWA If ⌬ s 2G )ޓ for some homology sphere ޓ and ⌺ is a subcomplex of ⌬, then ⌺ ( iy1 y1 ⌬ y1 H˜˜dyiy1Ž.; k HsŽ. Ž., s Ž.del⌬ Ž.G ; k , where d y 1 s dim ⌬. Starting with the right-hand side, we have iy1 y1 ⌬ y1 s iy1 y1 ⌬ y1 Ѩ⌬ Hs˜ Ž.Ž., s Ž.del⌬ Ž.G ; k Hs˜ Ž.Ž., s Ž .; k ( H˜iy1 Ž.55⌬ y 555⌺ , Ѩ⌬ 5y 55⌺ ; k ( ⌺ H˜dyiy1Ž.; k , where the second-to-last isomorphism uses Proposition 6.5, and the last isomorphism is the disguised Alexander duality lemmaŽ. Lemma 6.8 . 6. APPENDIX: TOOLS FROM COMBINATIONAL TOPOLOGY In this section we collect various definitions and facts from combinato- rial topology needed in the paper. Good references for much of this material arewx 15, 3, Section 4.7; 2 . Let ⌬ be a simplicial complex on vertex set V, that is, a collection of subsets F Ž.called faces of V which is closed under inclusion. The dimension dim F of a face F is one less than its cardinality < Ž.6.1 star⌬ Ž.F [ Ä4G g ⌬ : G j F g ⌬ , del⌬Ž.F [ Ä4G g ⌬ : F G , lk⌬Ž.F [ Ä4G g ⌬ : G j F g ⌬, G l F s л . Note that lk⌬Ž.F is a simplicial complex on the vertex set V y F. Given ⌬ ⌬ two simplicial complexes 12and on disjoint vertex sets V12and V , their ⌬ )⌬ @ simplicial join 12is the simplicial complex on vertex set V1V 2 @ ⌬ s having a face F12F for every Fiiin , i 1, 2. One can also define the LOCAL COHOMOLOGY OF STANLEY-REISNER RINGS 731 corresponding topological join X )Y of two spaces X, Y Žseew 15, Section 62x. so that 555555⌬ )⌬ ( ⌬ ) ⌬ 12 1 2. With these definitions, one can check that Ž.6.2 star⌬⌬Ž.F j del Ž.F s ⌬, star⌬⌬Ž.F l del Ž.F s Ѩ F)lk ⌬ Ž.F , F star⌬⌬Ž.F s 2 )lk Ž.F , where Ѩ F denotes the boundary complex of F, i.e., all proper subsets of F. The suspension, Susp ⌬, of a simplicial complexŽ. or a topological space ⌬ is the join ޓ00)⌬, where ޓ is a complexŽ. space having two discon- ⌬ nected vertices. More generally, when 1 triangulates a d-sphere, one has ⌬ )⌬ ( dq1⌬ 12Susp 2. For a subspace Y of a topological space X, we let XrY denote the quotient space that shrinks Y to a point. The following fact is well known. PROPOSITION 6.1. For any subcomplex ⌬Ј of a simplicial complex ⌬, we ha¨e H˜ ⅷ Ž.Ž⌬, ⌬Ј; k ( H˜ ⅷ 5555⌬ , ⌬Ј ; k .( H˜ ⅷ Ž. 5555⌬ r ⌬Ј ; k where the cohomology in the leftmost expression is simplicial, and the other two cohomologies are singular. The following proposition is a straightforward and well-known conse- quence of Eq.Ž. 6.2 . PROPOSITION 6.2. For any face F in a simplicial complex ⌬, we ha¨e 555 5 < F < 5 5 Ž.i ⌬ r del⌬⌬Ž.F ( Susp lk Ž.F i ⌬ ( Ž.ii H˜ Ž, del⌬ ŽF .; k .H˜iy< F < Žlk⌬ F; k .. Let Sd denote the barycentric subdivision operator on simplicial com- plexesŽwx 15, Section 15. . Given a subcomplex ⌺ of a simplicial complex ⌬, let SdŽ.⌬ y ⌺ denote the subcomplex of Sd ⌬ induced on the vertices which are not barycenters of faces in ⌺. PROPOSITION 6.3Ž seewx 3, Lemma 4.7.27. . Let ⌬ be a simplicial complex and let ⌬Ј and ⌺ be two subcomplexes. Then the pair of spaces Ž55⌬ y 5555⌺ , ⌬Ј y 55⌺ .Žis relati¨ely homotopy equi¨alent to the geometric realiza- tions. of the pair Ž.SdŽ.Ž.⌬ y ⌺ ,Sd ⌬Ј y ⌺ . We sometimes wish to refer to the topology of a finite poset P, by which we mean the geometric realization of its order complex, the simplicial complex on vertex set P having the totally ordered subsets of P as faces. 732 REINER, WELKER, AND YANAGAWA LEMMA 6.4Ž. Relative Quillen Fiber Lemma . Let f: P ª Qbean order-preser¨ing map of posets, and let QЈ : Q be an induced subposet. If the y1 y1 X ¨ g Ј g Ј posets fŽ. QF q and fŽ QF qЈ. are contractible for e ery q Q, q Q , then f induces a relati¨e homotopy equi¨alence of the pairsŽ P, fQy1 Ž..Ј and Ž.Q, QЈ . Proof. By the usual Quillen fiber lemmaŽw 2,Ž. 10.11 ; 3, Lemma 4.7.29x. , f induces homotopy equivalences between P and Q and between fQy1 Ž.Ј and QЈ. But this is sufficient for f to induce a relative homotopy equiva- lence of the pairs bywx 5, Lemma 3 . We recall some terminology used in Section 3. Let ⌺ be a subcomplex of ⌬ wx s q a simplicial complex on vertex set n , with I⌺ Ž.m1,...,m I⌬; i.e., ⌬ y ⌺ m1,...,m are monomials corresponding to the minimal faces of . Define the order-preserving map s 2w xwª 2 nx ¬s F supp Ł mi . igF Note that since s is order-preserving, sy1 Ž.⌬ is a simplicial complex on vertex set wx ,asis sy1 Ž.⌬Ј for any other subcomplex ⌬Ј of ⌬. When we wish to emphasize the dependence of this map s on the pair Ž.⌬, ⌺ ,we ⌬Ј ⌬ write s⌬, ⌺. However, note that if is a subcomplex of , then those ⌬Ј monomials in Ä4miis1 which correspond to faces of generate the image wx of I⌺ l ⌬Ј in k ⌬Ј , and consequently, y1 ⌬Ј s y1 ⌬Ј Ž.6.3 s⌬ , ⌺⌬ЈŽ.s , ⌺ l ⌬Ј Ž.. ⌬Ј ⌬ y1 ⌬ y1 ⌬Ј PROPOSITION 6.5. If is a subcomplex of , thenŽ s⌬, ⌺⌬Ž., s , ⌺Ž..is relati¨ely homotopy equi¨alent to the pair of spaces Ž55⌬ y 5555⌺ , ⌬Ј y 55⌺ .. y1 ⌬ ¨ 55⌬ y 55⌺ In particular, s⌬, ⌺Ž.is homotopy equi alent to . Proof. The first assertion specializes to the second for ⌬Ј s л.To prove the first assertion, note that by Proposition 6.3 it is equivalent to y1 ⌬ y1 ⌬Ј show that Žs⌬, ⌺⌬Ž., s , ⌺Ž..is relatively homotopy equivalent to the pair ŽŽSd ⌬ y ⌺ .,Sd Ž⌬Ј y ⌺ ... By the definition of barycentric subdivision, the latter is a pair of order complexes for the posets of faces in Ž⌬ y ⌺, ⌬Ј y ⌺.. Thus if we momentarily regard ⌬ y ⌺, ⌬Ј y ⌺ as posets, then the result follows from Lemma 6.4 if we can show that y y ⅷ 1 ⌬ s 1 ⌬ y ⌺ ⌬Ј s⌬, ⌺⌬Ž.s , ⌺ Ž .Žand similarly for .and y ⅷ ⌬ ⌺ 1 ⌬ y ⌺ for each face F in but not in , the subposet s⌬, ⌺ŽŽ .: F . is contractibleŽ. and similarly for ⌬Ј . : wx y1 ⌬ To see the first assertion, note that if G lies in s⌬, ⌺Ž., then this ⌬ ⌺ means that suppŽ.Ł jg Gjm is a face F of which does not lie in Žsince LOCAL COHOMOLOGY OF STANLEY-REISNER RINGS 733 its support contains the support of at least one mj in I⌺ .. For the second y1 ⌬ y ⌺ assertion, note that s⌬, ⌺ŽŽ .: F . contains a unique maximum element, g wx N : namely Ä j supp mj F4. The next proposition follows immediately using Eq.Ž. 6.2 and excision. i y1 y1 i y1 PROPOSITION 6.6. Hs˜ ŽŽ.Ž⌬ , s del⌬⌬ Ž...F ; k ( Hs˜ ŽŽstar F ., y1 ŽŽ.. s delstar⌬ Ž F . F ; k . The next proposition is used in Section 5 in showing the equivalence of Corollary 5.2 and Theorem 3.2 when kwx⌬ is Gorenstein. PROPOSITION 6.7. Let ⌽ be a simplicial complex and let ⌽Ј : ⌽ be a subcomplex. Let ⌬ and ⌬Ј be simplicial complexes which are joins ⌬ [ 2 F )⌽ and ⌬Ј [ 2 F )⌽Ј, in which F is a simplex on a ¨ertex set disjoint from that of ⌽, ⌽Ј. Assume that ⌺ is a subcomplex of ⌬ containing the face F. Then the pair y 1 ⌬ y 1 ⌬Ј ¨ ¨ Ž s⌬ , ⌺⌬Ž., s , ⌺Ž..is relati ely homotopy equi alent to the pair Ž y1 Ž.⌽ y1 Ž..⌽Ј s⌽,lk⌺⌺Ž F . , s⌽,lk Ž F . . Equi¨alently, ¨ia Proposition 6.5, the pair Ž55⌬ y 5555⌺ , ⌬Ј y 55⌺ . is 55 5 55 5 5 5 relati¨ely homotopy equi¨alent to the pair Ž ⌽ y lk ⌺⌺Ž.F , ⌽Ј y lk Ž.F .. Proof. By Proposition 6.3, it suffices to show thatŽŽ Sd ⌬ y ⌺ .,Sd Ž⌬Ј y ⌺..is relatively homotopy equivalent to Ž Sd Ž⌽ y lk ⌺ ŽF ..,Sd Ž⌽Ј y lk ⌺ŽF .... Regarding these as pairs of order complexes for the pairs of posets Ž.ŽŽ.Ž..⌬ y ⌺, ⌬Ј y ⌺ and ⌽ y lk ⌺⌺F , ⌽ y lk F , we can try to apply Lemma 6.4. To this end, define an order-preserving poset map ⌬ y ⌺ ªf ⌬ G ¬f G j F. Every face in the image of f clearly contains F, so one can define a map on the image in the reverse direction that sends H ¬ H y F. It is easy to check that this gives an isomorphism between the image poset fŽ.⌬ y ⌺ Žresp. f Ž⌬Ј y ⌺ .. and the poset ⌽ y lk ⌺⌺Ž.ŽF resp. ⌽Ј y lk ŽF ... Thus it only remains to show that f induces a relative homotopy equivalence, i.e., y1 ⌬ ⌬ y ⌺ the subposet f Ž.: H of is contractible for every H in the image y1 ⌬ of f. But f Ž.: H has H as its unique maximum element H. Our last result is a disguised form of Alexander duality, which turns out to be the crux of the equivalence of Corollary 5.2 and Theorem 3.2 when kwx⌬ is Gorenstein. 734 REINER, WELKER, AND YANAGAWA LEMMA 6.8. Let ޓ be a simplicial homology sphereŽ o¨er some coefficient ring., and let ⌬ s 2 F )ޓ for some simplex FŽ. possibly empty on a ¨ertex set disjoint from ޓ. Let d s dim ⌬ and let ⌺ be subcomplex of ⌬. Then ⌺ ( j 55⌬ y 555⌺ Ѩ⌬ 5y 55⌺ H˜˜i Ž.; k H Ž , ; k. for i q j s d y 1, where Ѩ⌬ s Ѩ F)ޓ Žand the coefficient ring has been suppressed for the sake of bre¨ity.. Proof. First, deal with the easy case where F is empty. Then ⌬ s ޓ is a homology d-sphere, and Ѩ⌬ s л. Consequently, H˜ j Ž.55⌬ y 555⌺ , Ѩ⌬ 5y 55⌺ ; k ( H˜ j Ž. 55ޓ y 55⌺ , л; k ( H˜ j Ž.55ޓ y 55⌺ ; k ( ⌺ q s y H˜i Ž.; k for i j d 1, where the last isomorphism is Alexander duality within ޓ Žw15, Sec- tion 71x. . If F / л, then Ѩ F is a sphere, and hence Ѩ⌬ s Ѩ F)ޓ is another homology sphere. The key idea is to enlarge ⌬ in a controlled way, so as to obtain yet another homology sphere ޓˆ, use excision, and then rely on Alexander duality within ޓˆˆ. Define this new homology sphere ޓ by adding a cone vertex ¨ over the subcomplex Ѩ⌬: ޓˆ [ ⌬ j Ž.¨ )Ѩ⌬ . PROPOSITION 6.9. ޓˆ is a homology d-sphere. Assuming this proposition for the moment, we can complete the proof of Lemma 6.8. We have the following isomorphisms, which are justified later: H˜ j Ž.55⌬ y 555⌺ , Ѩ⌬ 5y 55⌺ ; k ( H˜ j Ž. 55ޓˆ y 555⌺ , ¨ )Ѩ⌬ 5y 55⌺ ; k ( H˜ j Ž.55ޓˆ y 55⌺ ; k ( ⌺ q s y H˜i Ž.; k for i j d 1. The first isomorphism comes from excision, we excise away Ž.555555¨ )Ѩ⌬ y Ѩ⌬ y ⌺ from the pair on the right-hand side. The second isomorphism is due to the fact that 5555¨ )Ѩ⌬ y ⌺ is a contractible subspace of 55ޓˆ y 55⌺ ; it is star-shaped with respect to ¨, LOCAL COHOMOLOGY OF STANLEY-REISNER RINGS 735 since ⌺ can intersect only the cone ¨ )Ѩ⌬ in a subset of its base Ѩ⌬. Consequently, the homology of 5555¨ )Ѩ⌬ y ⌺ vanishes, and then the long exact sequence for the pair Ž55ޓˆ y 555⌺ , ¨ )Ѩ⌬ 5y 55⌺ . gives the isomor- phism. The third isomorphism is Alexander duality within ޓˆ, using Proposi- tion 6.9. Proof. We must show for every face G in ޓˆ, that lk ޓˆŽ.G has the homology of a sphere of the same dimension as lk ޓˆŽ.G . We first deal with the case where G s л. That is, we must show that ޓˆ itself has the homology of a d-sphere. This follows from the Mayer᎐Vieto- ris exact sequence applied to ޓˆ s ⌬ j Ž.¨ )Ѩ⌬ , Ѩ⌬ s ⌬ l Ž.¨ )Ѩ⌬ , and the fact that Ѩ⌬ s Ѩ F)ޓ is a homology sphere. We now deal with two other cases for G, depending on whether or not it contains ¨. Case 1. ¨ f G. If we write G s FЈ@GЈ, where FЈ : F and GЈ is a face of ޓ, then we can easily check that lk ޓˆ G s ⌬Ј j Ž.¨ )Ѩ⌬Ј , FyF Ј where ⌬Ј is the joint 2 )lk ޓGЈ. Since ⌬Ј is again a simplex joined with a homology sphereŽ. like ⌬ was , this link has the same form as ޓˆ and hence has the correct homology by the G s л case that we just explored. Case 2. ¨ g G. Then we can easily check that lk ޓˆ G s lk ¨ ) Ѩ⌬G s lkѨ⌬Ž.G y ¨ . Hence we are done in this case, because Ѩ⌬ is a homology Ž.d y 1 -sphere. ACKNOWLEDGMENTS The authors thank Professor Gennady Lyubeznik for valuable comments. The first author is partially supported by the NSF. 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