Advances in Mathematics AI1596 advances in mathematics 124, 332333 (1996) article no. 0086 Alexander Duality Theorem and Second Betti Numbers of StanleyReisner Rings Naoki Terai* Department of Mathematics, Faculty of Education, Saga University, Saga 840, Japan and Takayuki Hibi- Department of Mathematics, Faculty of Science, Osaka University, Toyonaka 560, Japan View metadata, citation and similar papers at core.ac.uk brought to you by CORE Received January 1, 1995; accepted June 25, 1995 provided by Elsevier - Publisher Connector A simplicial complex 2 on the vertex set V=[x1, x2, ..., xv] is a collec- tion of subsets of V such that (i) [xi] # 2 for every 1iv and (ii) _ # 2, {/_ O { # 2. Given a subset W of V, the restriction of 2 to W is the sub- complex 2W=[_ # 2 | _/W] of 2. Let H i (2; k) denote the ith reduced simplicial homology group of 2 with the coefficient field k. Note that H &1(2; k)=0 if 2{[<], H &1([<]; k)$k, and H i ([<]; k)=0 for each i0. We write |2| for the geometric realization of 2. Let A=k[x1 , x2 , ..., xv] be the polynomial ring in v-variables over a field k. Here, we identify each xi # V with the indeterminate xi of A. Define I2 to be the ideal of A which is generated by square-free monomials xi 1 xi 2 }}}xir,1i1<i2<}}}<irv, with [xi1 , xi2 , ..., xi r ] Â 2. We say that the quotient algebra k[2]:=AÂI2 is the StanleyReisner ring of 2 over k. In what follows, we consider A to be the graded algebra A=n0 An with the standard grading, i.e., each deg xi=1, and may regard k[2]= n0 (k[2])n as a graded module over A with the quotient grading. Let Z denote the set of integers. We write A( j ), j # Z, for the graded module A( j )=n# Z [A(j)]n over A with [A( j )]n := A n + j . We study a graded minimal free resolution ; . h . 2 ; . 1 . 0 0 wÄ A (&j) hj wÄ }}} wÄ A (&j) 1j wÄ A wÄ k [ 2 ] wÄ 0 j # Z j # Z * E-mail address: teraiÄcc.saga-u.ac.jp. - E-mail address: hibiÄmath.sci.osaka-u.ac.jp. 332 0001-8708Â96 18.00 Copyright 1996 by Academic Press All rights of reproduction in any form reserved. File: 607J 159601 . By:CV . Date:26:12:96 . Time:09:59 LOP8M. V8.0. Page 01:01 Codes: 4089 Signs: 1815 . Length: 50 pic 3 pts, 212 mm ALEXANDER DUALITY THEOREM AND RINGS 333 of k[2] over A. Here h is the homological dimension of k[2] over A and A ;i=;i (k[2]) :=j # Z ;ij is the ith Betti number of k[2] over A.Itis known [2, Theorem (5.1)] that ;i j = : dimk H j&i&1(2W; k), W/V, >(W)=j where >(W) is the cardinality of a finite set W. Thus, in particular, A ;i (k[2])= : dimk H >(W)&i&1(2W; k). (1) W/V Lemma. Let 2 be a simplicial complex on the vertex set V with >(V)=v and k a field. Then dimk H v&3(2; k) is independent of k. Proof. Let 2V denote the set of all subsets of V. Thus, the geometric realization X of the simplicial complex 2V&[V] is the (v&2)-sphere. We may assume that V Â 2; in particular, |2| is a subspace of X. Note that v&3 Hv&3(|2|; k)$H (|2|; k) since k is a field. Now, the Alexander duality v&3 theorem of topology guarantees that H (|2|; k)$H 0(X&|2|; k). On the other hand, dimk H 0(X&|2|; k)+1 is equal to the number of connected components of X&|2|. Thus, dimk H v&3(2; k)=dimk H 0(X&|2|; k)is independent of the base field k as required. Q.E.D. Theorem. The second Betti number of the StanleyReisner ring of a simplicial complex is independent of the base field. A Proof. By virtue of Eq. (1), the second Betti number ;2 (k[2]) of k[2] over A is equal to W/V dimk H >(W)&3(2W; k), which is independent of k by the above lemma. Q.E.D. A ring-theoretical proof of the above theorem is also given in [1]. REFERENCES 1. W. Bruns and J. Herzog, On multigraded resolutions, Math. Proc. Cambridge Philos. Soc. 118 (1995), 245257. 2. M. Hochster, CohenMacaulay rings, combinatorics, and simplical complexes, in ``Ring Theory II'' (B. R. McDonald and R. Morris, Eds.), Lecture Notes in Pure and Appl. Math., Vol. 26, pp. 171223, Dekker, New York, 1977. File: 607J 159602 . By:CV . Date:26:12:96 . Time:10:02 LOP8M. V8.0. Page 01:01 Codes: 2337 Signs: 1706 . Length: 45 pic 0 pts, 190 mm.