Alexander Duality Theorem and Second Betti Numbers of Stanley–Reisner Rings

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

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

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.

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