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

✍ Scribed by Naoki Terai; Takayuki Hibi

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 229 KB
• Volume: 124
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1006/aima.1996.0086

✦ Synopsis

A simplicial complex 2 on the vertex set V=[x 1 , x 2 , ..., x v ] is a collection of subsets of V such that (i)

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 i 0. We write |2| for the geometric realization of 2.

Let A=k[x 1 , x 2 , ..., x v ] be the polynomial ring in v-variables over a field k. Here, we identify each x i # V with the indeterminate x i of A. Define I 2 to be the ideal of A which is generated by square-free monomials

In what follows, we consider A to be the graded algebra A= n 0 A n with the standard grading, i.e., each deg x i =1, and may regard k

) 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

We study a graded minimal free resolution 0 wÄ j # Z A(&j ) ; h j wÄ . h } } } wÄ . 2 j # Z A(&j ) ; 1 j wÄ . 1 A wÄ . 0 k[2] wÄ 0 article no. 0086 332