RESOLUTIONS OF STANLEY-REISNER RINGS AND ALEXANDER DUALITY

Associated to any simplicial complex Delta on n vertices is a square-f ree monomial ideal I-Delta in the polynomial ring A = k[x(l),...,x(n)] , and its quotient k[Delta] =A/I-Delta known as the Stanley-Reisner ri ng. This note considers a simplicial complex Delta which is in a sens e a canonical Alexander dual to Delta. previously considered in [1, 5] . Using Alexander duality and a result of Hochster computing the Betti numbers dim(k) Tor(i)(A)(k[Delta],k) it is shown (Proposition 1) that these Betti numbers are computable from the homology of links of face s in Delta. As corollaries, we prove that I-Delta has a linear resolu tion as A-module if and only if Delta is Cohen-Macaulay over k, and s how how to compute the Betti numbers dim(k)Tor(i)(A)(k[Delta],k) in so me cases where Delta is well-behaved (shellable, Cohen-Macaulay, or B uchsbaum). Some other applications of the notion of shellability are a lso discussed. (C) 1998 Elsevier Science B.V. All rights reserved.