Resolutions of Stanley-Reisner rings and Alexander duality

Abstract

Associated to any simplicial complex Δ on n vertices is a square-free monomial ideal I Δ in the polynomial ring A = k[ x 1, …, x n ], and its quotient k[Δ] = A I Δ known as the Stanley-Reisner ring. This note considers a simplicial complex Δ ∗ which is in a sense a canonical Alexander dual to Δ, previously considered in [1, 5]. Using Alexander duality and a result of Hochster computing the Betti numbers dim k Tor i A ( k[ Δ], k), it is shown (Proposition 1) that these Betti numbers are computable from the homology of links of faces in Δ ∗. As corollaries, we prove that I Δ has a linear resolution as A-module if and only if Δ ∗ is Cohen-Macaulay over k, and show how to compute the Betti numbers dim k Tor i A ( k[ Δ], k) in some cases where Δ ∗ is wellbehaved (shellable, Cohen-Macaulay, or Buchsbaum). Some other applications of the notion of shellability are also discussed.