Associated to any simplicial complex Δ on n vertices is a square-free monomial ideal IΔ in the polynomial ring A = k[x1, …, xn], and its quotient 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 dimk ToriA(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 IΔ 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 dimk ToriA (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.
