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 k[Δ] = A/IΔ 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 dimkToriA(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 dimkToriA(k[Δ],k) in some cases where Δ* is well-behaved (shellable, Cohen-Macaulay, or Buchsbaum). Some other applications of the notion of shellability are also discussed.