Free resolutions of simplicial posets

A simplicial poset, a poset with a minimal element and whose every interval is a Boolean algebra, is a generalization of a simplicial complex. Stanley defined a ring A, associated with a simplicial poset P that generalizes the face-ring of a simplicial complex. If V is the set of vertices of P, then A(p) is a k[V]-module; we find the Betti polynomials of a free resolution of A(p), and the local cohomology modules of A(p), generalizing Hochster's corresponding results for simplicial complexes. The proofs involve splitting certain chain or cochain complexes more finely than in the simplicial complex case. Corollaries are that the depth of A(p) is a topological invariant, and that the depth may be computed in terms of the Cohen-Macaulayness of skeleta of P, generalizing results of Munkres and Hibi. (C) 1997 Academic Press.