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_P 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.