Free Resolutions of Simplicial Posets Original Research Article

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 ringAPassociated with a simplicial posetPthat generalizes the face-ring of a simplicial complex. IfVis the set of vertices ofP, thenAPis ak[V]-module; we find the Betti polynomials of a free resolution ofAP, and the local cohomology modules ofAP, generalizing Hochsterʹs 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 ofAPis a topological invariant, and that the depth may be computed in terms of the Cohen-Macaulayness of skeleta ofP, generalizing results of Munkres and Hibi.