h-Vectors of Gorenstein* simplicial posets

As is well known, h-vectors of simplicial convex polytopes are characterized. Those h-vectors satisfy Dehn-Sommerville equations and some inequalities conjectured by P. McMullen and first proved by R. Stanley using toric geometry. The boundary of a simplicial convex polytope determines a Gorenstein* simplicial poset but there are many Gorenstein* simplicial posets which do not arise this way. However, it is known that h-vectors of Gorenstein* simplicial posets still satisfy Dehn-Sommerville equations and that every component in the h-vectors is non-negative. In this paper we prove that h-vectors of Gorenstein* simplicial posets must satisfy one more subtle condition conjectured by R. Stanley and complete the characterization of h-vectors of Gorenstein* simplicial posets. Our proof is purely algebraic but the idea of the proof stems from topology.