H-vectors of simplicial complexes with Serre’s conditions

Abstract

We study h-vectors of simplicial complexes which satisfy Serre's condition (Sr). Let r be a positive integer. We say that a simplicial complex satisfies Serre's condition (Sr) if Hi (lk (F ); K) = 0 for all F and for all i < min{r-1, dim lk (F )}, where lk (F ) is the link of with respect to F and where Hi (; K) is the reduced homology groups of over a field K. The main result of this paper is that if satisfies Serre's condition (Sr) then (i) h k () is non-negative for k = 0, 1, . . . , r and (ii) P kr h k () is non-negative.