H-vectors of simplicial complexes with Serre's conditions

We study h-vectors of simplicial complexes which satisfy Serre's condition (S_r). Let r be a positive integer. We say that a simplicial complex △ satisfies Serre's condition (S_r) if H̃ _i(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 H̃_i(△;K) is the reduced homology groups of △ over a field K. The main result of this paper is that if △ satisfies Serre's condition (S_r) then (i) h_k(△) is non-negative for k = 0, 1, . . ., r and (ii) ∑_k≥r h_k(△) is non-negative.