H-vectors of simplicial complexes with Serre's conditions

Abstract

We study $h$-vectors of simplicial complexes which satisfy Serre's condition ($S_r$). We say that a simplicial complex $Δ$ satisfies Serre's condition ($S_r$) if $\tilde H_i(\lk_Δ(F);K)=0$ for all faces $F \in Δ$ and for all $i < \min \{r-1,\dim \lk_Δ(F)\}$, where $\lk_Δ(F)$ is the link of $Δ$ with respect to $F$ and where $\tilde H_i(Δ;K)$ is the reduced homology groups of $Δ$ over a field $K$. The main result of this paper is that if $Δ$ satisfies Serre's condition ($S_r$) then (i) $h_k(Δ)$ is non-negative for $k =0,1,...,r$ and (ii) $\sum_{k\geq r}h_k(Δ)$ is non-negative.

To appear in Math. Res. Lett