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 $\Delta$ satisfies Serre’s condition ($S_r$) if $\tilde H_i(\lk_\Delta(F);K)=0$ for all $F \in \Delta$ and for all $i <\min \{r-1,\dim \lk_\Delta(F)\}$, where $\lk_\Delta(F)$ is the link of $\Delta$ with respect to $F$ and where $\tilde H_i(\Delta;K)$ is the reduced homology groups of $\Delta$ over a field $K$. The main result of this paper is that if $\Delta$ satisfies Serre’s condition ($S_r$) then (i) $h_k(\Delta)$ is non-negative for $k =0,1,\dots,r$ and (ii) $\sum_{k\geq r}h_k(\Delta)$ is non-negative.

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Published 1 January 2009