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 $\Delta$ satisfies Serre's condition ($S_r$) if $\tilde H_i(\lk_Δ(F);K)=0$ for all faces $F\in \Delta$ and for all $i < \min \{r-1,\dim \lk_Δ(F)\}$, where $\lk_Δ(F)$ 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 $k=0,1,...,r$ and (ii) ${\sum }_{k\ge r}{h}_k(\Delta )$ is non-negative.

To appear in Math. Res. Lett

Cited by in corpus (1)

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