H-vectors of simplicial complexes with Serre's conditions

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