Lexsegment ideals and Simplicial Cohomology groups

Let $V$ be a $k-$vector space with basis $e_1,\ldots, e_n$ and let $E$ be the exterior algebra over $V$. For any subset $\sigma=\{i_1,\ldots,i_d\}$ of $\{1,\ldots,n\}$ with $i_1<i_2<\ldots<i_d$ we call $e_\sigma=e_{i_1}\wedge\ldots\wedge e_{i_d}$ a monomial of degree $d$ and we denote the set of all monomials of <a class="inlineAdmedialink" href="#">degree</a> $d$ by $M_d$. We order the monomials lexicographically so that $e_1>e_2>\ldots>e_n$. Then a lexsegment ideal is an ideal generated by a subset of $M_d$ of the form $L(u,v)=\{w\in M_d:u\geq w\geq v\}$, where $u,\; v\in M_d$ and $u\geq v.$ We describe all lexsegment ideals with linear resolution in the exterior algebra. Then we study the vanishing and non vanishing of reduced simplicial cohomology groups of a simplicial complex $\Delta$ and of certain subcomplexes of $\Delta$ with coefficients in a field $k.$ Finally we give an idea of the applicative aspects of our results.