Topological applications of Stanley-Reisner rings of simplicial complexes

Abstract:

Methods of commutative and homological algebra yield information on the Stanley-Reisner ring $\Bbbk [K]$ of a simplicial complex $K$. Consider the following problem: describe topological properties of simplicial complexes with given properties of the ring $\Bbbk [K]$. It is known that for a simplicial complex $K=\partial P^*$, where $P^*$ is a polytope dual to the simple polytope $P$ of dimension $n$, the depth of $\operatorname {depth}\Bbbk [K]$ equals $n$. A recent construction allows us to associate a simplicial complex $K_P$ to any convex polytope $P$. As a consequence, one wants to study the properties of the rings $\Bbbk [K_P]$. In this paper, we report on the obtained results for both of these problems. In particular, we characterize the depth of $\Bbbk [K]$ in terms of the topology of links in the complex $K$ and prove that $\operatorname {depth}\Bbbk [K_P] = n$ for all convex polytopes $P$ of dimension $n$. We obtain a number of relations between bigraded betti numbers of the complexes $K_P$. We also establish connections between the above questions and the notion of a $k$-Cohen-Macaulay complex, which leads to a new filtration on the set of simplicial complexes.