Homological algebra of Stanley-Reisner rings and modules

Abstract

Associated to each simplicial complex $\Delta$ and each field $\field$ is the Stanley--Reisner ring $\field[\Delta]$. The answers to a multitude of questions related to simplicial complexes have historically been found through a thorough examination of the algebraic structure of $\field[\Delta]$. There is a rich pre-existing body of literature equating combinatorial and topological statements about the structure of a simplicial complex with statements about $\field[\Delta]$; this dissertation expands upon the dictionary translating such statements by examining algebraic structures derived from $\field[\Delta]$. In particular, we mainly focus on the local cohomology modules $H_\mideal^i(\field[\Delta])$ and the Ext modules $\Ext^i(\field, \field[\Delta])$. Roughly speaking, a simplicial complex is called Buchsbaum if its geometric realization is similar to a manifold. In Chapter 2, we study the homological structure of $\field[\Delta]$ and some of its quotients by linear forms when $\Delta$ fails to be Buchsbaum in a way that may be considered ``minimal.'' We obtain a large family of rings with interesting combinations of the (ring-theoretic) properties of Buchsbaumness and quasi-Buchsbaumness, while developing a geometric interpretation of their presence. In Chapter 3, we turn our attention to complexes that exhibit some degree of symmetry via group actions. Here it is shown that the induced action on $H_\mideal^i(\field[\Delta])$ can be described in a similar manner to the one induced on the simplicial cohomology modules of $\Delta$ and some of its subcomplexes. Some applications to the study of face numbers are provided. If the definition of a simplicial complex is slightly relaxed, then one arrives at the notion of a simplicial poset. Chapter 4 is devoted to the study of these objects and their associated face rings. We provide extensions of well-known results describing the structure of the Ext and local cohomology modules of simplicial complexes to this larger class and further examine the Buchsbaum property. In Chapter 5, we study the class of balanced triangulations of manifolds and obtain lower bounds on entries in the $h$-vector phrased in terms of topological invariants. This proves a conjecture of Klee and Novik.