Serre's Condition and Depth of Stanley-Reisner Rings

Abstract

The aim of this work is to garner a deeper understanding of the relationship between depth of a ring and connectivity properties of the spectrum of that ring. We examine with particular interest the case where our ring is a Stanley- Reisner ring. In this circumstance, we consider the simplicial complex that corresponds to the spectrum of R. We examine properties of simplicial complexes whose Stanley-Reisner rings satisfy depth conditions such as Cohen-Macaulay and Serre's condition (S_l). We leverage these properties to use algebraic tools to examine combinatorial problems. For example, the gluing lemma in (Hol18) allows us to construct bounds on the diameter of a class of graphs acting as a generalization of the 1-skeleton of polytopes. Throughout, we give special consideration to Serre's condition (S_l). We create a generalized Serre's condition (S_l^j) and prove equivalent homological, topological, and combinatorial properties for this condition. We generalize many well-known results pertaining to (S_l) to apply to (S_l^j). This work also explores a generalization of the nerve complex and considers the correlation between the homologies of the nerve complex of a Stanley-Reisner ring and depth properties of that ring. Finally we explore rank selection theorems for simplicial complexes. We prove many results on depth properties of simplicial complexes. In particular, we prove that rank selected subcomplexes of balanced (S_l) simplicial complexes retain (S_l). The primary focus of this work is on Stanley-Reisner rings, however, other commutative, Noetherian rings are also considered.