Algebra, Combinatorics, and Computation of Certain Tight Closure Invariants in Stanley-Reisner Rings
Date
2019
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Abstract
Tight closure was first introduced in the 1980's \cite{HHOrigin} by Hochster and Huneke to answer questions about invariant theory and the Brian\c{c}on-Skoda theorem. It has since come into its own as a fairly robust theory. The tight closure $I^*$ of an ideal $I$ is named as such because it is, in general, contained in, but not equal to, the integral closure $I^-$ of the same ideal, so it is a ``tighter'' closure operator than integral closure. Tight closure is notoriously difficult to compute for an arbitary ideal, but with certain rings, this task is less arduous. In this dissertation, we build a a bridge between tight closure theory and combinatorics by way of simplicial complexes and Stanley-Reisner rings. We discuss the specifics of tight closure theory and Stanley-Reisner rings and make special effort to focus on the standard results of both topics that will be most useful to our purposes. We discuss the analogous notions for $*$-reductions and reductions of ideals for tight and integral closure repectively. When we focus our attention on the maximal ideal, $\mathfrak{m}$, of the Stanley-Reisner ring $k[\Delta]$ that is generated by the variables of the ring, we observe that if $I$ is a reduction of $\mathfrak{m}$, then it is also a $*$-reduction of $\mathfrak{m}$. We will determine the the minimal number of generaters of a $*$-reduction of $\mathfrak{m}$, called the $*$-spread of $\mathfrak{m}$, and the intersection of all minimally generated $*$-reductions of $\mathfrak{m}$, called the $*\core$ of $\mathfrak{m}$. These notions were introduced by Epstein \cite{nme*spread} and Fouli and Vassilev \cite{FoVa-core} respectively. We endeavor to describe both in terms of the Stanley-Reisner ring and the simplicial complex of the Stanley-Reisner ring. Finally, we examine $*\core{\mathfrak{m}}$ in specific examples and in slightly more general cases of Stanley-Reisner rings. These include dimension 1 Stanley-Reisner rings, Stanley-Reisner rings with disconnected simplicial complexes, and Stanley-Reisner rings with a graph for a simplicial complex.