The $*$-core of the graded maximal ideal in a Stanley-Reisner ring

TL;DR

This paper investigates ideals in Stanley-Reisner rings whose tight closure equals the maximal ideal, determining minimal generators and exploring their intersections and relation to integral closure.

Contribution

It characterizes ideals with tight closure equal to the maximal ideal in Stanley-Reisner rings, including their minimal generators and intersection properties.

Findings

Minimal number of generators is dimension of complex plus one.

Bounded the intersection of all such ideals.

Connected the results to integral closure concepts.

Abstract

We consider ideals $I$ in a Stanley-Reisner ring $k[\Delta]$ over the simplical complex $\Delta$, such that the tight closure of $I$, $I^*$, is equal to $\mathfrak{m}$, the standard graded maximal ideal of $k[\Delta]$. We determine the minimal number of generators of $I$ to be the $\dim \Delta+1$ and note the important role this value plays in bounding the intersection of all such ideals $I$. We make mention of this intersection in special cases of Stanley-Reisner rings. We conclude with a description of how this work relates to integral closure.