Tight Closure of powers of ideals and tight Hilbert polynomials

TL;DR

This paper introduces the tight Hilbert function and polynomial for ideals in prime characteristic rings, showing how $F$-rationality relates to these polynomials and computing them for specific classes of rings.

Contribution

It defines the tight Hilbert polynomial, links its coefficients to $F$-rationality, and computes these polynomials for certain classes of rings, advancing understanding of tight closure theory.

Findings

$F$-rationality characterized by vanishing of the first coefficient of $P_I^*(n)$

Computed tight Hilbert polynomials for parameter ideals in hypersurface and Stanley-Reisner rings

Established connections between tight closure properties and Hilbert functions in positive characteristic

Abstract

Let $(R,\mathfrak m)$ be an analytically unramified local ring of positive prime characteristic $p.$ For an ideal $I$, let $I^*$ denote its tight closure. We introduce the tight Hilbert function $H^*_I(n)=\ell(R/(I^n)^*)$ and the corresponding tight Hilbert polynomial $P_I^*(n)$ where $I$ is an $\mathfrak m$-primary ideal. It is proved that $F$-rationality can be detected by the vanishing of the first coefficient of $P_I^*(n).$ We find the tight Hilbert polynomial of certain parameter ideals in hypersurface rings and Stanley-Reisner rings of simplicial complexes.