On unmixed and equi-dimensional associated graded rings

TL;DR

This paper investigates the properties of associated graded rings of certain local rings, establishing conditions under which integral and tight closures coincide or behave polynomially, with bounds on Hilbert coefficients.

Contribution

It proves that the integral closure function is polynomial of degree d-1 or the closures coincide, extending to tight closure in characteristic p, with bounds for Hilbert coefficients.

Findings

The function $ar{I^n}/I^n$ behaves polynomially of degree $d-1$ or closures coincide.

Analogous results hold for tight closure in characteristic p.

Bounds are provided for first Hilbert coefficients in specific cases.

Abstract

Let $(A,\mathfrak{m})$ be an analytically un-ramified Noetherian local ring of dimension $d \geq 1$, $I$ a regular $\mathfrak{m}$-primary ideal of $A$ and let $\overline{I}$ be integral closure ideal of $I$. If $A$ is of characteristic $p > 0$ then let $I^*$ denote the tight closure of $I$. Let $G_I(A)=\bigoplus_{n\geq 0}I^n/I^{n+1}$ be the associated graded ring of $A$ with respect to $I$. Assume $G_I(A)$ is unmixed and equi-dimensional. We show that either the function $P_{\overline{I}} :\,n\mapsto \lambda(\overline{I^n}/I^n)$ is a polynomial type of degree $d-1$ or $\overline{I^n}=I^n$ for all $n\geq 1.$ We prove an analogus result for the tight closure filtration if $A$ is of characteristic $p > 0$. When $A$ is generalized Cohen-Macaulay and $I$ is generated by standard system of parameters we give bounds for the first Hilbert coefficients of the integral closure filtration of $I$ and the tight closure filtration of $I$.