Associated graded rings of the filtration of tight closure of powers of parameter ideals

TL;DR

This paper investigates the properties of the associated graded rings of the tight closure filtration of parameter ideals in excellent Cohen-Macaulay and Buchsbaum local domains, establishing Cohen-Macaulay and Buchsbaum properties and computing tight closures in specific hypersurface rings.

Contribution

It proves that the associated graded ring of the tight closure filtration is Cohen-Macaulay in excellent Cohen-Macaulay domains and Buchsbaum in Buchsbaum domains, providing new insights and simplified proofs.

Findings

$G^*(I)$ is Cohen-Macaulay in excellent Cohen-Macaulay domains.

$G^*(I)$ is a Buchsbaum module over the Rees ring in Buchsbaum domains.

Explicit computation of tight closure in hypersurface rings with homogeneous parameters.

Abstract

Let $I$ be an ideal generated by a system of parameters in an excellent Cohen-Macaulay local domain. We show that the associated graded ring $G^*(I)$ of the filtration $\{(I^n)^*: n\in \mathbb{N}\}$ is Cohen-Macaulay. We prove that if $R$ is an excellent Buchsbaum local domain then $G^*(I)$ is a Buchsbaum module over the Rees ring $\mathcal R^*(I)=\oplus_{n\in \mathbb{N}}(I^n)^*.$ We provide quick proofs of well-known results of I. Aberbach, Huneke-Itoh and Huneke-Hochster about the filtration $\{(I^n)^*: n\in \mathbb{N}\}$ in excellent local domains. An important tool used in the proofs is a deep result due to M. Hochster and C. Huneke which states that the absolute integral closure of an excellent local domain is a big Cohen-Macaulay algebra. We compute the tight closure of $I^n$ where $I$ is generated by homogeneous system of parameters having the same degree $e$ in the hypersurface ring $R=\mathbb{F}_p[X_0,\ldots ,X_d]/(X_0^r+\cdots+X_d^r).$ In such cases we prove that $G^*(I)$ is Cohen-Macaulay. We provide conditions on $r, d, e$ for the Rees algebra $\mathcal R^*(I)$ to be Cohen-Macaulay.