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

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 $G^{\ast }(I)$ of the filtration $\{(I^n{)}^{\ast }:n\in \mathbb{N}\}$ is Cohen-Macaulay. We prove that if $R$ is an excellent Buchsbaum local domain then $G^{\ast }(I)$ is a Buchsbaum module over the Rees ring ${\mathcal{R}}^{\ast }(I)={\oplus }_{n\in \mathbb{N}}(I^n{)}^{\ast }.$ We provide quick proofs of well-known results of I. Aberbach, Huneke-Itoh and Huneke-Hochster about the filtration $\{(I^n{)}^{\ast }:n\in \mathbb{N}\}$ 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 $R={\mathbb{F}}_p[X_0,\dots ,X_d]/(X_0^r+\cdots +X_d^r).$ In such cases we prove that $G^{\ast }(I)$ is Cohen-Macaulay. We provide conditions on $r,d,e$ for the Rees algebra ${\mathcal{R}}^{\ast }(I)$ to be Cohen-Macaulay.