Some characterizations of Gorenstein Rees Algebras

TL;DR

This paper explores the conditions under which Gorenstein Rees algebras are characterized via the graded canonical module of extended Rees algebras, relaxing previous Cohen-Macaulay assumptions and applying to quasi-Gorenstein rings.

Contribution

It provides a new characterization of Gorenstein Rees algebras using the graded canonical module of the extended Rees algebra without requiring Cohen-Macaulay conditions.

Findings

Characterization of Gorenstein Rees algebras via graded canonical modules.

Application to Kawasaki's arithmetic Cohen-Macaulayfication becoming Gorenstein.

Extension of results to quasi-Gorenstein rings with finite local cohomology.

Abstract

The aim of this paper is to elucidate the relationship between the Gorenstein Rees algebra $\R(I):=\bigoplus_{i\ge 0}I^i$ of an ideal $I$ in a complete Noetherian local ring $A$ and the graded canonical module of the extended Rees algebra $\R'(I):=\bigoplus_{i\in\Z}I^i$. It is known that the Gorensteinness of $\R(I)$ is closely related to the property of the graded canonical module of the associated graded ring $\G(I):=\bigoplus_{i\ge 0}I^i/I^{i+1}$. However, there appears to be a shortage of satisfactory references analyzing the relationship between $\R(I)$ and $\R'(I)$ unless the ring $\G(I)$ is Cohen-Macaulay. This paper provides a characterization of the Gorenstein property of $\R(I)$ using the graded canonical module of $\R'(I)$ without assuming that the base ring $A$ is Cohen-Macaulay. Applying our criterion, we demonstrate that a certain Kawasaki's arithmetic Cohen-Macaulayfication becomes a Gorenstein ring when $A$ is a quasi-Gorenstein local ring with finite local cohomology.