Some characterizations of Gorenstein Rees Algebras

Abstract

The aim of this paper is to elucidate the relationship between the Gorenstein Rees algebra $\mathbb{R}(I):={⨁}_{i\ge 0}{I}^i$ of an ideal $I$ in a complete Noetherian local ring $A$ and the graded canonical module of the extended Rees algebra ${\mathbb{R}}^{\prime }(I):={⨁}_{i\in \mathbb{Z}}{I}^i$. It is known that the Gorensteinness of $\mathbb{R}(I)$ 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 $\mathbb{R}(I)$ and ${\mathbb{R}}^{\prime }(I)$ unless the ring $\G(I)$ is Cohen-Macaulay. This paper provides a characterization of the Gorenstein property of $\mathbb{R}(I)$ using the graded canonical module of ${\mathbb{R}}^{\prime }(I)$ 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.