Some characterizations of Gorenstein Rees Algebras

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 ${\mathbb{R}}^{\mathrm{\prime }}(I):={⨁}_{i\in \mathbb{Z}}{I}^i$. 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 $\text{G}(I):={⨁}_{i\ge 0}{I}^i\mathrm{/}{I}^{i+1}$. However, there appears to be a shortage of satisfactory references analyzing the relationship between $\mathbb{R}(I)$ and ${\mathbb{R}}^{\mathrm{\prime }}(I)$ unless the ring $\text{G}(I)$ is Cohen-Macaulay. This paper provides a characterization of the Gorenstein property of $\mathbb{R}(I)$ using the graded canonical module of ${\mathbb{R}}^{\mathrm{\prime }}(I)$ 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.