2024/05/29 by Iai, Shin-ichiro
#Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.2405.18963
The aim of this paper is to elucidate the relationship between the Gorenstein Rees algebra \R(I):=\bigoplusi≥ 0Ii of an ideal I in a complete Noetherian local ring A and the graded canonical module of the extended Rees algebra \R'(I):=\bigoplusi∈\ZIi. 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):=\bigoplusi≥ 0Ii/Ii+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.