On the Gorenstein defect categories

Abstract

The main theme of this paper is to study different “Gorenstein defect categories” and their connections. This is done by studying rings for which ${\mathbb{𝕂}}_{ac}\left(\text{ Prj-}R\right)={\mathbb{𝕂}}_{tac}\left(\text{ Prj-}R\right)$, that is, rings enjoying the property that every acyclic complex of projectives is totally acyclic. Such studies have been started by Iyengar and Krause over commutative Noetherian rings with a dualizing complex. We show that a virtually Gorenstein Artin algebra is Gorenstein if and only if it satisfies the above mentioned property. Then, we introduce recollements connecting several categories which help in providing categorical characterizations of Gorenstein rings. Finally, we study relative singularity categories that lead us to some more “Gorenstein defect categories”.