Noncommutative Gorenstein Projective Schemes and Gorenstein-Injective Sheaves

Abstract

We prove that if a positively-graded ring $R$ is Gorenstein and the associated torsion functor has finite cohomological dimension, then the corresponding noncommutative projective scheme ${\rm Tails}(R)$ is a Gorenstein category in the sense of \cite{EEG}. Moreover, under this condition, a (right) recollement relating Gorenstein-injective sheaves in ${\rm Tails}(R)$ and (graded) Gorenstein-injective $R$-modules is given.