Noncommutative Gorenstein Projective Schemes and Gorenstein-Injective Sheaves

TL;DR

This paper establishes conditions under which noncommutative projective schemes derived from Gorenstein graded rings are Gorenstein categories, and relates Gorenstein-injective sheaves to modules via a recollement.

Contribution

It proves that Gorenstein properties of a graded ring extend to the associated noncommutative projective scheme and constructs a recollement linking sheaves and modules.

Findings

${ m Tails}(R)$ is Gorenstein under specified conditions.

A recollement relates Gorenstein-injective sheaves and modules.

Conditions involve finite cohomological dimension of the torsion functor.

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.