The derived category of a graded Gorenstein ring

TL;DR

This paper generalizes Orlov's theorem to a broader class of graded Gorenstein rings, including non-commutative cases, and establishes an equivalence involving derived categories and matrix factorizations.

Contribution

It extends Orlov's theorem to non-negatively graded Gorenstein rings with non-commutative degree zero components, broadening its applicability.

Findings

Theorem holds for non-commutative rings of finite global dimension.

Establishes an equivalence between derived categories and homotopy categories of matrix factorizations.

Provides foundational results for local cohomology and Grothendieck duality in this context.

Abstract

We give an exposition and generalization of Orlov's theorem on graded Gorenstein rings. We show the theorem holds for non-negatively graded rings which are Gorenstein in an appropriate sense and whose degree zero component is an arbitrary non-commutative right noetherian ring of finite global dimension. A short treatment of some foundations for local cohomology and Grothendieck duality at this level of generality is given in order to prove the theorem. As an application we give an equivalence of the derived category of a commutative complete intersection with the homotopy category of graded matrix factorizations over a related ring.