Categorical resolutions of a class of derived categories

TL;DR

This paper demonstrates that under certain conditions on an Artin algebra, its bounded derived category admits a categorical resolution and desingularization, with special cases being weakly crepant for CM-finite Gorenstein algebras.

Contribution

It establishes conditions under which the bounded derived category of an Artin algebra admits a categorical resolution and desingularization using relative derived categories.

Findings

Existence of categorical resolutions for specific Artin algebras.

Conditions for weakly crepant resolutions in Gorenstein cases.

Results extend to derived categories of modules, not just finitely generated ones.

Abstract

By using the relative derived categories, we prove that if an Artin algebra $A$ has a module $T$ with ${\rm inj.dim}T<\infty$ such that $^\perp T$ is finite, then the bounded derived category $D^b(A\mbox{-}{\rm mod})$ admits a categorical resolution in the sense of [Kuz], and a categorical desingularization in the sense of [BO]. For CM-finite Gorenstein algebra, such a categorical resolution is weakly crepant. The similar results hold also for $D^b(A\mbox{-}{\rm Mod})$.