Categorical resolutions of a class of derived categories

TL;DR

This paper demonstrates that certain Artin algebras have their bounded derived categories admit categorical resolutions, with some cases being weakly crepant, using relative derived categories.

Contribution

It introduces a method to construct categorical resolutions of derived categories for specific classes of Artin algebras using relative derived categories.

Findings

Categorical resolutions exist for Artin algebras with finite injective dimension modules.

For CM-finite Gorenstein algebras, these resolutions are weakly crepant.

The approach leverages properties of relative derived categories.

Abstract

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 $D^b({\rm mod}A)$ admits a categorical resolution; and that for CM-finite Gorenstein algebra, such a categorical resolution is weakly crepant.