Linearisations of triangulated categories with respect to finite group actions

TL;DR

This paper explores how to construct linearized triangulated categories under finite group actions, especially in geometric contexts like derived categories of coherent sheaves on varieties and stacks.

Contribution

It provides conditions and methods for linearizing triangulated categories with finite group actions, extending to geometric cases involving smooth projective varieties and stacks.

Findings

Constructs linearized categories using DG-enhancements.

Produces derived categories of quotient varieties and stacks.

Applies to actions by automorphisms, tensoring with torsion bundles, and spherical objects.

Abstract

Given an action of a finite group on a triangulated category, we investigate under which conditions one can construct a linearised triangulated category using DG-enhancements. In particular, if the group is a finite group of automorphisms of a smooth projective variety and the category is the bounded derived category of coherent sheaves, then our construction produces the bounded derived category of coherent sheaves on the smooth quotient variety resp. stack. We also consider the action given by the tensor product with a torsion canonical bundle and the action of a finite group on the category generated by a spherical object.