Filtrations via tensor actions

TL;DR

This paper generalizes the concept of filtrations in tensor triangulated categories to actions of such categories on other categories, applying this to Gorenstein schemes and refining filtrations for local complete intersections.

Contribution

It extends Balmer's filtrations to tensor actions, providing new proofs and refinements for Gorenstein schemes and local complete intersections.

Findings

Filtrations can be constructed via tensor actions in triangulated categories.

The approach yields new proofs for existing results on Gorenstein injective sheaves.

Refined filtrations and spectral sequences are developed for local complete intersections.

Abstract

We extend work of Balmer, associating filtrations of essentially small tensor triangulated categories to certain dimension functions, to the setting of actions of rigidly-compactly generated tensor triangulated categories on compactly generated triangulated categories. We show that the towers of triangles associated to such a filtration can be used to produce filtrations of Gorenstein injective quasi-coherent sheaves on Gorenstein schemes. This extends and gives a new proof of a result of Enochs and Huang. In the case of local complete intersections a further refinement of this filtration is given and we comment on some special properties of the associated spectral sequence in this case.