Support theory via actions of tensor triangulated categories

TL;DR

This paper introduces a new framework for actions of tensor triangulated categories on triangulated categories, extending existing support theories and establishing a local-to-global principle with implications for the telescope conjecture.

Contribution

It defines the action of tensor triangulated categories on triangulated categories and extends support theory, including a local-to-global principle and a relative telescope conjecture.

Findings

Supports categorify previous work of Benson, Iyengar, and Krause

A version of the local-to-global principle is proved generally

Provides a sufficient condition for the relative telescope conjecture

Abstract

We give a definition of the action of a tensor triangulated category T on a triangulated category K. In the case that T is rigidly-compactly generated and K is compactly generated we show this gives rise to a notion of supports which categorifies work of Benson, Iyengar, and Krause and extends work of Balmer and Favi. We prove that a suitable version of the local-to-global principle holds very generally. A relative version of the telescope conjecture is formulated and we give a sufficient condition for it to hold.