Formal extension of noncommutative tensor-triangular support varieties

TL;DR

This paper extends support variety theory from compact to non-compact objects in monoidal triangulated categories, generalizing previous frameworks and providing conditions for zero object detection in noncommutative settings.

Contribution

It introduces a method to extend support varieties to non-compact objects and characterizes when these extended theories detect zero objects, especially in noncommutative contexts.

Findings

Extended support varieties detect zero objects under certain conditions.

Generalized support theory applies to noncommutative tensor categories.

Confirms a conjecture on cohomological support in finite tensor categories.

Abstract

Given a support variety theory defined on the compact part of a monoidal triangulated category, we define an extension to the non-compact part following the blueprint of Benson--Carlson--Rickard, Benson--Iyengar--Krause, Balmer--Favi, and Stevenson. We generalize important aspects of the theory of extended support varieties to the noncommutative case, and give characterizations of when an extended support theory detects the zero object, under certain assumptions. In particular, we show that when the original support variety theory is based on a Noetherian topological space, detects the zero object, satisfies a generalized tensor product property, and comes equipped with a comparison map, then the extended support variety also detects the zero object. In the case of stable categories of finite tensor categories, this gives conditions under which the central cohomological support admits an extension that detects the zero object, confirming part of a recent conjecture made by the second author together with Nakano and Yakimov.