Support theories for non-Noetherian tensor triangulated categories

TL;DR

This paper extends support theory to non-Noetherian tensor triangulated categories, enabling stratification of categories with non-Noetherian cohomology rings and comparing different support theories.

Contribution

It introduces a new small support notion, compares support theories, and establishes conditions under which they coincide or fail to stratify categories.

Findings

Stable module category is stratified by Tate cohomology ring.

Support theories coincide when the spectrum is homeomorphic to the Zariski spectrum.

Support theory can only stratify spectra that are weakly Noetherian.

Abstract

We extend the support theory of Benson--Iyengar--Krause to the non-Noetherian setting by introducing a new notion of small support for modules. This enables us to prove that the stable module category of a finite group is canonically stratified by the action of the Tate cohomology ring, despite the fact that this ring is rarely Noetherian. In the tensor triangular context, we compare the support theory proposed by W. Sanders (which extends the Balmer--Favi support theory beyond the weakly Noetherian setting) with our generalized BIK support theory. When the Balmer spectrum is homeomorphic to the Zariski spectrum of the endomorphism ring of the unit, the two support theories coincide as do their associated theories of stratification. We also prove a negative result which states that the Balmer--Favi--Sanders support theory can only stratify categories whose spectra are weakly Noetherian. This provides additional justification for the weakly Noetherian hypothesis in the work of Barthel, Heard and B. Sanders. On the other hand, the detection property and the local-to-global principle remain interesting in the general setting.