Homological stratification and descent

TL;DR

This paper develops a new notion of stratification for tensor-triangulated categories based on the homological spectrum, proving it has strong descent properties and applying it to equivariant module spectra for compact Lie groups.

Contribution

It introduces a homological stratification framework with descent properties and extends tensor triangular geometry results from finite to compact Lie groups.

Findings

Stratification admits a general form of descent.

Provides a uniform approach to recent stratification results.

Extends tensor triangular geometry to compact Lie groups.

Abstract

We introduce a notion of stratification for rigidly-compactly generated tensor-triangulated categories relative to the homological spectrum and develop the fundamental features of this theory. In particular, we demonstrate that it exhibits excellent descent properties. In conjunction with Balmer's Nerves of Steel conjecture, we conclude that stratification admits a general form of descent. This gives a uniform treatment of several recent stratification results and provides a complete answer to the question: When does stratification descend? As a new application, we extend earlier work on the tensor triangular geometry of equivariant module spectra from finite groups to compact Lie groups.