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Homological stratification and descent

Tobias Barthel, Drew Heard, Beren Sanders, Changhan Zou

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.

Homological stratification and descent

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.

Paper Structure

This paper contains 12 sections, 67 theorems, 122 equations, 1 figure.

Table of Contents

  1. Homological support and cosupport
  2. The homological local-to-global principle
  3. Homological stratification
  4. Homological stratification via cosupport
  5. Comparison with the Balmer--Favi support
  6. Base change for homological support
  7. Weakly descendable functors and descent
  8. Comparison between h-stratification and tt-stratification
  9. Descent for the Nerves of Steel conjecture
  10. Applications to tt-stratification
  11. Equivariant applications
  12. Open questions

Key Result

Theorem A

Let $(f_i^*\colon \mathscr{T} \to \mathscr{S}_i)_{i \in I}$ be a family of geometric functors that jointly detect when an object of $\mathscr{T}$ is zero. Suppose that $\mathscr{S}_i$ is tt-stratified for all $i \in I$. If $\mathscr{T}$ satisfies the Nerves of Steel conjecture and has a weakly noeth

Figures (1)

  • Figure 1: Schematic overview of the relations between the various notions of stratification.

Theorems & Definitions (197)

  • Theorem A
  • Theorem B: \ref{['thm:hstratfundamental']} and \ref{['thm:hstratcosupp']}
  • Theorem C: \ref{['cor:ttfields1', 'prop:ttfields2']}
  • Theorem D: \ref{['thm:h-descent']}
  • Theorem E: \ref{['thm:tt=h+NS']}
  • Theorem F: \ref{['prop:ttstratdescent']}$(c)$ and \ref{['rem:ttdescent-finite']}
  • Theorem G: \ref{['cor:ttstratfinitemonodescent']}
  • Theorem H: \ref{['prop:ttstratdescent']}$(a)(b)$
  • Theorem I: \ref{['thm:infleq_stratification']}
  • Remark 1.1
  • ...and 187 more