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(Co)stratification of stable categories for infinite groups

Gregory Kendall

Abstract

We classify the localising tensor ideal and colocalising hom-closed subcategories of the stable module category for $\mathrm{LH}\mathfrak{F}$ groups. Along the way we develop techniques to provide similar classifications for other categories of infinite groups.

(Co)stratification of stable categories for infinite groups

Abstract

We classify the localising tensor ideal and colocalising hom-closed subcategories of the stable module category for groups. Along the way we develop techniques to provide similar classifications for other categories of infinite groups.

Paper Structure

This paper contains 14 sections, 50 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Setup
  3. Localising, tensor ideals
  4. Reducing to subgroups
  5. The detection property
  6. Linking to the spectrum
  7. An isomorphism of frames
  8. Colocalising, hom-closed subcategories
  9. Examples
  10. Finite groups
  11. Module category
  12. Stable module category
  13. Homotopy category of projectives
  14. Derived category and variations

Key Result

Theorem 1.1

(See ohyea) Suppose $k$ is a commutative noetherian ring of finite global dimension and $G$ is an $\textup{LH}\Phi_k$ group. Then there are bijective maps between the following sets.

Theorems & Definitions (118)

  • Theorem 1.1
  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Example 2.5
  • Example 2.6
  • Lemma 2.7
  • proof
  • Definition 3.1
  • ...and 108 more