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The Balmer spectrum and telescope conjecture for infinite groups

Gregory Kendall

TL;DR

The paper advances tensor-triangular geometry for stable module categories of infinite groups by (i) computing the Balmer spectrum of dualisable objects for $ extup{H}_1 extup{F}$ groups of type $ extup{FP}_{ extinfty}$ as $ extup{Spc}( ilde{ extup{Mod}}(kG)^d)\cong extup{Proj}(H^*(G,k))$, (ii) establishing a corresponding spectral description for certain infinite free products and illustrating when stratification by dualisable spectrum fails, (iii) showing that smashing localisations are generated by dualisable objects and that the telescope conjecture holds in the $ extup{FP}_{ extinfty}$ case while failing in many infinite free-product examples, and (iv) providing a detailed analysis of support and Colocalisation theory in non-rigid, perfectly generated tensor-triangulated settings. These results illuminate how tensor-triangular geometry behaves beyond rigidly generated settings and clarify the role of dualisable objects in governing localising subcategories and smashing localisations in infinite-group contexts.

Abstract

We determine the Balmer spectrum of dualisable objects in the stable module category for $\mathrm{H}_1\mathfrak{F}$ groups of type $\mathrm{FP}_{\infty}$ and show that the telescope conjecture holds for these categories. We also determine the spectrum of dualisable objects for certain infinite free products of finite groups. Using this, we give examples where the stable category is not stratified by the spectrum of dualisable objects and where the telescope conjecture does not hold.

The Balmer spectrum and telescope conjecture for infinite groups

TL;DR

The paper advances tensor-triangular geometry for stable module categories of infinite groups by (i) computing the Balmer spectrum of dualisable objects for groups of type as , (ii) establishing a corresponding spectral description for certain infinite free products and illustrating when stratification by dualisable spectrum fails, (iii) showing that smashing localisations are generated by dualisable objects and that the telescope conjecture holds in the case while failing in many infinite free-product examples, and (iv) providing a detailed analysis of support and Colocalisation theory in non-rigid, perfectly generated tensor-triangulated settings. These results illuminate how tensor-triangular geometry behaves beyond rigidly generated settings and clarify the role of dualisable objects in governing localising subcategories and smashing localisations in infinite-group contexts.

Abstract

We determine the Balmer spectrum of dualisable objects in the stable module category for groups of type and show that the telescope conjecture holds for these categories. We also determine the spectrum of dualisable objects for certain infinite free products of finite groups. Using this, we give examples where the stable category is not stratified by the spectrum of dualisable objects and where the telescope conjecture does not hold.

Paper Structure

This paper contains 9 sections, 61 theorems, 11 equations.

Table of Contents

  1. Introduction
  2. Background on tensor triangular geometry
  3. The stable module category
  4. Support
  5. Smashing subcategories
  6. The telescope conjecture
  7. Infinite free products of finite groups
  8. $\mathrm{H}_1\mathfrak{F}$ groups of type $\mathrm{FP}_{\infty}$
  9. The telescope conjecture for $\mathrm{H}_1\mathfrak{F}$ groups of type $\mathrm{FP}_{\infty}$

Key Result

Theorem 1.1

(See bspec2 and specfortrees) Let $k$ be a field of characteristic $p > 0$ and $G$ be an $\textup{H}_1\mathfrak{F}$ group of type $\textup{FP}_{\infty}$. The Balmer spectrum of $\widetilde{\textup{Mod}}(kG)^d$ is homeomorphic to $\textup{Proj}(H^*(G,k))$. Let $\mathcal{H}$ be a finite collection of

Theorems & Definitions (131)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • proof
  • ...and 121 more