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Permutation twisted cohomology, remixed

Sam K. Miller

TL;DR

This work extends Balmer–Gallauer's tensor-triangular framework from elementary abelian $p$-groups to all finite $p$-groups by constructing forerunners, enabling a refined open-cover description of the Balmer spectrum $\operatorname{Spc}(\mathscr{K}(G))$ via endotrivial line bundles. It introduces the remixed permutation twisted cohomology ring $\operatorname{H}^{\bullet\bullet}(G)$, locally identifies with cohomology on each open, and proves that the Balmer comparison map $\operatorname{comp}_G$ is injective with dense image; when $\operatorname{H}^{\bullet\bullet}(G)$ is Noetherian, $\operatorname{comp}_G$ is an open immersion, endowing $\operatorname{Spc}(\mathscr{K}(G))$ with a Dirac-scheme structure. The paper establishes Noetherianity for Dedekind $p$-groups and all $p$-groups of order at most $p^3$, and conjectures Noetherianity for all finite $p$-groups, outlining a strategy to prove it in general. Overall, the results provide a topological and algebraic framework connecting endotrivial complexes, permutation modules, and twisted cohomology to describe the tt-geometry of $p$-groups in a precise, locally-driven fashion.

Abstract

For each endotrivial complex arising from Bredon homology of a representation sphere, we construct $p$-local quasi-isomorphisms, called forerunners, enabling us to extend Balmer--Gallauer's results in arXiv:2307.04398 Part II concerning the tensor-triangular geometry of permutation modules for elementary abelian $p$-groups to all $p$-groups. We construct an open cover of the Balmer spectrum under which all endotrivials are line bundles, that is, every endotrivial is locally isomorphic to a shifted tensor unit. We define a 'remixed' permutation twisted cohomology ring for which the canonical comparison map from the Balmer spectrum to the homogeneous spectrum of the twisted cohomology ring is injective. If the twisted cohomology ring is Noetherian, the comparison map is an open immersion, and the open cover endows the Balmer spectrum with Dirac scheme structure. We prove Noetherianity holds for Dedekind groups and all $p$-groups of order at most $p^3$, and conjecture Noetherianity holds for all finite $p$-groups.

Permutation twisted cohomology, remixed

TL;DR

This work extends Balmer–Gallauer's tensor-triangular framework from elementary abelian -groups to all finite -groups by constructing forerunners, enabling a refined open-cover description of the Balmer spectrum via endotrivial line bundles. It introduces the remixed permutation twisted cohomology ring , locally identifies with cohomology on each open, and proves that the Balmer comparison map is injective with dense image; when is Noetherian, is an open immersion, endowing with a Dirac-scheme structure. The paper establishes Noetherianity for Dedekind -groups and all -groups of order at most , and conjectures Noetherianity for all finite -groups, outlining a strategy to prove it in general. Overall, the results provide a topological and algebraic framework connecting endotrivial complexes, permutation modules, and twisted cohomology to describe the tt-geometry of -groups in a precise, locally-driven fashion.

Abstract

For each endotrivial complex arising from Bredon homology of a representation sphere, we construct -local quasi-isomorphisms, called forerunners, enabling us to extend Balmer--Gallauer's results in arXiv:2307.04398 Part II concerning the tensor-triangular geometry of permutation modules for elementary abelian -groups to all -groups. We construct an open cover of the Balmer spectrum under which all endotrivials are line bundles, that is, every endotrivial is locally isomorphic to a shifted tensor unit. We define a 'remixed' permutation twisted cohomology ring for which the canonical comparison map from the Balmer spectrum to the homogeneous spectrum of the twisted cohomology ring is injective. If the twisted cohomology ring is Noetherian, the comparison map is an open immersion, and the open cover endows the Balmer spectrum with Dirac scheme structure. We prove Noetherianity holds for Dedekind groups and all -groups of order at most , and conjecture Noetherianity holds for all finite -groups.

Paper Structure

This paper contains 16 sections, 50 theorems, 85 equations, 18 figures.

Table of Contents

  1. Preliminaries: Tensor-triangular geometry and permutation modules
  2. Tensor-triangular geometry
  3. Permutation modules
  4. Preliminaries: Endotrivial complexes and Borel-Smith functions
  5. Borel-Smith functions and representation spheres
  6. Chain complexes over the orbit category
  7. Removing contractible summands
  8. An open cover of the Balmer spectrum via endotrivials
  9. Constructing forerunners
  10. An open cover of the spectrum
  11. The remixed twisted cohomology ring
  12. Twisted cohomology under localization and tt-functors
  13. An identification for the image of the comparison map
  14. Injectivity of the comparison map
  15. Towards Noetherianity of twisted cohomology
  16. ...and 1 more sections

Key Result

Theorem A

Let $G$ be a finite $p$-group. For every effective endotrivial complex (see def:decreasing) and subgroup $H \leq G$, there exists a forerunner homomorphism $\iota^H_C: k \to C[h_C(H)]$ such that $\Psi^H(\iota^H_C)$ is a quasi-isomorphism (thm:construction). Here, $h_C$ denotes the h-mark homomorphis

Figures (18)

  • Figure 1: Subgroup lattice of $3^{1+2}_+$ and $3^{1+2}_-$ respectively, with conjugate subgroups merged. Purple indicates normal, blue indicates characteristic, subscript counts number of conjugates if non-normal. Diagram obtained from GND32.
  • Figure :
  • Figure :
  • Figure :
  • Figure :
  • ...and 13 more figures

Theorems & Definitions (126)

  • Theorem A
  • Theorem B
  • Theorem
  • Definition : \ref{['def:twistedcohomology']}
  • Theorem C
  • Theorem D
  • Conjecture
  • Definition 1.1
  • Definition 1.2
  • Example 1.3
  • ...and 116 more