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Relatively endotrivial complexes

Sam K. Miller

TL;DR

Extends Lassueur's relative endotrivial theory to chain complexes by introducing weakly/strongly $V$-endotrivial and $V$-endosplit-trivial complexes, with $V$ absolutely $p$-divisible. Develops equivalent local characterizations via $h$-marks, proves finite generation of the corresponding Picard groups, and establishes Borel-Smith constraints on the local data. Demonstrates a chain-complex Green correspondence and compatibilities with restriction, induction, and Brauer construction, enabling a reduction to the $p$-group case and yielding a decomposition of endotrivial complexes into a $G$-stable part over a Sylow subgroup plus a torsion Hom$(G,k^ imes)$ component. Specializes to $V=kG$ to recover classical endotrivial theory for modules and connects endosplit $p$-permutation resolutions to $V$-endosplit-trivial complexes. The work includes conjectural realizations of Borel-Smith functions via $h$-marks and outlines restriction/induction phenomena for subgroups containing Sylow $p$-subgroups with a view toward classifying endotrivial complexes in the relative setting.

Abstract

Let $G$ be a finite group and $k$ be a field of characteristic $p > 0$. In prior work, we studied endotrivial complexes, the invertible objects of the bounded homotopy category $K^b({}_{kG}\mathbf{triv})$ of $p$-permutation $kG$-modules. Using the notion of projectivity relative to a $kG$-module, we expand on this study by defining notions of "relatively" endotrivial chain complexes, analogous to Lassueur's construction of relatively endotrivial $kG$-modules. We obtain equivalent characterizations of relative endotriviality and find corresponding local homological data which almost completely determine the isomorphism class of a relatively endotrivial complex. We show this local data must partially satisfy the Borel-Smith conditions, and consider the behavior of restriction to subgroups containing Sylow $p$-subgroups $S$ of $G$.

Relatively endotrivial complexes

TL;DR

Extends Lassueur's relative endotrivial theory to chain complexes by introducing weakly/strongly -endotrivial and -endosplit-trivial complexes, with absolutely -divisible. Develops equivalent local characterizations via -marks, proves finite generation of the corresponding Picard groups, and establishes Borel-Smith constraints on the local data. Demonstrates a chain-complex Green correspondence and compatibilities with restriction, induction, and Brauer construction, enabling a reduction to the -group case and yielding a decomposition of endotrivial complexes into a -stable part over a Sylow subgroup plus a torsion Hom component. Specializes to to recover classical endotrivial theory for modules and connects endosplit -permutation resolutions to -endosplit-trivial complexes. The work includes conjectural realizations of Borel-Smith functions via -marks and outlines restriction/induction phenomena for subgroups containing Sylow -subgroups with a view toward classifying endotrivial complexes in the relative setting.

Abstract

Let be a finite group and be a field of characteristic . In prior work, we studied endotrivial complexes, the invertible objects of the bounded homotopy category of -permutation -modules. Using the notion of projectivity relative to a -module, we expand on this study by defining notions of "relatively" endotrivial chain complexes, analogous to Lassueur's construction of relatively endotrivial -modules. We obtain equivalent characterizations of relative endotriviality and find corresponding local homological data which almost completely determine the isomorphism class of a relatively endotrivial complex. We show this local data must partially satisfy the Borel-Smith conditions, and consider the behavior of restriction to subgroups containing Sylow -subgroups of .
Paper Structure (17 sections, 55 theorems, 47 equations, 3 figures)

This paper contains 17 sections, 55 theorems, 47 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Projectivity relative to subgroups for chain complexes
  3. Relative projectivity for modules and chain complexes with respect to modules
  4. Absolute $p$-divisibility
  5. $V$-stable categories
  6. $V$-endotrivial modules
  7. $V$-endotrivial complexes
  8. An equivalent definition of weak $V$-endotriviality
  9. Endosplit $p$-permutation resolutions
  10. h-mark homomorphisms
  11. h-mark homomorphisms for weakly $V$-endotrivial complexes
  12. h-marks for endosplit $p$-permutation resolutions and $V$-endosplit-trivial complexes
  13. Borel-Smith conditions
  14. The case of $V = kG$
  15. Induction and restriction for subgroups containing Sylow $p$-subgroups
  16. ...and 2 more sections

Key Result

Theorem 1.2

Let $C \in Ch^b({}_{kG}\mathbf{triv})$, let $V$ be a $p$-permutation $kG$-module which is absolutely $p$-divisible, and let $\mathcal{X}_V$ be the set of $p$-subgroups $P$ of $G$ for which $V(P) = 0$.

Figures (3)

  • Figure :
  • Figure :
  • Figure :

Theorems & Definitions (142)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Theorem 2.5
  • ...and 132 more