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The classification of endotrivial complexes

Sam K. Miller

TL;DR

The paper provides a complete classification of endotrivial complexes over a field of prime characteristic, identifying the Picard group $\mathcal{E}_k(G)$ as an extension of $\operatorname{Hom}(G,k^\times)$ by the Borel-Smith superclass function group $CF_b(G,p)$, with a Higuchi-type map $h$ yielding $\mathcal{E}_k(G) \cong CF_b(G,p)$ for $p$-groups. The authors introduce the $V(\mathcal{F}_G)$-endosplit-trivial framework and connect endotrivial complexes to the generalized Dade group $D_k(G)$ via a short exact sequence, then transport the rational $p$-biset functor structure to these objects. In the $p$-group case they show $h = \Psi \circ h$ and prove Bouc’s homomorphism kernel equals the Borel-Smith oriented subgroup, $\ker(\Psi)=CF_{ba^+}(G,p)$, while establishing the surjectivity of the Lefschetz map $\Lambda: \mathcal{E}_k(G) \to O(T(kG))$ and, consequently, that every $p$-permutation autoequivalence of $kG$ is induced by a splendid Rickard autoequivalence. For arbitrary finite groups, the paper gives a refined decomposition of endotrivial complexes, describes kernels via oriented Artin conditions, and situates the results within the tensor-triangulated framework, offering new tools for understanding the Balmer spectrum and related cohomological structures.

Abstract

Let $G$ be a finite group and $k$ a field of prime characteristic $p$. We give a complete classification of endotrivial complexes, i.e. determine the Picard group $\mathcal{E}_k(G)$ of the tensor-triangulated category $K^b({}_{kG}\mathbf{triv})$, the bounded homotopy category of $p$-permutation modules, which Balmer and Gallauer recently considered. For $p$-groups, we identify $\mathcal{E}_k(-)$ with the rational $p$-biset functor $CF_b(-)$ of Borel-Smith functions and recover a short exact sequence of rational $p$-biset functors constructed by Bouc and Yalçin. As a consequence, we prove that every $p$-permutation autoequivalence of a $p$-group arises from a splendid Rickard autoequivalence. Additionally, we give a positive answer to a question of Gelvin and Yalçin, showing the kernel of the Bouc homomorphism for an arbitrary finite group $G$ is described by superclass functions $f: s_p(G) \to \mathbb{Z}$ satisfying the oriented Artin-Borel-Smith conditions.

The classification of endotrivial complexes

TL;DR

The paper provides a complete classification of endotrivial complexes over a field of prime characteristic, identifying the Picard group as an extension of by the Borel-Smith superclass function group , with a Higuchi-type map yielding for -groups. The authors introduce the -endosplit-trivial framework and connect endotrivial complexes to the generalized Dade group via a short exact sequence, then transport the rational -biset functor structure to these objects. In the -group case they show and prove Bouc’s homomorphism kernel equals the Borel-Smith oriented subgroup, , while establishing the surjectivity of the Lefschetz map and, consequently, that every -permutation autoequivalence of is induced by a splendid Rickard autoequivalence. For arbitrary finite groups, the paper gives a refined decomposition of endotrivial complexes, describes kernels via oriented Artin conditions, and situates the results within the tensor-triangulated framework, offering new tools for understanding the Balmer spectrum and related cohomological structures.

Abstract

Let be a finite group and a field of prime characteristic . We give a complete classification of endotrivial complexes, i.e. determine the Picard group of the tensor-triangulated category , the bounded homotopy category of -permutation modules, which Balmer and Gallauer recently considered. For -groups, we identify with the rational -biset functor of Borel-Smith functions and recover a short exact sequence of rational -biset functors constructed by Bouc and Yalçin. As a consequence, we prove that every -permutation autoequivalence of a -group arises from a splendid Rickard autoequivalence. Additionally, we give a positive answer to a question of Gelvin and Yalçin, showing the kernel of the Bouc homomorphism for an arbitrary finite group is described by superclass functions satisfying the oriented Artin-Borel-Smith conditions.
Paper Structure (12 sections, 39 theorems, 98 equations, 9 figures)

This paper contains 12 sections, 39 theorems, 98 equations, 9 figures.

Table of Contents

  1. Introduction
  2. $V$-endosplit-trivial complexes
  3. The module $V(\mathcal{F}_G)$
  4. Capped endosplit $p$-permutation resolutions and the Dade group of a finite group
  5. The case of $p$-groups
  6. Transporting rational $p$-biset functor structure
  7. Results for arbitrary finite groups
  8. The classification of endotrivial complexes
  9. The kernel of the Bouc homomorphism
  10. Extending virtual modules to invertible complexes for $p$-groups
  11. Surjectivity of the Lefschetz homomorphism
  12. Every $p$-permutation autoequivalence of a $p$-group arises from a splendid Rickard equivalence

Key Result

Theorem 1

Let $G$ be a finite group. We have a split exact sequence of abelian groups Here, $CF_b(G,p)$ denotes the additive group of Borel-Smith superclass functions $f: s_p(G) \to \mathbb{Z}$ valued on $p$-subgroups (see Definition def:bsf), and $h$ denotes the h-mark homomorphism (see Proposition hmarkhoms). In particular, if $G$ is a $p$-group, we have an isomorphism $h: \mathcal

Figures (9)

  • Figure 1: The induced surjective homomorphism $\gamma: CF_b(G) \to B(G)^\times$.
  • Figure :
  • Figure :
  • Figure :
  • Figure :
  • ...and 4 more figures

Theorems & Definitions (98)

  • Theorem : Corollary \ref{['thm:borelsmithfornonpgroups']}
  • Theorem : Theorem \ref{['bigtheorem']}
  • Theorem : Theorem \ref{['thm:bouchomker']}
  • Theorem : Theorem \ref{['thm:surjetivityoflefschetz']}
  • Theorem : Theorem \ref{['thm:ppermsliftforpgroups']}
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • Definition 2.4
  • ...and 88 more