The classification of torsion endo-trivial modules

TL;DR

This paper advances the classification of endo-trivial modules over p-groups, showing the structure of their isomorphism classes is mostly torsion-free except for specific group types, with implications for module and block theory.

Contribution

It proves that the group of endo-trivial modules modulo projectives is torsion-free for all but cyclic, quaternion, or semi-dihedral p-groups, extending understanding of module classifications.

Findings

The group of endo-trivial modules is torsion-free except for cyclic, quaternion, or semi-dihedral groups.

Tensor powers of modules with a specific decomposition imply the module is a sum of the trivial and a projective.

The proof reduces to extraspecial p-groups and uses support variety theory.

Abstract

This paper is a major step in the classification of endotrivial modules over p-groups. Let G be a finite p-group and k be a field of characteristic p. A kG-module M is an endo-trivial module if {\End_k(M)\cong k\oplus F} as kG-modules, where F is a free module. The classification of endo-trivial modules is the crucial step for understanding the more general class of endo-permutation modules. The endo-permutation modules play an important role in module theory, in particular as source modules, and in block theory where they appear in the description of source algebras. Endo-trivial modules are also important in the study of both derived equivalences and stable equivalences of group algebras and block algebras. The collection of isomorphism classes of endo-trivial modules modulo projectives is an abelian group under tensor product. The main result of this paper is that this group is torsion free except in the case that G is cyclic, quaternion or semi-dihedral. Hence for any p-group which is not cyclic, quaternion or semi-dihedral and any finitely generated kG-module M, if M \otimes_k M \otimes_k ... \otimes_k M \cong k \oplus P for some projective module P and some finite number of tensor products, then M \cong k \oplus Q for some projective module Q. The proof uses a reduction to the cases in which G is an extraspecial or almost extraspecial p-group, proved in a previous paper of the authors, and makes extensive use of the theory of support varieties for modules.