The classification of torsion endo - trivial modules

This paper settles a problem raised at the end of the seventies by J.L. Alperin [Al1], E.C. Dade [Da] and J.F. Carlson [Ca1], namely the classification of torsion endo-trivial modules for a finite p-group over a field of characteristic p. Our results also imply, at least when p is odd, the complete classification of torsion endo-permutation modules. We refer to [CaTh] and [BoTh] for an overview of the problem and its importance in the representation theory of finite groups. Let us only mention that the classification of endo-trivial modules is the crucial step for understanding the more general class of endo-permutation modules, and that endopermutation modules play an important role in module theory, in particular as source modules, in block theory where they appear in the description of source algebras, and in both derived equivalences and stable equivalence of block algebras, for which many new developments have appeared recently. Let G be a finite p-group and k be a field of characteristic p. Recall that a (finitely generated) kG-module M is called endo-trivial if Endk(M) ∼= k ⊕ F as kG-modules, where F is a free module. Typical examples of endo-trivial modules are the Heller translates Ωn(k) of the trivial module. Any endo-trivial kG-module M is a direct sum M = M0 ⊕ L, where M0 is an indecomposable endo-trivial kG-module and L is free. Conversely, by adding a free module to an endo-trivial module, we always obtain an endo-trivial module. This defines an equivalence relation among endo-trivial modules and each equivalence class contains exactly one indecomposable module up to isomorphism. The set T (G) of all equivalence classes of endo-trivial kG-modules is a group with multiplication induced by tensor product, called simply the group of endo-trivial kG-modules. Since scalar extension of the coefficient field induces an injective map between the groups of endo-trivial modules, we can replace k by its algebraic closure. So we assume that k is algebraically closed. We refer to [CaTh] for more details about T (G).