tailieunhanh - Đề tài " The classification of torsion endo-trivial modules "

This paper settles a problem raised at the end of the seventies by . Alperin [Al1], . Dade [Da] and . 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. | Annals of Mathematics The classification of torsion endo-trivial modules By Jon F. Carlson and Jacques Th evenaz Annals of Mathematics 162 2005 823-883 The classification of torsion endo-trivial modules By Jon F. Carlson and Jacques Thevenaz 1. Introduction This paper settles a problem raised at the end of the seventies by . Alperin All . Dade Da and . Carlson Cal 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 Endfe M k F as kG-modules where F is a free module. Typical examples of endo-trivial modules are the Heller translates Qn 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 .