On Galois coverings and tilting modules

Patrick Le Meur (1, 2)

1 CMLA - Centre de Mathématiques et de Leurs Applications
2 IMJ - Institut de Mathématiques de Jussieu

Let A be a basic connected finite dimensional algebra over an algebraically closed field, let G be a group, let T be a basic tilting A-module and let B the endomorphism algebra of T. Under a hypothesis on T, we establish a correspondence between the Galois coverings with group G of A and the Galois coverings with group G of B. The hypothesis on T is expressed using the Hasse diagram of basic tilting A-modules and is always verified if A is of finite representation type. Then, we use the above correspondence to prove that A is simply connected if and only if B is simply connected, under the same hypothesis on T. Finally, we prove that if a tilted algebra B of type Q is simply connected, then Q is a tree and the first Hochschild cohomology group of B vanishes

Mots clés: algèbre; dimension finie; revêtement galoisien; module basculant; simplement connexe

Domaines: Théorie des représentations [math.RT]

Patrick Le Meur. On Galois coverings and tilting modules. Journal of Algebra, 2008, 319 (12), pp.4961--4999. ⟨10.1016/j.jalgebra.2008.03.003⟩. ⟨hal-00097962v4⟩