On Galois coverings and tilting modules

Abstract

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