On Galois coverings and tilting modules

Let A be a basic connected finite dimensional algebra over an algebraically closed field k. Let T be a basic tilting A-module with arbitrary finite projective dimension. For a fixed group G we compare the set of isoclasses of connected Galois coverings of A with group G and the set of isoclasses of connected Galois coverings of EndA(T) with group G. Using the Hasse diagram (see [D. Happel, L. Unger, On a partial order of tilting modules, Algebr. Represent. Theory 8 (2) (2005) 147–156] and [C. Riedtmann, A. Schofield, On a simplicial complex associated with tilting modules, Comment. Math. Helv. 66 (1) (1991) 70–78]) of basic tilting A-modules, we give sufficient conditions on T under which there is a bijection between these two sets (these conditions are always verified when A is of finite representation type). Then we apply these results to study when the simple connectedness of A implies the one of EndA(T) (see [I. Assem, E.N. Marcos, J.A. de la Peña, The simple connectedness of a tame tilted algebra, J. Algebra 237 (2) (2001) 647–656]). Finally, using an argument due to W. Crawley-Boevey, we prove that the type of any simply connected tilted algebra is a tree and that its first Hochschild cohomology group vanishes.