No abstract

Let A be a basic finite dimensional and connected algebra over an algebraically closed field k with zero characteristic. If the ordinary quiver of A has no double bypasses, we show that A admits a Galois covering which satisfies a universal property with respect to the Galois coverings of A. This universal property is similar to the one of the universal cover of a connected topological space.Comment: This text (21 pages) gives detailed proofs of the results announced in a previous note of the author (The fundamental group of a triangular algebra without double bypasses) and extends the study of this previous note to the Galois coverings of an algebr

Abstract. We investigate the Galois coverings of piecewise algebras and more particularly their behaviour under derived equivalences. Under a technical assumption which is satisfied if the algebra is derived equivalent to a hereditary algebra, we prove that there exists a universal Galois covering whose group of automorphisms is free and depends only on the derived category of the algebra. As a corollary, we prove that the algebra is simply connected if and only if its first Hochschild cohomology vanishes.

Let A be a basic connected finite dimensional algebra over a field k and let Q be the ordinary quiver of A. To any presentation of A with Q and admissible relations, R. Martinez-Villa and J. A. de La Pena have associated a group called the fundamental group of this presentation. There may exist different presentations of A with non isomorphic fundamental groups. In this note, we show that if the field k has characteristic zero, if Q has no oriented cycles and if Q has no double bypasses then there exists a privileged presentation of A such that the fundamental group of any other presentation is the quotient of the fundamental group of this privileged presentation.Comment: The proofs of the results in this note will be given in a subsequent paper where the framework will be extended to galois coverings of an algebr