Topological invariants of piecewise hereditary algebras

Abstract

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.

Coverings of laura algebras: The standard case

“…Let B be a product of tilted algebras and n be the rank of its Grothendieck group. In [31,Cor. 4.5], it is proved that tilting modules are of the first kind with respect to any Galois covering of B.…”

“…In particular, Skowroński posed the following problem: for which algebras A do we have HH 1 (A) = 0 if and only if A is simply connected? This problem has been the subject of several investigations: notably this equivalence holds true for algebras derived equivalent to hereditary algebras [31], weakly shod algebras [30] (see also [7]), large classes of selfinjective algebras [34] and schurian cluster-tilted algebras [10]. It was proved in [15] that, for a representation-finite algebra, the first Hochschild cohomology group vanishes if and only if its Auslander-Reiten quiver is simply connected.…”

“…Our approach, already used in [30,31], uses coverings. Covering theory was introduced by Gabriel and his school (see, for instance, [14,21,36]) and consists in replacing an algebra by a locally bounded category, called its covering, which is sometimes easier to study.…”

Galois Coverings of Weakly Shod Algebras

“…Let be the full subcategory of ind generated by g T i i ∈ 1 n and g ∈ G . So and have equivalent derived categories (see the proof of [22,Lem. 4.8]).…”

“…( ) Up to now, there have been partial answers to (regardless the tame assumption): For algebras derived equivalent to a hereditary algebra in [22] (and therefore for tilted algebras), for tame quasi-tilted algebras in [3] and for tame weakly shod algebras in [5]. Therefore, it is natural to try to answer for laura algebras.…”