Galois Coverings of Weakly Shod Algebras

Abstract

We investigate the Galois coverings of weakly shod algebras. For a weakly shod algebra not quasi-tilted of canonical type, we establish a correspondence between its Galois coverings and the Galois coverings of its connecting component. As a consequence we show that a weakly shod algebra which is not quasi-tilted of canonical type is simply connected if and only if its first Hochschild cohomology group vanishes.

Coverings of laura algebras: The standard case

“…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.…”

“…Since q : Γ → Γ is surjective on vertices and F λ Φ = Φ k(q), then X ∈ Ω lies in the image of F λ . Also, k(q) and Φ commute with the translation, and so does F λ (see [30,Lem. 2.1]).…”

“…Finally k(q) maps meshes to meshes, and Φ maps meshes to almost split sequences. So Φ maps meshes to almost split sequences (see [30,Lem. 2.2]).…”

“…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.…”

The first Hochschild (co)homology when adding arrows to a bound quiver algebra

“…A. de la Peña and M. Saorín in [14] obtained formulas computing the dimension of HH 1 (B), see also [9,10,12,20]. For several families of algebras, results concerning the first cohomology vector space are given for instance in [2,3,4,25,26,27,31,32]. In case the algebra B is split, a canonical decomposition of HH 1 (B) into four direct summands is obtained in [11].…”