Universal Galois coverings of self-injective algebras by repetitive algebras and Hochschild cohomology
The paper deals with various related subjects. If one starts with one algebra \(B\) and one considers the repetive algebra, then a natural question, already considered by various authors, is which algebras have the repetive algebra as a covering, this kind of considerations goes back to Chr. Riedtmann. Here this is used to get information about the universal Galois covering, the first fundamental group and the first Hochschild cohomology group. A self-injective algebra \(A\) is called a repetitive covering self-injective algebra if there is an ideal \(I\) such that \(B=A/I\) has no oriented cycles, \(IeI=0\) and \(Ie\) is an injective cogenerator in mod-\(B\), where \(e\) is the residual identity of \(B\). A theorem of Skowroński and Yamagata states that for these algebras there is a Galois covering \(F\colon\widehat B\to A\) with infinite cyclic Galois automorphism group. Various examples of such algebras are given in the paper, one of them is the trivial extension of a triangular algebra \(B\), a second example is the so called truncated quiver algebra of a cycle. Presentations of such algebras are given relating the coverings. One of the principal theorems of the paper is Theorem 3.11, which states the following: let \(A\) be a repetitive covering self-injective algebra and \(F\colon\widehat B\to A\) be the Galois covering, mentioned above, with Galois group \(G\), moreover assume \(B\) is Schurian. Then there is a short exact sequence of groups: \[ 1\to\pi_1(B)\to\pi_1(A)\to G\to 1. \] An immediate corollary is that in the case above the covering is universal if and only if \(B\) is simply connected. In particular this applies for a trivial extension of a Schurian triangular algebra, this is also applied in the paper to the other examples given. The last section relates all the material to the study of the first Hochschild cohomology group. Theorem 4.12 is similar to a proposition proved by Assem and de la Peña and as a consequence she gets the following Corollary 4.13. Let \(A\) be a repetitive covering self-injective algebra and \(F\colon\widehat B\to A\) be the mentioned Galois covering. Then there is an exact sequence of \(k\) vector spaces of the form: \[ 0\to\Hom(\pi_1(B),k^+)\to H^1(A)\to k\to 0. \] As a corollary she gets that if \(B\) is simply connected then \(H^1(A)=k\). Finally I have the following general remarks: The paper is very well written and almost self contained. I think that in the definition of Galois covering one should require that the covering is connected. This makes the existence of universal coverings possible. In Example 2.6 we need the assumption that \(B\) is triangular. Also the Galois covering is not finite dimensional and is not an algebra with unit in general.
scientific article; zbMATH DE number 1740509