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Galois coverings of selfinjective algebras by repetitive algebras
Authors:Andrzej Skowronski   Kunio Yamagata
Affiliation:Faculty of Mathematics and Informatics, Nicholas Copernicus University, Chopina 12/18, 87-100 Torun, Poland ; Department of Mathematics, Tokyo University of Agriculture and Technology, Fuchu, Tokyo 183, Japan
Abstract:In the representation theory of selfinjective artin algebras an important role is played by selfinjective algebras of the form $widehat {B}/G$ where $widehat {B}$ is the repetitive algebra of an artin algebra $B$ and $G$ is an admissible group of automorphisms of $widehat {B}$. If $B$ is of finite global dimension, then the stable module category $ underline{operatorname{mod}} widehat {B}$ of finitely generated $widehat {B}$-modules is equivalent to the derived category $D^{b} (operatorname{mod} B)$ of bounded complexes of finitely generated $B$-modules. For a selfinjective artin algebra $A$, an ideal $I$ and $B=A/I$, we establish a criterion for $A$ to admit a Galois covering $F: widehat {B}to widehat {B}/G=A$ with an infinite cyclic Galois group $G$. As an application we prove that all selfinjective artin algebras $A$ whose Auslander-Reiten quiver $Gamma _{A}$ has a non-periodic generalized standard translation subquiver closed under successors in $Gamma _{A}$ are socle equivalent to the algebras $widehat {B}/G$, where $B$ is a representation-infinite tilted algebra and $G$ is an infinite cyclic group of automorphisms of $widehat{B}$.

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