Galois coverings of selfinjective algebras by repetitive algebras

Skowroński, Andrzej; Yamagata, Kunio

doi:10.1090/S0002-9947-99-02362-4

Galois coverings of selfinjective algebras by repetitive algebras
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by Andrzej Skowroński and Kunio Yamagata PDF

Trans. Amer. Math. Soc. 351 (1999), 715-734 Request permission

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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