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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Galois coverings of selfinjective algebras
by repetitive algebras


Authors: Andrzej Skowronski and Kunio Yamagata
Journal: Trans. Amer. Math. Soc. 351 (1999), 715-734
MSC (1991): Primary 16D50, 16G10, 16G70, 16S99
MathSciNet review: 1615962
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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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Additional Information

Andrzej Skowronski
Affiliation: Faculty of Mathematics and Informatics, Nicholas Copernicus University, Chopina 12/18, 87-100 Toruń, Poland
Email: skowron@mat.uni.torun.pl

Kunio Yamagata
Affiliation: Department of Mathematics, Tokyo University of Agriculture and Technology, Fuchu, Tokyo 183, Japan
Email: yamagata@cc.tuat.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02362-4
PII: S 0002-9947(99)02362-4
Received by editor(s): February 4, 1997
Article copyright: © Copyright 1999 American Mathematical Society