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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
DOI: https://doi.org/10.1090/S0002-9947-99-02362-4
MathSciNet review: 1615962
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Abstract | References | Similar Articles | Additional Information

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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  • [1] I. Assem, J. Nehring and A. Skowronski, Domestic trivial extensions of simply connected algebras, Tsukuba J. Math. 13 (1989), 31-72. MR 90j:16043
  • [2] I. Assem and A. Skowronski, On tame repetitive algebras, Fund. Math. 142 (1993), 59-84. MR 94a:16022
  • [3] M. Auslander, I. Reiten and S. O. Smalø, Representation Theory of Artin Algebras, Cambridge Studies in Adv. Math. 36 (Cambridge Univeristy Press, 1995). MR 96c:16015
  • [4] K. Bongartz, Tilted algebras, in: Representations of Algebras, Lecture Notes in Math. 903, 26-38. MR 83g:16053
  • [5] K. Bongartz and P. Gabriel, Covering spaces in representation theory, Inventiones Math. 65 (1981), 331-378. MR 84i:16030
  • [6] O. Bretscher, C. Läser and C. Riedtmann, Self-injective and simply connected algebras, Manuscripta Math. 36 (1981), 253-307. MR 84i:16021
  • [7] P. Dowbor, H. Lenzing and A. Skowronski, Galois coverings of algebras by locally support-finite categories, in: Representation Theory I. Finite Dimensional Algebras, Lecture Notes in Math. 1177, 91-93. MR 87j:16012
  • [8] P. Dowbor and A. Skowronski, Galois coverings of representation-infinite algebras, Commentarii Math. Helvetici 62 (1987), 311-337. MR 88m:16020
  • [9] K. Erdmann, Blocks of tame representation type and related algebras, Lecture Notes in Math. 1428. MR 91c:20016
  • [10] K. Erdmann, O. Kerner and A. Skowronski, Self-injective algebras of wild tilted type, J. Pure Appl. Algebra, in press.
  • [11] K. Erdmann and A. Skowronski, On Auslander-Reiten components of blocks and biserial algebras, Transactions Amer. Math. Soc. 330 (1992), 165-189. MR 93b:16022
  • [12] P. Gabriel, The universal cover of a representation-finite algebra, in: Representations of Algebras, Lecture Notes in Math. 903, 68-105. MR 83f:16036
  • [13] D. Happel, On the derived category of a finite-dimensional algebra, Commentari Math. Helvetici 62 (1987), 339-389. MR 89c:16029
  • [14] D. Happel and C. M. Ringel, Tilted algebras, Transactions Amer. Math. Soc. 274 (1982), 399-443. MR 84d:16027
  • [15] D. Hughes and J. Waschbüsch, Trivial extensions of tilted algebras, Proc. London Math. Soc. 46 (1983), 347-364. MR 84m:16023
  • [16] S. Liu, Degrees of irreducible maps and the shapes of the Auslander-Reiten quivers, J. London Math. Soc. 45 (1992), 32-54. MR 93f:16015
  • [17] S. Liu, Semi-stable components of an Auslander-Reiten quiver, J. London Math. Soc. 47 (1993), 405-416. MR 94a:16024
  • [18] L. Peng and J. Xiao, Invariability of repetitive algebras of tilted algebras under stable equivalence, J. Algebra 170 (1994), 54-68. MR 95k:16023
  • [19] C. Riedtmann, Representation-finite self-injective algebras of class $\mathbb{D}_{n}$, Compositio Math. 49 (1983), 231-282. MR 85e:16049
  • [20] C. M. Ringel, Tame algebras and integral quadratic forms, Lecture Notes in Math. 1099. MR 87f:16027
  • [21] C. M. Ringel, The regular components of the Auslander-Reiten quiver of a tilted algebra, Chinese Ann. Math. 9B (1988), 1-18. MR 89e:16036
  • [22] A. Skowronski, Selfinjective algebras of polynomial growth, Math. Annalen 285 (1989), 177-199. MR 90k:16024
  • [23] A. Skowronski, Generalized standard Auslander-Reiten components without oriented cycles, Osaka J. Math. 30 (1993), 515-527. MR 94k:16026
  • [24] A. Skowronski, Generalized standard Auslander-Reiten components, J. Math. Soc. Japan 46 (1994), 517-543. MR 95d:16022
  • [25] A. Skowronski, Regular Auslander-Reiten components containing directing modules, Proc. Amer. Math. Soc. 120 (1994), 19-26. MR 94b:16021
  • [26] A. Skowronski and K. Yamagata, Socle deformations of self-injective algebras, Proc. London Math. Soc. 72 (1996), 545-566. MR 96m:16004
  • [27] K. Yamagata, Extensions over hereditary Artinian rings with self-dualities I, J. Algebra 73 (1981), 386-433. MR 83g:16030
  • [28] K. Yamagata, Representations of nonsplittable extension algebras, J. Algebra 115 (1988), 32-45. MR 89d:16029
  • [29] K. Yamagata, Frobenius Algebras, in: Handbook of Algebra, vol. 1, (North-Holland, 1996), pp. 841-887. MR 97k:16022
  • [30] Y. Zhang, The structure of stable components, Canadian J. Math. 43 (1991), 652-672. MR 92f:16017

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

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