Abstract
We describe the irreducible morphisms in the category of modules over a repetitive algebra. We find three special canonical forms: The first canonical form happens when all the component morphisms are split monomorphisms, the second when all the component morphisms are split epimorphisms and the third when there is exactly one irreducible component map. Also, we obtain the same result for the irreducible homomorphisms in the stable category of modules over a repetitive algebra.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arnesen, K.K., Grimeland, Y.: The Auslander-Reiten components of Kb(P r o jΛ), where Λ is a cluster tilted algebra of type Ã. J Algebra Appl. 14(01), 1550005 (2015)
Auslander, M., Reiten, I.: Representation theory of artin algebras I I I. Commun. Algebra 3, 239–294 (1975)
Bautista, R., Salorio, M.J.S.: Irreducible morphisms in the bounded derived category. J Pure Appl. Algebra 215(5), 866–884 (2011)
Bobinski, G.: The almost split triangles for perfect complexes over gentle algebras. J. Pure Appl. Algebra 215(4), 642–654 (2011)
Giraldo, H., Marcos, E.: Heart of irreducible morphisms of bounded complexes. Commun. Algebra 44(01), 354–370 (2016)
Giraldo, H., Merklen, H.: Irreducible morphisms of categories of complexes. J Algebra 321(10), 2716–2736 (2009)
Happel, D.: Triangulated categories in the representation theory of finite dimensional algebras. Cambridge University Press, Cambridge (1988)
Happel, D.: Auslander-Reiten triangles in derived categories of finite-dimensional algebras. Proc. Amer. Math. Soc. 112(3), 641–648 (1991)
Happel, D., Keller, B., Reiten, I.: Bounded derived categories and repetitive algebras. J Algebra 319(4), 1611–1635 (2008)
Hughes, D., Waschbüsch, J.: Trivial extensions of tilted algebras. Proc. London Math. Soc. s3-46(2), 347–364 (1983)
Acknowledgments
We acknowledge the important collaboration and many, very helpful comments and suggestions of Raymundo Bautista for this work. The results presented here were obtained while the author were visiting the Centro de Ciencias de Matemáticas, UNAM, Unidad Morelia, for whose hospitality I am very grateful. We are deeply thankful to the reviewer(s) for his(hers) (theirs) remarks and suggestions, wich enabled us to improve this article.
Author information
Authors and Affiliations
Corresponding author
Additional information
Presented by Yuri Drozd.
This research was partly supported by the Centro de Ciencias de Matemáticas, UNAM, Unidad Morelia (Mexico), CODI and Estrategia de Sostenibilidad 2016-2017 (Universidad de Antioquia), and COLCIENCIAS-ECOPETROL (Contrato RC. No. 0266-2013).
Rights and permissions
About this article
Cite this article
Giraldo, H. Irreducible Morphisms Between Modules over a Repetitive Algebras. Algebr Represent Theor 21, 683–702 (2018). https://doi.org/10.1007/s10468-017-9733-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10468-017-9733-9