Abstract
Assume that k is an algebraically closed field and A is a finite-dimensional wild k-algebra. Recently, L. Gregory and M. Prest proved that in this case the width of the lattice of all pointed A-modules is undefined. Hence the result of M. Ziegler implies that there exists a super-decomposable pure-injective A-module, if the base field k is countable. Here we give a different and straightforward proof of this fact. Namely, we show that there exists a special family of pointed A-modules, called an independent pair of dense chains of pointed A-modules. This also yields the existence of a super-decomposable pure-injective A-module.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Crawley-Boevey, W.W.: Tame algebras and modules generic. Proc. London Math. Soc. 63, 241–264 (1991)
Drozd, Yu. A.: Tame and wild matrix problems. In: Representations and Quadratic Forms, Kiev, 39–74 (in Russian) (1979)
Erdmann, K., Skowroński, A.: Weighted surface algebras. J. Algebra 505, 490–558 (2018)
Gregory, L., Prest, M.: Representation embeddings, interpretation functors and controlled wild algebras. J. Lond. Math. Soc. (2) 94(3), 747–766 (2016)
Huisgen-Zimmermann, B.: Purity, Algebraic Compactness, Direct Sum Decompositions, and Representation Type. In: Infinite Length Modules (Bielefeld, 1998), pp 331–367. Birkhäuser, Basel, Trends Math. (2000)
Jaworska, A., Skowroński, A.: The component quiver of a self-injective Artin algebra. Colloq. Math. 122(2), 233–239 (2011)
Jensen, Ch.U., Lenzing, H.: Model Theoretic Algebra with particular emphasis on Fields, Rings, Modules. Algebra, Logic and Applications, vol. 2. Gordon and Breach, New York (1989)
Kasjan, S., Pastuszak, G.: On two tame algebras with super-decomposable pure-injective modules. Colloq. Math. 123, 249–276 (2011)
Kasjan, S., Pastuszak, G.: On the existence of super-decomposable pure-injective modules over strongly simply connected algebras of non-polynomial growth. Colloq. Math. 136, 179–220 (2014)
Kasjan, S., Pastuszak, G.: Super-decomposable pure-injective modules over algebras with strongly simply connected Galois coverings. J. Pure Appl. Algebra 220(8), 2985–2999 (2016)
Kiełpiński, R.: On Γ-pure-injective modules. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 15, 127–131 (1967)
Mac Lane, S.: Categories for the working mathematician graduate texts in mathematics, vol. 5. Springer-Verlag, New York (1998)
Nörenberg, R., Skowroński, A.: Tame minimal non-polynomial growth simply connected algebras. Colloq. Math. 73, 301–330 (1997)
Pastuszak, G.: On Krull-Gabriel dimension and Galois coverings. Adv. Math. 349, 959–991 (2019)
Pastuszak, G.: Strongly simply connected algebras with super-decomposable pure-injective modules. J. Pure Appl. Algebra 219(8), 3314–3321 (2015)
Prest, M.: Model theory and modules. London Mathematical Society Lecture Note Series, vol. 130. Cambridge University Press, Cambridge (1988)
Prest, M.: Purity, spectra and localization Encyclopedia of mathematics and its applications, vol. 121. Cambridge University Press, Cambridge (2009)
Prest, M.: Superdecomposable pure-injective modules. In: Advances in Representation Theory of Algebras, EMS Series of Congress Reports 263–296. (2013)
Puninski, G.: When a super-decomposable pure-injective module over a serial ring exists. Rings, modules, algebras, and abelian groups, 449–463, Lecture Notes in Pure and Appl Math., vol. 236. Dekker, New York (2004)
Puninski, G.: How to construct a ‘concrete’ superdecomposable pure-injective module over a string algebra. J. Pure Appl. Algebra 212, 704–717 (2008)
Puninski, G.: Superdecomposable pure-injective modules exist over some string algebras. Proc. Amer. Math. Soc. 132, 1891–1898 (2004)
Puninski, G., Puninskaya, V., Toffalori, C.: Krull-Gabriel dimension and the model-theoretic complexity of the category of modules over group rings of finite groups. J. Lond. Math. Soc. 78, 125–142 (2008)
Simson, D., Skowroński, A.: Elements of the representation theory of associative algebras 3: Representation-Infinite Tilted Algebras, London Mathematical Society Student Texts 72, Cambridge University Press (2007)
Skowroński, A.: Algebras of Polynomial Growth Topics in Algebra, Banach Center Publ. 26, Part, vol. 1, pp 535–568. PWN, Warsaw (1990)
Skowroński, A.: Module categories over tame algebras, in: Representations Theory of Algebras ad Related Topics. CMS Conf. Proc. 19, 218–313 (1996)
Skowroński, A.: Selfinjective algebras of polynomial growth. Math. Ann. 285, 177–199 (1989)
Skowroński, A.: Selfinjective algebras: finite and tame type. Contemp. Math. 406, 169–238 (2006)
Skowroński, A.: Simply connected algebras Hochschild cohomologies in: Representations of Algebras. CMS Conf. Proc. 14, 431–447 (1993)
Skowroński, A.: Tame module categories of finite dimensional algebras, in: Trends in Ring Theory. CMS Conf. Proc. 22, 187–219 (1998)
Stenström, B., Pure submodules: Ark. Mat.,7, 159–171 (1967)
Ziegler, M.: Model theory of modules. Annals of Pure and Applied Logic 26, 149–213 (1984)
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
The authors did not receive support from any organization for the submitted work. The authors have no relevant financial or non-financial interests to disclose. Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
Additional information
Presented by: Kenneth Goodearl
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Dedicated to Professor Daniel Simson on the occasion of his 80th birthday
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Pastuszak, G. On Wild Algebras and Super-Decomposable Pure-Injective Modules. Algebr Represent Theor 26, 957–965 (2023). https://doi.org/10.1007/s10468-022-10117-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10468-022-10117-6