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Transactions of the American Mathematical Society

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Classification of modules for infinite-dimensional string algebras

Author: William Crawley-Boevey
Journal: Trans. Amer. Math. Soc. 370 (2018), 3289-3313
MSC (2010): Primary 16D70; Secondary 13C05
Published electronically: December 20, 2017
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Abstract: We relax the definition of a string algebra to also include infinite-dimensional algebras such as $ k[x,y]/(xy)$. Using the functorial filtration
method, which goes back to Gelfand and Ponomarev, we show that finitely generated modules and artinian modules (and more generally finitely controlled and pointwise artinian modules) are classified in terms of string and band modules. This subsumes the known classifications of finite-dimensional modules for string algebras and of finitely generated modules for $ k[x,y]/(xy)$. Unlike in the finite-dimensional case, the words parameterizing string modules may be infinite.

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Additional Information

William Crawley-Boevey
Affiliation: Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom
Address at time of publication: Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany

Received by editor(s): March 7, 2016
Received by editor(s) in revised form: July 26, 2016
Published electronically: December 20, 2017
Additional Notes: This material is based upon work supported by the National Science Foundation under grant No. 0932078 000 while the author was in residence at the Mathematical Science Research Institute (MSRI) in Berkeley, California, during the spring semester 2013.
Article copyright: © Copyright 2017 American Mathematical Society

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