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Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Cuntz, Joachim / Huybrechts, Daniel / Hwang, Jun-Muk

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The homology of string algebras I

B. Huisgen-Zimmermann1 / S. O. Smalø2

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Citation Information: Journal für die reine und angewandte Mathematik. Volume 2005, Issue 580, Pages 1–37, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: 10.1515/crll.2005.2005.580.1, July 2005

Publication History

Received:
2. November 2001
Published Online:
2005-07-27

Abstract

We show that string algebras are ‘homologically tame’ in the following sense: First, the syzygies of arbitrary representations of a finite dimensional string algebra Λ are direct sums of cyclic representations, and the left finitistic dimensions, both little and big, of Λ can be computed from a finite set of cyclic left ideals contained in the Jacobson radical. Second, our main result shows that the functorial finiteness status of the full subcategory P <∞(Λ-mod) consisting of the finitely generated left Λ-modules of finite projective dimension is completely determined by a finite number of, possibly infinite dimensional, string modules—one for each simple Λ-module—which are algorithmically constructible from quiver and relations of Λ. Namely, P <∞(Λ-mod) is contravariantly finite in L-mod precisely when all of these string modules are finite dimensional, in which case they coincide with the minimal P <∞(Λ-mod)-approximations of the corresponding simple modules. Even when P <∞(Λ-mod) fails to be contravariantly finite, these ‘characteristic’ string modules encode, in an accessible format, all desirable homological information about Λ-mod.

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[1]
Silvana Bazzoni and Dolors Herbera
Israel Journal of Mathematics, 2009, Volume 174, Number 1, Page 119

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