The homology of string algebras I
| Autor: | B. Huisgen-Zimmermann, Sverre O. Smalo |
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| Rok vydání: | 2005 |
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| Zdroj: | Journal für die reine und angewandte Mathematik (Crelles Journal). 2005:1-37 |
| ISSN: | 1435-5345 0075-4102 |
| DOI: | 10.1515/crll.2005.2005.580.1 |
| Popis: | We show that string algebras are `homologically tame' in the following sense: First, the syzygies of arbitrary representations of a finite dimensional string algebra $\Lambda$ are direct sums of cyclic representations, and the left finitistic dimensions, both little and big, of $\Lambda$ 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 $\Cal P$ consisting of the finitely generated left $\Lambda$-modules of finite projective dimension is completely determined by a finite number of, possibly infinite dimensional, string modules -- one for each simple $\Lambda$-module -- which are algorithmically constructible from quiver and relations of $\Lambda$. Namely, $\Cal P$ is contravariantly finite in $\Lambda$-mod precisely when all of these string modules are finite dimensional, in which case they coincide with the minimal $\Cal P$-approximations of the corresponding simple modules. Yet, even when $\Cal P$ fails to be contravariantly finite, these `characteristic' string modules encode, in an accessible format, all desirable homological information about $\Lambda$-mod. |
| Databáze: | OpenAIRE |
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