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B. Huisgen-Zimmermann, S. O. Smalø

The homology of string algebras I

We show that string algebras are ‘homologically tame’ in the following sense: First, the syzygies of arbitrary representations of a finite dimensional string algebra ? are direct sums of cyclic representations, and the left finitistic dimensions, both little and big, of ? 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 ^{(?-mod) consisting of the finitely generated left ?-modules of finite projective dimension is completely determined by a finite number of, possibly infinite dimensional, string modules?one for each simple ?-module?which are algorithmically constructible from quiver and relations of ?. Namely, P (?-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 (?-mod)-approximations of the corresponding simple modules. Even when P (?-mod) fails to be contravariantly finite, these ‘characteristic’ string modules encode, in an accessible format, all desirable homological information about ?-mod.}

Journal fur die reine und angewandte Mathematik (Crelles Journal), Walter de Gruyter

Print ISSN: 0075-4102
Volume: 2005, 03/2005
Pages: 1 - 37

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