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The homology of string algebras I | | B. Huisgen-Zimmermann
; S.O. Smalo
; | | Date: |
31 Oct 2001 | | Subject: | Representation Theory; Rings and Algebras MSC-class: 16D70, 16D90, 16E10, 16G10, 16G20 | math.RT math.RA | | 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 $Lambda$ 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. | | Source: | arXiv, math.RT/0111001 | | Services: | Forum | Review | PDF | Favorites | |
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