2001/10/31 by Birge Huisgen-Zimmermann, B. Huisgen-Zimmermann, Huisgen-Zimmermann, B. +3 · 1 citation
Mathematics · Physics and Astronomy · #16D70 #16D90 #16E10 #16G10 #16G20 #Algebraic structures and combinatorial models #Black Holes and Theoretical Physics #FOS: Mathematics #Nonlinear Waves and Solitons #Representation Theory (math.RT) #Rings and Algebras (math.RA) #math.RA #math.RT #msc:16D70 #msc:16D90 #msc:16E10 #msc:16G10 #msc:16G20
paper · pdf · doi:10.48550/arxiv.math/0111001
arxiv created 2001/10/31 · openalex publication_date 2001/10/31 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \Cal P 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, \Cal P is contravariantly finite in Λ-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 Λ-mod.