The homology of string algebras I

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.