The homology of string algebras I

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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 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 is contravariantly finite in Λ-mod precisely when all of these string modules are finite dimensional, in which case they coincide with the minimal P-approximations of the corresponding simple modules. Yet, even when P fails to be contravariantly finite, these `characteristic' string modules encode, in an accessible format, all desirable homological information about Λ-mod.