The homology of string algebras I

Abstract

Dedikert til vår venn og kollega Idun Reiten i andledning hennes sekstiårsdag 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 Λ -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. Yet, 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.

On Extensions for Gentle Algebras

“…We note that gentle algebras are string algebras and that there is a large body of work on string algebras. In particular, projective resolutions and syzygies, have been considered before, see for example [15,16]. In [24], minimal projective presentations of string and band modules were given in terms of string combinatorics, which in the case of gentle algebras can be formulated in terms of homotopy string combinatorics.…”

Higher extensions for gentle algebras

“…In this section, we determine syzygies of indecomposable modules over gentle algebras and deduce some consequences for higher extensions. Since gentle algebras are monomial and hence string algebras, it already follows from [24,14] that in a projective resolution, from the second syzygy onwards, all indecomposable summands are uniserial. However, we need a more precise description of the decomposition of the syzygies into direct sums of indecomposable modules.…”

“…In this paper, we determine the higher extension spaces between indecomposable modules over gentle algebras by a detailed study of their syzygies. We note that syzygies of indecomposable modules over string algebras have been studied in [14] and over gentle algebras in [15]. However, for our purposes, we need a more precise understanding of the syzygies.…”

Cotorsion pairs generated by modules of bounded projective dimension

“…By [24] and [26], P(R-mod) is contravariantly finite and f.dim R =F. dim R = 2, so every module in P 2 is a direct limit of objects in P 2 (R-mod).…”