A Quillen Model Structure Approach to the Finitistic Dimension Conjectures

We explore the interlacing between model category structures attained to classes of modules of finite X -dimension, for certain classes of modules X . As an application we give a model structure approach to the Finitistic Dimension Conjectures and present a new conceptual framework in which these conjectures can be studied. Let Λ be a finite dimensional algebra over a field k (or more generally, let Λ be an artin ring). The big finitistic dimension of Λ, Findim(Λ), is defined as the supremum of the projective dimensions of all modules having finite projective dimension. And the little finitistic dimension of Λ, findim(Λ), is defined in a similar way by restricting to the subclass of all finitely generated modules of finite projective dimension. In 1960, Bass stated the so-called Finitistic Dimension Conjectures: (I) Findim(Λ) = findim(Λ), and (II) findim(Λ) is finite. The first conjecture was proved to be false by Huisgen-Zimmermann in [19], but the second one still remains open. It has been proved to be true, for instance, for finite-dimensional monomial algebras [15], for Artin algebras with vanishing cube radical [18], or Artin algebras with representation dimension bounded by 3 [21]. In [4] (see also [20]), Auslander proved that the finitistic dimension conjectures hold for Artin algebras in which the class P of all finitely generated modules of finite projective dimension is contravariantly finite (equivalently, it is a precovering class in the sense of [10, 14]). In general, P does not need to be contravariantly finite, even for Artin algebras satisfying the finitistic dimension conjectures. But, as Angeleri-Hügel and Trlifaj have noticed in [3], it cogenerates a cotorsion pair (F , C) in which the class F is precovering in R -Mod. By means of this idea, the authors are able to extend Auslander’s approach to arbitrary artinian rings and obtain a general criterium for an artinian ring to satisfy the finitistic dimension conjectures in terms of Tilting Theory (see [3]). This type of arguments has also been recently extended to more general homologies induced by arbitrary hereditary cotorsion pairs (see [1]). On the other hand, Hovey has recently shown in [16] that there exists a quite strong relation between the construction of hereditary cotorsion pairs in module categories and the existence of model structures in the sense of Quillen in the 2000 Mathematics Subject Classification. Primary: 16D90, 16E30. Secondary: 55U35, 18G35.