The authors consider the finitistic dimension conjectures in terms of questions about a certain model category structure which is defined from the cotorsion pair cogenerated by the modules of finite projective dimension.
Given a hereditary cotorsion theory \((\mathcal{F,C})\) in the category of \(R\)-modules which is cogenerated by a set, the authors obtain a cofibrantly generated model category structure on the category of chain complexes of \(R\)-modules, somewhat as in [J. Gillespie, Math. Z. 257, No. 4, 811-843 (2007; Zbl 1134.55016)] but, in view of the applications, without any assumption that \(\mathcal F\) is closed under direct limits. They define a notion of \(\mathcal F\)-cofibrant dimension in terms of resolutions and show that this equals a notion of projective dimension relative to \(\mathcal C\) defined in terms of \(\text{Ext}^n\). They consider cotorsion pairs defined using this relative projective dimension (considering, for each \(n\), the cotorsion pair cogenerated by the modules of relative dimension \(\leq n\)), obtaining model structures on the category of unbounded chain complexes in which the weak equivalences are the homology isomorphisms.
Taking \(\mathcal F\) to consist of the projective modules, so \(\mathcal C=R\text{-Mod}\), they obtain a criterion, in terms of the cotorsion pair cogenerated by the modules of finite projective dimension, for the big finitistic dimension of any ring \(R\) to be finite, as well as an analogous result for the little finitistic dimension.