An approach to the finitistic dimension conjecture

Abstract: Let R be a finite dimensional k-algebra over an algebraically closed field k and modR be the category of all finitely generated left R-modules. For a given full subcategory of modR, we denote by the projective finitistic dimension of . That is, . It was conjectured by H. Bass in the 60''s that the projective finitistic dimension has to be finite. Since then, much work has been done toward the proof of this conjecture. Recently, K. Igusa and G. Todorov defined in [K. Igusa, G. Todorov, On the finitistic global dimension conjecture for artin algebras, in: Representations of Algebras and Related Topics, in: Fields Inst. Commun., vol. 45, Amer. Math. Soc., Providence, RI, 2005, pp. 201–204] a function , which turned out to be useful to prove that is finite for some classes of algebras. In order to have a different approach to the finitistic dimension conjecture, we propose to consider a class of full subcategories of modR instead of a class of algebras. That is, we suggest to take the class of categories , of θ-filtered R-modules, for all stratifying systems in modR. We prove that the Finitistic Dimension Conjecture holds for the categories of filtered modules for stratifying systems with one or two (and some cases of three) modules of infinite projective dimension. [Copyright &y& Elsevier]