An approach to the finitistic dimension conjecture

Abstract

Let $R$ be a finite dimensional $k$-algebra over an algebraically closed field $k$ and $\mathrm{mod}R$ be the category of all finitely generated left $R$-modules. For a given full subcategory $\mathcal{X}$ of $\mathrm{mod}R,$ we denote by $\pfd \mathcal{X}$ the projective finitistic dimension of $\mathcal{X}.$ That is, $\pfd \mathcal{X}:=\mathrm{sup} \{\pd X : X\in\mathcal{X} \text{and} \pd X<\infty\}.$ \ It was conjectured by H. Bass in the 60's that the projective finitistic dimension $\pfd (R):=\pfd (\mathrm{mod} R)$ has to be finite. Since then, much work has been done toward the proof of this conjecture. Recently, K. Igusa and J. Todorov defined a function $\Psi :\mathrm{mod}R\to \mathbb{N},$ which turned out to be useful to prove that $\pfd (R)$ 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 $\mathrm{mod}R$ instead of a class of algebras, namely to take the class of categories $\F(θ)$ of $\theta$-filtered $R$-modules for all stratifying systems $(\theta ,\le )$ in $\mathrm{mod}R.$