An approach to the finitistic dimension conjecture

Abstract

Let R be a finite dimensional k-algebra over an algebraically closed field k and mod R be the category of all finitely generated left R-modules. For a given full subcategory X of mod R, we denote by pfd X the projective finitistic dimension of X . That is, pfd X := sup{pd X: X ∈ X and pd X < ∞}.It was conjectured by H. Bass in the 60's that the projective finitistic dimension pfd(R) := pfd(mod R) 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 Ψ : mod R → 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 mod R instead of a class of algebras. That is, we suggest to take the class of categories F (θ), of θ-filtered R-modules, for all stratifying systems (θ, ) in mod R. 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.

Relative Igusa-Todorov Functions and Relative Homological Dimensions

“…In what follows, we generalise (for the relative ones) some basic properties of the Igusa-Todorov functions given in [33,29]. Some of them can be proven in a similar way, but for completeness, we include a proof.…”

Recollements and homological dimensions

“…Indeed, φ(M) = pd(M) if if pd(M) < ∞; and φ(M) is always finite even if pd(M) = ∞ (Ref. [13,14]). Now we introduce the natural concept of φ-dimension of a finite dimensional algebra A.…”

“…These Igusa-Todorov functions determine new homological measures, generalising the notion of projective dimension, and have become a powerful tool in the understanding of the finitistic dimension conjecture [14,22,25]. From [12,13], the φ-dimension of an algebra A is…”

“…Moreover, the φ-dimension can be used to describe selfinjective algebras: an algebra A is selfinjective if and only if φdim(A) = 0 [12]. Recently, various works were dedicated to study and generalise the properties of Igusa-Todorov function and the φdimension [7,12,13,18,25]. In particular, the φ-dimension of an algebra A was characterised in terms of the bi-functors Ext i A (−, −) and Tor A i (−, −) in [7].…”

Igusa–Todorov functions for Artin algebras

“…They also prove that the finiteness of the this dimension is invariant for derived equivalence. Recently various works were dedicated to study and generalize the properties of these functions, see for instance [9], [10], [16]. In particular in [16] the Igusa-Todorov functions were defined for the derived category of an Artin algebra.…”