An Approach to the Finitistic Dimension Conjecture

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 X of modR, we denote by pfdX the projective finitistic dimension of X . That is, pfdX := sup {pdX : X ∈ X and pdX < ∞}. It was conjectured by H. Bass in the 60’s that the projective finitistic dimension pfd (R) := pfd (modR) has to be finite. Since then, much work has been done toward the proof of this conjecture. Recently, K. Igusa and J. Todorov defined in [9] a function Ψ : modR → 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 modR instead of a class of algebras, namely to take the class of categories F(θ) of θ-filtered R-modules for all stratifying systems (θ,≤) in modR. 1. Preliminaries 1.1. Basic notations. Throughout this paper, R will denote a finite dimensional k-algebra over a fixed algebraically closed field k, and modR will be the category of all finitely generated left R-modules. Only finitely generated left R-modules will be considered. Given a class C of R-modules, we consider the following: (a) the full subcategory F(C) of modR whose objects are the zero R-module and the C-filtered R-modules, that is, 0 6= M ∈ F(C) if there is a finite chain 0 = M0 ⊆ M1 ⊆ · · · ⊆ Mm = M of submodules of M such that each quotient Mi/Mi−1 is isomorphic to some object in C; (b) the projective dimension pd C := sup {pdC : C ∈ C} ∈ N ∪ {∞} of the class C; (c) the projective finitistic dimension pfd C := sup {pdC : C ∈ C and pdC < ∞} of the class C; (d) the class P(C) := {X : ExtR(X,−)|F(C) = 0} of C-projective R-modules; and (e) the class I(C) := {X : ExtR(−, X)|F(C) = 0} of C-injective R-modules. The global dimension gldim (R) := pd (modR) and the projective finitistic dimension pfd (R) := pfd (modR) are important homological invariants of R. The projective finitistic dimension was introduced in the 60’s to study commutative 1 2 FRANÇOIS HUARD, MARCELO LANZILOTTA, OCTAVIO MENDOZA noetherian rings; however, it became a fundamental tool for the study of noncommutative artinian rings. The finitistic dimension conjecture states that pfd (A) is finite for any left artinian ring A. This conjecture is also closely related with other famous homological conjectures, see for example in [6] and [15]. 1.2. Stratifying systems. For any positive integer t ∈ Z, we set by definition [1, t] := {1, 2, · · · , t}. Moreover, the natural total order ≤ on [1, t] will be considered throughout the paper. Definition 1.1. [10] A stratifying system (θ,≤), in modR, of size t consists of a set θ = {θ(i) : i ∈ [1, t]} of indecomposable R-modules satisfying the following homological conditions: (a) HomR(θ(j), θ(i)) = 0 for j > i, (b) ExtR(θ(j), θ(i)) = 0 for j ≥ i. 1.3. Canonical stratifying systems. Once we have recalled the definition of stratifying system, it is important to know whether such a system exists for a given algebra R. Consider the set [1, n] which is in bijective correspondence with the iso-classes of simple R-modules. For each i ∈ [1, n], we denote by S(i) the simple R-module corresponding to i, and by P (i) the projective cover of S(i). Following V. Dlab and C. M. Ringel in [7], we recall that the standard module R∆(i) is the maximal quotient of P (i) with composition factors amongst S(j) with j ≤ i. So, by Lemma 1.2 and Lemma 1.3 in [7], we get that the set of standard modules R∆ := {R∆(i) : i ∈ [1, n]} satisfies Definition 1.1. We will refer to this set as the cannonical stratifying system of R. Moreover, in case RR ∈ F(R∆) following I. Agoston,V. Dlab and E. Lukacs in [1], we say that R is a standardly stratified algebra (or a ss-algebra for short). 1.4. The finitistic dimension conjecture for ss-algebras. It was shown in [2] that the finitistic dimension conjecture holds for ss-algebras. In this paper, it is shown that pfd (R) ≤ 2n − 2 where n is the number of iso-classes of simple R-modules. Of course, not only the number n but also the category F(R∆) is closely related to pfd (R) as can be seen in the following result (see the proof in the Corollary 6.17(j) in [13]). Theorem 1.2. [13] Let R be a ss-algebra and n be the number of iso-classes of simple R-modules. Then pfd (R) ≤ pdF(R∆) + resdimF(R∆) (F(R∆) ) ≤ 2n− 2. 2. The stratifying system approach In Theorem 1.2, we saw that if R is a ss-algebra, then the category F(R∆) is closely related to pfd (R). Assume now that R is not a ss-algebra. We know from 1.3 that R still admits at least one stratifying system. Thus it makes sense to study pfd (R) from the point of view of stratifying systems. The following theorem was shown in [13]: Theorem 2.1. [13] Let R be an algebra, n be the number of iso-classes of simple R-modules and (θ,≤) be a stratifying system of size t. If I(θ) is coresolving, then pdF(θ) ≤ t ≤ n and pfd (R) ≤ pdF(θ) + resdimY (Y ), AN APPROACH TO THE FINITISTIC DIMENSION CONJECTURE 3 where Y := {X : ExtR(X,−)|I(θ) = 0} ⊇ F(θ). Remark 2.2. (1) The condition of I(θ) being coresolving given in the Theorem 2.1 is strong and has as a consequence that pdF(θ) is finite. However, we do not know whether the resolution dimension resdimY (Y) is finite. (2) In general, we can have that pdF(θ) = ∞ as can be seen in the following example. Example 2.3. Let R be the quotient path algebra given by the quiver