On the finitistic global dimension conjecture for Artin algebras

Abstract

Abstract. We find a simple condition which implies finiteness of (little) finitistic global dimension for artin algebras. As a consequence we obtain a short proof of the finitistic global dimension conjecture for radical cubed zero algebras. The same condition also holds for algebras of representation dimension less then or equal to three. Hence the conjecture holds in that case as well.Let Λ be an Artin algebra (an algebra of finite length over a commutative Artinian ring). We will consider only finitely generated (f.g.) modules over Λ, and hence we will only address the little finitistic global dimension conjecture which states that there exists a uniform bound called f indimΛ for the finite projective dimensions of all f.g. (left) Λ-modules of finite projective dimension. This conjecture implies the Nakayama conjecture. Some of the known cases in which the finitistic global dimension conjecture holds are the radical cubed zero case [GZ] and the monomial relation case [GKK] (see also [IZ], [BFGZ]). The conjecture is also true in the case the category of modules of finite projective dimension is contravariantly finite in the category of all f.g. modules [AR]. However, the converse is not true [IST]. In this paper we give a short proof that sup{pdM | M f.g. with rad 2 M = 0 and pdM < ∞} is finite. This is a generalization of the radical cubed zero case since all syzygies have radical square zero in that case. A thorough overview of the state of both little and big finitistic global dimension conjectures is given by B. Huisgen-Zimmerman [H].As another consequence of the main theorem we prove the finitistic dimension conjecture for algebras with representation dimension repdimΛ ≤ 3. The notion of representation dimension was introduced by M. Auslander [A1]. O. Iyama showed

Auslander-Reiten sequences and global dimensions

“…The following result is a consequence of Theorem 3.11 and a result in [9]. Recall that the finitistic dimension of an Artin algebra A is by definition the supremum of the projective dimensions of all finitely generated (left) A-modules of finite projective dimension.…”

“…It was shown in [9] that if an Artin algebra Γ has global dimension at most three, then for any projective Γ-module P , the finitistic dimension of End Γ (P ) is finite. By Theorem 3.11, we know that the global dimension of End A (M ⊕ X ⊕ Z) is at most three.…”

On syzygies over 2-Calabi–Yau tilted algebras

“…These functions were introduced in [22] to study the relation between the representation dimension and the finitistic dimension of an algebra. The supremum of φ, or ψ respectively, is called the Igusa-Todorov φ-dimension, respectively ψ-dimension, of the algebra.…”

Infinite dimensional tilting modules over finite dimensional algebras

“…The first of these conjectures was disproved by in [41], and the second one has been proven for (i) monomial algebras in [36] (again in [44]), (ii) radical cube zero (and even more generally for algebras with Loewy length 2n q 1 and ΛT r n of finite representation type, see [32]) in [37] ( [43]), (iii) when P I mod Λ¡ is contravariantly finite in [9], (iv) when the representation dimension of Λ is at most 3 in [43] (all special biserial algebras has representation dimension at most 3, see [33]). None of the above proofs of the finiteness of findim Λ directly involve a tilting or a cotilting module.…”