On the finitistic global dimension conjecture for Artin algebras

Abstract

We find a simple condition which implies finiteness of 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 finitistic dimension conjecture holds in that case as well. Let Λ be an Artin algebra (an algebra of finite length over a commutative Artinian ring). Then the finitistic global dimension conjecture states that there exists a uniform bound called findimΛ for the finite projective dimensions (pd) of all f.g. (left) Λ-modules of finite pd. This conjecture would imply 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 when the category of modules of finite pd 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 of the finitistic gl dim conjecture for all modules of radical square zero over any Artin algebra. 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 the finitistic global dimension conjecture can be found in [Z-H]. As another consequence of the main theorem we prove the finitistic dimension conjecture for algebras with weak representation dimension at most 3, and consequently for algebras with representation dimension repdimΛ ≤ 3. The notion of representation dimension was introduced by M. Auslander in his Queen Mary Notes [A1], and he and many others expect this dimension to be bounded by 3. O. Iyama showed that it is always finite [I], many classes of algebras are known to have repdimΛ = 3, the most recent class being subalgebras of algebras of finite representation type with the same radical [EHIS]. The proof of the main theorem is based on the following well-known elementary observation. Lemma 1 (Fitting’s Lemma). a) Let M be a module over a Noetherian ring R and let f : M → M be an endomorphism of M . Then for any Research supported by NSF 90 02512 Research supported by NSF 90 09590