Algebras that satisfy Auslander’s condition on vanishing of cohomology

Abstract

Auslander conjectured that every Artin algebra satisfies a certain condition on vanishing of cohomology of finitely generated modules. The failure of this conjecture-by a 2003 counterexample due to Jorgensen andŞega-motivates the consideration of the class of rings that do satisfy Auslander's condition. We call them AC rings and show that an AC Artin algebra that is left-Gorenstein is also right-Gorenstein. Furthermore, the Auslander-Reiten Conjecture is proved for AC rings, and Auslander's G-dimension is shown to be functorial for AC rings that are commutative or have a dualizing complex.

Auslander bounds and homological conjectures

“…In their paper, it was shown that, if gdO = d < 1, then GrepR = d − 1, in particular, the finitistic dimension of R is finite. Combining these with results in this paper and [4], we also know that in case O has finite global dimension, the algebra O/πO also satisfies the Auslander conjecture, the conjecture (FAC), the Auslander-Reiten conjecture, Gorenstein Symmetry conjecture and the conjecture (EWTC).…”

“…Moreover, it is pointed out in [4] that there is a commutative notherian ring R with infinite Krull dimension such that lAbM < ∞ for every M ∈ modR but gAbR = ∞.…”

“…Rings satisfying the assertion in the Auslander conjecture are studied in [4,9,16]. In [4] the authors investigate in detail the relationship between such rings and some homological conjectures, for instance, the Auslander-Reiten conjecture, which we recall as follows. In this paper, we continue the study and focus on the number b M , which we called the left Auslander bound of M and denote by lAbM, for the fixed module M. Note that the Auslander bound can be ∞.…”

On the vanishing of cohomology in triangulated categories

“…A priori, the complexity of a pair is not finite. However, finiteness holds when Hom this, let n 0 be an integer such that the R-module Hom Such algebras were studied in [15]. Motivated by this, we say that a full subcategory C of T satisfies the left Auslander condition if for every object X ∈ C, there exists an integer d X , depending only on X, such that the following holds: if Hom * T (X, Y ) is eventually zero for some object…”

‘Ext-Finite Modules for Weakly Symmetric Algebras With Radical Cube Zero’

“…There is also the 'generalized Auslander-Reiten condition', GARC, which has been introduced in [1] in the context of homological conjectures, which has attracted a lot of interest, see for example [6,7,13,8,9]. The condition GARC for a ring R states:…”