Auslander bounds and homological conjectures

Abstract

Inspired by recent works on rings satisfying Auslander's conjecture, we study invariants, called Auslander bounds, and prove that they have strong relations to some homological conjectures.

Derived invariance by syzygy complexes

“…It is also hard to achieve the above result in the scope of tilting theory. In case that M is an R-module and C = E = modR, the notions given here coincide with the usual ones in module categories [4,20], i.e., Auslander bound of M, global Auslander bound of the algebra R and the finitistic Auslander bound of the algebra R, respectively.…”

“…We refer to [4] for the list of AC-algebras. In [20], the author suggests a revisited version of Auslander conjecture, named the finitistic Auslander conjecture, which asserts that the finitistic Auslander bound of every algebra is finite. Note that the finitistic Auslander conjecture implies the finitistic dimension conjecture.…”

The Auslander-Reiten conjecture for certain non-Gorenstein Cohen-Macaulay rings

“…The Auslander-Reiten conjecture and several related conjectures are problems about the vanishing. For a guide to these conjectures, one can consult [7, Appendix A] and [6,16,24,25]. These conjectures originate from the representation theory of rings.…”

Auslander–Reiten conjecture and Auslander–Reiten duality

“…The Auslander-Reiten conjecture is closely related to other important conjectures such as the Tachikawa conjecture [30] and the finitistic dimension conjecture [18]. Indeed it is rooted in a conjecture of Nakayama [26] and the one known as the generalized Nakayama conjecture [6]: If R is an Artin algebra, then every indecomposable injective R-module occurs as a direct summand of one of the terms of the minimal injective resolution of R. We refer the reader to [7,10,13,16,17,[32][33][34] for more information on the relation of these homological conjectures.…”

“…The main tool he used for the proof of his result is a duality theorem for Gorenstein local rings, which is known as the Auslander-Reiten duality theorem (see [4,9,35]). On the other hand, as a natural generalization of (AC), the following generalized Auslander-Reiten condition has also been investigated by several authors [16,32,33] (GARC) For every finitely generated R-module M, if there exists a nonnegative integer n such that Ext >n R (M, M ⊕ R) = 0, then M has projective dimension at most n.…”