Auslander conditions and tilting-like cotorsion pairs

We study homological behavior of modules satisfying the Auslander condition. Assume that AC is the class of left R-modules satisfying the Auslander condition. It is proved that each cycle of an exact complex with each term in AC belongs to AC for any ring R. As a consequence, we show that for any left Noetherian ring R, AC is a resolving subcategory of the category of left R-modules if and only if RR satisfies the Auslander condition if and only if each Gorenstein projective left R-module belongs to AC. As an application, we prove that, for an Artinian algebra R satisfying the Auslander condition, R is Gorenstein if and only if AC coincides with the class of Gorenstein projective left R-modules if and only if (AC<∞,(AC<∞)⊥) is a tilting-like cotorsion pair if and only if (AC<∞,I) is a tilting-like cotorsion pair, where AC<∞ is the class of left R-modules with finite AC-dimension and I is the class of injective left R-modules. This leads to some criteria for the validity of the Auslander and Reiten conjecture which says that an Artinian algebra satisfying the Auslander condition is Gorenstein.