Let R be a local Cohen-Macaulay ring with canonical module ωR. We investigate the following question of Huneke: If the sequence of Betti numbers {β i (ωR)} has polynomial growth, must R be Gorenstein? This question is well-understood when R has minimal multiplicity. We investigate this question for a more general class of rings which we say are homologically of minimal multiplicity. We provide ...

The main theorem of this paper complements the tame-wild dichotomy for commutative Noetherian rings, obtained by Klingler and Levy [14]–[16]. They gave a complete classification of all finitely generated modules over Dedekind-like rings (cf. Definition 1.1) and showed that, over any ring that is not a homomorphic image of a Dedekind-like ring, the category of finite-length modules has wild repr...

Recently, Takahashi established a new approximation theory for finitely generated modules over commutative Noetherian rings, which unifies the spherical approximation theorem due to Auslander and Bridger and the Cohen–Macaulay approximation theorem due to Auslander and Buchweitz. In this paper we generalize these results to much more general case over non-commutative rings. As an application, w...

We introduce the notion of relative singularity category with respect to any self-orthogonal subcategory ω of an abelian category. We introduce the Frobenius category of ω-Cohen-Macaulay objects, and under some reasonable conditions, we show that the stable category of ω-Cohen-Macaulay objects is triangle-equivalent to the relative singularity category. As applications, we relate the stable cat...