Foxby equivalence over associative rings

Abstract

We extend the definition of a semidualizing module to associative rings. This enables us to define and study Auslander and Bass classes with respect to a semidualizing bimodule C. We then study the classes of C-flats, C-projectives, and C-injectives, and use them to provide a characterization of the modules in the Auslander and Bass classes. We extend Foxby equivalence to this new setting. This paper contains a few results which are new in the commutative, noetherian setting.2000 Mathematics Subject Classification. 13D02, 13D07, 13D25, 16E05, 16E30.

AB-Contexts and Stability for Gorenstein Flat Modules with Respect to Semidualizing Modules

“…Since F C -pd R (M ) n, the proof of the implication (i) =⇒ (ii) shows that fd R (Hom R (C, M )) n. Since each R-module Hom R (C, X i ) is flat by Lemma 3.3, the exact complex Hom R (C, X ′ ) is a truncation of an augmented flat resolution of Hom R (C, M ). It follows that Hom R (C, K n ) is flat, and so K n ∈ F C (R) by [18,Thm. 1].…”

“…Similarly, we have M ∈ B C (R) if and only if Hom R (C, M ) ∈ A C (R) by [24, (2.8.a)]. From [18,Thm. 6.1] we know that every module in B C (R) has a P C -proper P C -resolution.…”

$\V\W$-Gorenstein categories

“…If X (R) is a subcategory of Mod R , then X f (R) is the subcategory of finitely generated modules in X (R) . Definition 2.5 [13] An (R, S)-bimodule C = R C S is called semidualizing if it satisfies the following:…”

Balance for Relative Homology With Respect to Semidualizing Modules

“…More examples of semidualizing modules can be found in [1,11,12,19]. In the remainder of the paper, let C be a fixed semidualizing module.…”

“…In Section 2, we review some basic notation and notions. Also, some necessary facts appeared in [11,20] are listed. We focus on in Section 3 comparison of different relative homologies.…”