Tilting modules and Auslander's Gorenstein property

Abstract

The notion of (generalized) tilting modules is introduced over arbitrary rings. It is shown that many results on tilting modules over Artin algebras are naturally generalized. The Auslander's Gorenstein property for tilting modules is considered over noetherian rings.

Totally reflexive modules with respect to a semidualizing bimodule

“…First, we employ some notions used by S. Sather-Wagstaff, T. Wakamatsu and D. White in [14], [20], [21]. Definition 1.1.…”

“…A subcategory or a class of right R-modules (left S-modules) is a full subcategory of the category of right R-modules (left S-modules), which is closed under isomorphisms. For unexplained concepts and notation, we refer the reader to [13], [20], [14].…”

Relative syzygies and grade of modules

“…If R is a left Noetherian ring, S is a right Noetherian ring and R ω S is a faithfully balanced and n-selforthogonal bimodule for all n, then R ω is just a generalized tilting module with S = End( R ω) in the sense of Wakamatsu [4,5]. In this case, ω S is also a generalized tilting module with R = End(ω S ) by [5,Corollary 3.2].…”

The socle of the last term in a minimal injective resolution

“…We use mod Λ to denote the category of finitely generated left Λ-modules, and Mod Λ to denote the category of left Λ-modules. Definition 2.1 [10].…”

“…The following result gives some equivalent characterizations of k-Gorenstein modules, among which the equivalence of (1) and (1) op is due to [10,Theorem 7.5], and the other implications are contained in…”