Duality pairs induced by Auslander and Bass classes

Let R and S be arbitrary rings and let C S R {{}_{R}C_{S}} be a semidualizing bimodule, and let 𝒜 C ⁢ ( R op ) {\mathcal{A}_{C}(R^{\mathrm{op}})} and ℬ C ⁢ ( R ) {\mathcal{B}_{C}(R)} be the Auslander and Bass classes, respectively. Then both pairs ( 𝒜 C ⁢ ( R op ) , ℬ C ⁢ ( R ) ) and ( ℬ C ⁢ ( R ) , 𝒜 C ⁢ ( R op ) ) (\mathcal{A}_{C}(R^{\mathrm{op}}),\mathcal{B}_{C}(R))\quad\text{and}\quad(% \mathcal{B}_{C}(R),\mathcal{A}_{C}(R^{\mathrm{op}})) are coproduct-closed and product-closed duality pairs and both 𝒜 C ⁢ ( R op ) {\mathcal{A}_{C}(R^{\mathrm{op}})} and ℬ C ⁢ ( R ) {\mathcal{B}_{C}(R)} are covering and preenveloping; in particular, the former duality pair is perfect. Moreover, if ℬ C ⁢ ( R ) {\mathcal{B}_{C}(R)} is enveloping in Mod ⁡ R {\operatorname{Mod}R} , then 𝒜 C ⁢ ( S ) {\mathcal{A}_{C}(S)} is enveloping in Mod ⁡ S {\operatorname{Mod}S} . Also, some applications to the Auslander projective dimension of modules are given.

Homological Behavior of Relative Cotorsionfree and Cosyzygy Modules

As the dual of the Auslander transpose and the resulting k-torsionfree module, the cotranspose and k-cotorsionfree module with respect to a semidualizing bimodule have been introduced recently. In this paper we first investigate the relation between relative k-cotorsionfree modules and relative k-cosyzygy modules. Then we study the extension closure of these two classes of modules.

Semidualizing bimodules and related Gorenstein homological dimensions

Let [Formula: see text] be a semidualizing bimodule with [Formula: see text] left coherent and [Formula: see text] right coherent. For a non-negative integer [Formula: see text], it is shown that [Formula: see text]-[Formula: see text]-[Formula: see text] if and only if every finitely presented left [Formula: see text]-module has [Formula: see text]-projective dimension at most [Formula: see text] if and only if every finitely presented right [Formula: see text]-module has [Formula: see text]-projective dimension at most [Formula: see text]. As applications, some well-known results are extended.

WEAK AB-CONTEXT FOR FP-INJECTIVE MODULES WITH RESPECT TO SEMIDUALIZING MODULES

Let S and R be rings and SCR a semidualizing bimodule. We define and study GC-FP-injective R-modules, and these modules are exactly C-Gorenstein injective R-modules defined by Holm and Jørgensen provided that S = R is a commutative Noetherian ring. We mainly prove that the category of GC-FP-injective R-modules is a part of a weak AB-context, which is dual of weak AB-context in the terminology of Hashimoto. In particular, this allows us to deduce the existence of certain Auslander–Buchweitz approximations for R-modules with finite GC-FP-injective dimension. As an application, a new model structure in Mod R is given.