Related papers: Foxby equivalence relative to $C$-weak injective a…

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…

Let $R$ and $S$ be rings, $C= {}_SC_R$ a (faithfully) semidualizing bimodule, and $n$ a positive integer or $n=\infty$. In this paper, we introduce the concepts of $C$-$fp_n$-injective $R$-modules and $C$-$fp_n$-flat $S$-modules as a common…

We continue our work on adic semidualizing complexes over a commutative noetherian ring $R$ by investigating the associated Auslander and Bass classes (collectively known as Foxby classes), following Foxby and Christensen. Fundamental…

Let R be a local ring and C a semidualizing module of R. We investigate the behavior of certain classes of generalized Cohen-Macaulay R-modules under the Foxby equivalence between the Auslander and Bass classes with respect to C. In…

We define and study induced duality pairs under Foxby equivalences. Given a semidualizing $(S,R)$-bimodule ${}_S C_R$, if $(\mathcal{A}_C(R),\mathcal{B}_C(R^{\rm op}))$ and $(\mathcal{A}_C(S^{\rm op}),\mathcal{B}_C(S))$ denote the duality…

In this paper, we first introduce $\mathcal {W}_F$-Gorenstein modules to establish the following Foxby equivalence: $\xymatrix@C=80pt{\mathcal {G}(\mathcal {F})\cap \mathcal {A}_C(R) \ar@<0.5ex>[r]^{C\otimes_R-} & \mathcal {G}(\mathcal…

In this paper, we introduce the notions of Gorenstein weak injective and weak flat modules respectively in terms of weak injective and weak flat modules, which is larger than classical classes of Gorenstein injective and flat modules. In…

We investigate the properties of categories of G_C-flat R-modules where C is a semidualizing module over a commutative noetherian ring R. We prove that the category of all G_C-flat R-modules is part of a weak AB-context, in the terminology…

Let $R$ be a commutative noetherian ring with a semi-dualizing module $C$. The Auslander categories with respect to $C$ are related through Foxby equivalence: $\xymatrix@C=50pt{\mathcal {A}_C(R) \ar@<0.4ex>[r]^{C\otimes^{\mathbf{L}}_{R} -}…

A model structure on a category is a formal way of introducing a homotopy theory on that category, and if the model structure is abelian and hereditary, its homotopy category is known to be triangulated. So a good way to both build and…

In this paper we introduce and study the weak Gorenstein global dimension of a ring $R$ with respect to a left $R$-module $C$. We provide several characterizations of when this homological invariant is bounded. Two main applications are…

This paper aims to provide several relations between Bass and Betti numbers of a given module and its deficiency modules. Such relations and the tools used throughout allow us to generalize some results of Foxby, characterize Cohen-Macaulay…

Let $R$ be a commutative ring with identity, and let $S$ be a multiplicative subset of $R$. In this paper, we introduce the notion of $S$-injective modules as a weak version of injective modules. Among other results, we provide an…

For a dualizing module $D$ over a commutative Noetherian ring $R$ with identity, it is known that its Auslander class $\mathscr{A}_D\left(R\right)$ (respectively, Bass class $\mathscr{B}_D\left(R\right)$) is characterized as those…

Let $R$ be a Noetherian ring and let $C$ be a semidualizing $R$-module. In this paper, by using relative homological dimensions with respect to $C$, we impose various conditions on $C$ to be dualizing. First, we show that $C$ is dualizing…

Let $R$ and $S$ be any rings and $_RC_S$ a semidualizing bimodule, and let $\mathcal{A}_C(R^{op})$ and $\mathcal{B}_C(R)$ be the Auslander and Bass classes respectively. Then both the pairs $$(\mathcal{A}_C(R^{op}),\mathcal{B}_C(R))\ {\rm…

Let $R$ and $S$ be rings and $_R\omega_S$ a semidualizing bimodule. We prove that there exists a Morita equivalence between the class of $\infty$-$\omega$-cotorsionfree modules and a subclass of the class of $\omega$-adstatic modules. Also…

Let $S$ and $R$ be rings, $n, d\geq 0$ be two integers or $n=\infty$. In this paper, first we introduce special (faithfully) semidualizing bimodule $_S(K_{d-1})_R$, and then introduce and study the concepts of $K_{d-1}$-$(n,d)$-injective…

In this paper, we define a class of relative derived functors in terms of left or right weak flat resolutions to compute the weak flat dimension of modules. Moreover, we investigate two classes of modules larger than that of weak injective…

For a semidualizing module $C$ over a ring $R$, we study the following classes modulo exact zero divisors: $\g_C$--projectives, $\mathcal G_C$; the Auslander class $\mathcal A_C$; the Bass class $\mathcal B_C$; $\mathcal{P}_C$--projective;…