Let $S$ and $R$ be rings and $_SC_R$ a (faithfully) semidualizing bimodule. We introduce and study $C$-weak flat and $C$-weak injective modules as a generalization of $C$-flat and $C$-injective modules (J. Math. Kyoto Univ. 47(2007), 781-808) respectively, and use them to provide additional information concerning the important Foxby equivalence between the subclasses of the Auslander class $\mathcal{A}_C(R)$ and that of the Bass class $\mathcal{B}_C(S)$. Then we study the stability of Auslander and Bass classes, which enables us to give some alternative characterizations of the modules in $\mathcal{A}_C(R)$ and $\mathcal{B}_C(S)$. Finally we consider an open question which is closely relative to the main results (Proc. Edinb. Math. Soc. 48(2005), 75-90), and discuss the relationship between the Bass class $\mathcal{B}_C(S)$ and the class of Gorenstein injective modules.
In this paper, it is proved that a commutative noetherian local ring admitting a finitely generated module of finite projective and injective dimensions with respect to a semidualizing module is Gorenstein. This result recovers a celebrated theorem of Foxby.
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.
Gorensteinness from duality pairs induced via Foxby equivalences
Centro de Ciencias Matemáticas. Universidad Nacional Autónoma de México（墨西哥国立自治大学数学科学中心）；Instituto de Matemática y Estadística “Prof. Ing. Rafael Laguardia”. Facultad de Ingeniería. Universidad de la República（乌拉圭共和国大学工程学院拉夫卡迪亚教授数学与统计研究所）
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Let $R$ and $S$ be rings and $_Rω_S$ a semidualizing bimodule. We prove that there exists a Morita equivalence between the class of $\infty$-$ω$-cotorsionfree modules and a subclass of the class of $ω$-adstatic modules. Also we establish the relation between the relative homological dimensions of a module $M$ and the corresponding standard homological dimensions of ${\rm Hom}(ω,M)$. By investigating the properties of the Bass injective dimension of modules (resp. complexes), we get some equivalent characterizations of semi-tilting modules (resp. Gorenstein artin algebras). Finally we obtain a …
