Gorenstein Projective Dimension Relative to a Semidualizing Bimodule

Abstract

Let S and R be rings and _S C _R a semidualizing bimodule. We investigate the relation between the G _C -syzygy with the C-syzygy of a module as well as the relation between the G _C -projective resolution and the projective resolution of a module. As a consequence, we get that if

is an exact sequence of S-modules with all G _i , G ⁱ G _C -projective, such that Hom _S (𝔾, T) is still exact for any module T which is isomorphic to a direct summand of direct sums of copies of _S C, then Im(G ₀ → G ⁰) is also G _C -projective. We obtain a criterion for computing the G _C -projective dimension of modules. When _S C _R is a faithfully semidualizing bimodule, we study the Foxby equivalence between the subclasses of the Auslander class and that of the Bass class with respect to C.

Key Words:

• Auslander class
• Bass class
• Foxby equivalence
• (faithfully) semidualizing bimodules
• G _C -projective dimension
• G _C -projective modules

2010 Mathematics Subject Classification:

• 18G25, 18G20

ACKNOWLEDGMENTS

This research was partially supported by the Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20100091110034), NSFC (Grant No. 11171142), NSF of Jiangsu Province of China (Grant Nos. BK2010047, BK2010007), and a project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institution. The authors thank the referee for the useful suggestions.

Notes

Communicated by S. Bazzoni.