Finitistic dimension and orthogonal classes of Gorenstein projective modules with respect to a semidualizing module

Abstract

Let R be a commutative Noetherian ring and let C be a semidualizing R-module. It is proved that $\text{FPD}(R)=\sup \{\mathcal{G}{\mathcal{P}}_C{-\text{pd}}_R(M)|M\in {\mathcal{B}}_C(R)\text{and}\mathcal{G}{\mathcal{P}}_C{-\text{pd}}_R(M)<\infty \}$, which is a generalization of Holm, Gorenstein homological dimensions. J. Pure Appl Algebra 189(1–3):167–193, Theorem 2.28. Also, it is given a condition to verify certain relation between G_C-projectives and G_C-flats. Indeed, it is shown that every G_C-projective module is G_C-flat if and only if for every C-injective module I, the character module ${\text{Hom}}_{\mathbb{Z}}(I,\mathbb{Q}/\mathbb{Z})$ is in $\mathcal{G}{\mathcal{P}}_C{(R)}^{\perp }$.

Keywords:

• Finitistic projective dimension
• G_C-projective
• G_C-injective
• G_C-flat
• semidualizing

2010 Mathematics Subject Classification:

• 13D05
• 13D45
• 18G20

Acknowledgements

The author is very grateful to the referee for several comments that greatly improved the paper.