Relative grade and relative Gorenstein dimension with respect to a semidualizing module

Abstract

Let R be a commutative Noetherian ring, and let C be a semidualizing R-module. For R-modules M and N, the notions ${\text {grade}}_{{\mathcal {P}}_{C}} (M, N)$ and ${\text {grade}}_{{\mathcal {I}}_{C}} (M, N)$ are introduced as the relative setting of the notion ${\text {grade}}(M, N)$ with respect to C. Some results about ${\text {grade}}_{{\mathcal {P}}_{C}} (M, N)$, ${\text {grade}}_{{\mathcal {I}}_{C}} (M, N)$ and ${\text {grade}}(M, N)$ are mentioned. For finitely generated R-modules M and N, we show that ${\text {grade}}_{{\mathcal {P}}_{C}} (M, N)= {\text {grade}}(M, N)$ $({\text {grade}}_{{\mathcal {I}}_{C}} (M, N)= {\text {grade}}(M, N))$, provided we have some special conditions. Also, the notions of C-perfect and $G_{C}$-perfect R-modules are introduced as the relative setting of the notions of perfect and G-perfect R-modules with respect to C, and it is proven that several results for these new concepts are similar to the classical results. Finally, some results about relative grade of tensor and Hom functors with respect to C are given.