$\boldsymbol{n}$-coherent rings and semidualizing bimodules

Abstract

In this paper, we characterize right $n$-coherent rings by $C$-$FP_n$-injective modules and $C$-$FP_n$-flat modules, where $_SC_R$ is a semidualizing bimodule and $n$ is an integer with $n>1$. We investigate right derived functors of $-\otimes-$ defined via proper right $C$-$FP_n$-flat resolutions and proper right $C$-$FP_n$-injective resolutions, and study left derived functorsof Hom$(-,-)$ defined via proper right (left) $C$-$FP_n$-injective resolutions. As applications, we give some characterizations of relative homological dimensions defined by $C$-$FP_n$-injective modules and $C$-$FP_n$-flat modules.