Relative Tor Functors with Respect to a Semidualizing Module

Abstract

Let C be a semidualizing module over a commutative noetherian ring R. We exhibit an isomorphism \(\operatorname{Tor}^{{\mathcal{F}_C}\mathcal{M}}_{i}(-,-) \cong \operatorname{Tor}^{{\mathcal{P}_C}\mathcal{M}}_{i}(-,-)\) between the bifunctors defined via C-flat and C-projective resolutions. We show how the vanishing of these functors characterizes the finiteness of \({{\mathcal{F}_C}\text{-}\operatorname{pd}}\), and use this to give a relation between the \({{\mathcal{F}_C}\text{-}\operatorname{pd}}\) of a module and of a pure submodule. On the other hand, we show that other isomorphisms force C to be trivial.