Relative Tor functors with respect to a semidualizing module

TL;DR

This paper explores relative Tor functors associated with a semidualizing module over a noetherian ring, establishing isomorphisms, characterizations of homological dimensions, and limitations of certain relations.

Contribution

It introduces and compares relative Tor functors from different resolutions, characterizes homological dimensions via vanishing, and reveals constraints on relations involving semidualizing modules.

Findings

Isomorphism between Tor^{F_CM}_i and Tor^{P_CM}_i functors.

Vanishing of these functors characterizes F_C-pd finiteness.

Certain expected relations fail unless the semidualizing module is trivial.

Abstract

We consider relative Tor functors built from resolutions described by a semidualizing module C over a commutative noetherian ring R. We show that the bifunctors Tor^{F_CM}_i (-,-) and Tor^{P_CM}_i (-,-), defined using flat-like and projective-like resolutions, are isomorphic. We show how the vanishing of these functors characterizes the finiteness of the homological dimension F_C-pd, and we use this to give a relation between the F_C-pd of a given module and that of a pure submodule. On the other hand, we show that other relations that one may expect to hold similarly, fail in general. In fact, such relations force the semidualizing modules under consideration to be trivial.