  We study the vanishing of (co)homology along ring homomorphisms for modules
that admit certain filtrations, and generalize a theorem of O.
Celikbas-Takahashi. Our work produces new classes of rigid and test modules, in
particular over local rings of prime characteristic. It also gives applications
in the study of torsion in tensor products of modules, for example, concerning
the Huneke-Wiegand conjecture.
