On the vanishing of (co)homology for modules admitting certain filtrations

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.