Frobenius and homological dimensions of complexes

It is proved that a module M over a Noetherian local ring R of prime characteristic and positive dimension has finite flat dimension if ToriR(eR,M)=0$${\text {Tor}}_i<^>R({}<^>{e}\!R, M)=0$$\end{document} for dimR$${\text {dim}}\,R$$\end{document} consecutive positive values of i and infinitely many e. Here eR$${}<^>{e}\!R$$\end{document} denotes the ring R viewed as an R-module via the eth iteration of the Frobenius endomorphism. In the case R is Cohen-Macualay, it suffices that the Tor vanishing above holds for a single e > logpe(R)$$e\geqslant \log _p e(R)$$\end{document}, where e(R) is the multiplicity of the ring. This improves a result of Dailey et al. (J Commut Algebra), as well as generalizing a theorem due to Miller (Contemp Math 331:207-234, 2003) from finitely generated modules to arbitrary modules. We also show that if R is a complete intersection ring then the vanishing of ToriR(eR,M)$${\text {Tor}}_i<^>R({}<^>{e}\!R, M)$$\end{document} for single positive values of i and e is sufficient to imply M has finite flat dimension. This extends a result of Avramov and Miller (Math Res Lett 8(1-2):225-232, 2001).