Frobenius and homological dimensions of complexes

Abstract

It is proved that a module M over a Noetherian local ring R of prime characteristic and positive dimension has finite flat dimension if ${\text {Tor}}_i^R({}^{e}\!R, M)=0$ for ${\text {dim}}\,R$ consecutive positive values of i and infinitely many e. Here ${}^{e}\!R$ 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 $e\geqslant \log _p e(R)$, 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 ${\text {Tor}}_i^R({}^{e}\!R, M)$ 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).