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 Tor R i ( e R, M ) = 0 for 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 log p e(R), where e(R) is the multiplicity of the ring. This improves a result of D. Dailey, S. Iyengar, and the second author [6], as well as generalizing a theorem due to C. Miller [14] from finitely generated modules to arbitrary modules. We also show that if R is a complete intersection ring then the vanishing of Tor R i ( 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 L. Avramov and C. Miller [2].Throughout this paper (R, m, k) will denote a commutative Noetherian local ring with maximal ideal m and residue field k. In the case R has prime characteristic p, we let f : R → R denote the Frobenius endomorphism; i.e., f (r) = r p for every r ∈ R. For an integer e 1 we let e R denote the ring R viewed as an R-algebra via f e ; i.e., for r ∈ R and s ∈ e R, r · s := f e (r)s = r p e s. If e R is finitely generated as an R-module for some (equivalently, all) e > 0, we say that R is F -finite.