Douglas J. Dailey, Srikanth B. Iyengar, Thomas Marley. Source: Journal of Commutative Algebra, Volume 12, Number 1, 71--76.Abstract:
It is proved that a module [math] over a Noetherian ring [math] of positive characteristic [math] has finite flat dimension if there exists an integer [math] such that [math] for [math] and infinitely many [math] . This extends results of Herzog, who proved it when [math] is finitely generated. It is also proved that when [math] is a Cohen–Macaulay local ring, it suffices that the [math] vanishing holds for one [math] , where [math] is the multiplicity of [math] .