It is proved that a module over a Noetherian ring of positive characteristic has finite flat dimension if there exists an integer such that for and infinitely many . This extends results of Herzog, who proved it when is finitely generated. It is also proved that when is a Cohen–Macaulay local ring, it suffices that the vanishing holds for one , where is the multiplicity of .
"Detecting finite flat dimension of modules via iterates of the Frobenius endomorphism." J. Commut. Algebra 12 (1) 71 - 76, Spring 2020. https://doi.org/10.1216/jca.2020.12.71