Abstract
It is proved that a module M over a Noetherian ring R of positive characteristic p has finite flat dimension if there exists an integer t ≥ 0 such that Tor i R ( M , f e R ) = 0 for t ≤ i ≤ t + dim R and infinitely many e . This extends results of Herzog, who proved it when M is finitely generated. It is also proved that when R is a Cohen–Macaulay local ring, it suffices that the Tor vanishing holds for one e ≥ log p e ( R ) , where e ( R ) is the multiplicity of R .
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