Detecting finite flat dimension of modules via iterates of the Frobenius endomorphism. (English) Zbl 1442.13041

Summary: 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 \(t \geqslant 0\) such that \(\operatorname{Tor}_i^R (M,^{f^{e}}R)=0\) for \(t \leqslant i \leqslant t + \operatorname{dim} R\) 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 \(\operatorname{Tor}\) vanishing holds for one \(e \geqslant \operatorname{log}_p e(R)\), where \(e(R)\) is the multiplicity of \(R\).


13D05 Homological dimension and commutative rings
13D07 Homological functors on modules of commutative rings (Tor, Ext, etc.)
13A35 Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure
Full Text: DOI arXiv Euclid


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