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Dailey, Douglas J.; Iyengar, Srikanth B.; Marley, Thomas

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

J. Commut. Algebra 12, No. 1, 71-76 (2020).
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 \(R\).

MSC:

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

Keywords:

Frobenius map; flat dimension; homotopical Loewy length
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\textit{D. J. Dailey} et al., J. Commut. Algebra 12, No. 1, 71--76 (2020; Zbl 1442.13041)
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