The acyclicity of the Frobenius functor for modules of finite flat dimension

Marley, Thomas; Webb, Marcus

doi:10.1016/j.jpaa.2016.01.007

Cited by 4 publications

(4 citation statements)

References 19 publications

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“…We choose not to make use of this result in this paper, however.) In [15], the second author together with M. Webb proved the analogue of Peskine and Szpiro's result for modules of finite flat dimension; that is, if M has finite flat dimension then Tor R i ( e R, M ) = 0 for all positive integers i and e. Further, it was shown that that the analogue of Herzog's result holds for arbitrary modules as well. Subsequently, Dailey, Iyengar and the second author [6] showed that if Tor R i ( e R, M ) = 0 for dim R + 1 consecutive positive values of i and infinitely many e, then M has finite flat dimension.…”

Section: Introductionmentioning

confidence: 93%

Frobenius and homological dimensions of complexes

Funk

Marley

2019

Collect. Math.

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It is proved that a module M over a Noetherian local ring R of prime characteristic and positive dimension has finite flat dimension if Tor R i ( e R, M ) = 0 for dim R consecutive positive values of i and infinitely many e. Here e R denotes the ring R viewed as an R-module via the eth iteration of the Frobenius endomorphism. In the case R is Cohen-Macualay, it suffices that the Tor vanishing above holds for a single e log p e(R), where e(R) is the multiplicity of the ring. This improves a result of D. Dailey, S. Iyengar, and the second author [6], as well as generalizing a theorem due to C. Miller [14] from finitely generated modules to arbitrary modules. We also show that if R is a complete intersection ring then the vanishing of Tor R i ( e R, M ) for single positive values of i and e is sufficient to imply M has finite flat dimension. This extends a result of L. Avramov and C. Miller [2].Throughout this paper (R, m, k) will denote a commutative Noetherian local ring with maximal ideal m and residue field k. In the case R has prime characteristic p, we let f : R → R denote the Frobenius endomorphism; i.e., f (r) = r p for every r ∈ R. For an integer e 1 we let e R denote the ring R viewed as an R-algebra via f e ; i.e., for r ∈ R and s ∈ e R, r · s := f e (r)s = r p e s. If e R is finitely generated as an R-module for some (equivalently, all) e > 0, we say that R is F -finite.

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Section: Introductionmentioning

confidence: 93%

Frobenius and homological dimensions of complexes

Funk

Marley

2019

Collect. Math.

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“…We remark that part (a) in the case M = R and L a finitely generated R-module is a classic result of Peskine-Szpiro [20,Théorèm 1.7]. This was later generalized to the case L is an arbitrary module in [19,Theorem 1.1], and to the case L is an arbitrary complex in [8, Theorem 1.1]. Part (b) was proved in [20,Théorèm 4.15] in the case M = R and L a finitely generated module, and in [19,Corollary 3.5] in the case M = R and L an arbitrary module.…”

Section: (A) If There Exists An Integer T Inf H(m ) + Sup H(l) Such T...mentioning

confidence: 77%

“…This was later generalized to the case L is an arbitrary module in [19,Theorem 1.1], and to the case L is an arbitrary complex in [8, Theorem 1.1]. Part (b) was proved in [20,Théorèm 4.15] in the case M = R and L a finitely generated module, and in [19,Corollary 3.5] in the case M = R and L an arbitrary module. Before stating the next corollary, we set some terminology and notation.…”

Section: (A) If There Exists An Integer T Inf H(m ) + Sup H(l) Such Tmentioning

confidence: 99%

Characterizing Gorenstein rings using contracting endomorphisms

Falahola

Marley

2021

Journal of Algebra

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Dedicated to Craig Huneke on the occasion of his 65th birthday.Abstract. We prove several characterizations of Gorenstein rings in terms of vanishings of derived functors of certain modules or complexes whose scalars are restricted via contracting endomorphisms. These results can be viewed as analogues of results of Kunz (in the case of the Frobenius) and Avramov-Hochster-Iyengar-Yao (in the case of general contracting endomorphisms).

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“…Peskine and Szpiro [15] proved that (1)⇒(2), Herzog [10] proved that (3)⇒(1), and Koh and Lee [11] proved that (4)⇒ (1). Recently, the third author and M. Webb [14,Theorem 4.2] proved the equivalence of conditions (1), (2), and (3) for all R-modules, even infinitely generated ones. In their work, the argument for (3)⇒( 1) is quite technical and heavily dependent on results of Enochs and Xu [8] concerning flat cotorsion modules and minimal flat resolutions.…”

Section: Introductionmentioning

confidence: 99%

Detecting finite flat dimension of modules via iterates of the Frobenius endomorphism

Dailey¹,

Iyengar²,

Marley³

2016

Preprint

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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 R i (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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The acyclicity of the Frobenius functor for modules of finite flat dimension

Cited by 4 publications

References 19 publications

Frobenius and homological dimensions of complexes

Frobenius and homological dimensions of complexes

Characterizing Gorenstein rings using contracting endomorphisms

Detecting finite flat dimension of modules via iterates of the Frobenius endomorphism

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