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 .

/pdf/detecting-finite-flat-dimension-of-modules-via-iterates-of-1xv33wnw0h.pdf

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

use flat, cotorsion module theory to show that if R has finite Krull dimension, then an R-module M has finite flat dimension if and only if Tor R i (R (e) , M) = 0 for all i > 0 and infinitely many e > 0. Using methods involving the derived category, we show that one only needs vanishing for dim R + 1 consecutive values of i for infinitely many values of e to conclude that M has finite flat dimension. We also study a general notion of Matlis duality and prove a change of rings result for Matlis reflexive modules. Finally, we determine some properties of minimal flat resolutions and prove that if the Frobenius map is finite, then tensoring with R (e) preserves minimal flat resolutions. We also demonstrate a version of the New Intersection Theorem for complexes of flat, cotorsion modules. iii ACKNOWLEDGMENTS First, I wish to acknowledge Jesus Christ, the Savior of the world, through whom all things, including mathematics, were made and without whom nothing would exist. Second, I give my thanks to Tom Marley, without whose constant guidance and patient consideration I would never have been able to conduct this research and so enter into an academic career. Third, I must also thank those with whom I have had mathematical discussions, in particular Brittney Falahola, Peder Thompson, and Mark Webb. I hope the conversations were as fruitful for them as they were for me. Also, I am indebted to serving on my Ph.D. committee. Fourth, to the many professors and graduate students in UNL's Math Department who have prepared me so well to be a teacher of mathematics: Thank you for helping me to find my professional calling. To Nathan Wakefield, in particular, I owe a great deal of thanks for his many hours of mentorship and assistance. Fifth, I may have never studied mathematics were it not for my high school math teacher, Rita Ahmann. I hope to inspire my students as she once inspired me. Also, to my undergraduate professors who so strongly encouraged me in my academic pursuits I give heartfelt thanks. Sixth, to my friends and colleagues in the Mathematics Department, whose companionship made graduate school much more enjoyable and who have bought me more beer than I have bought for them: I hope I wasn't a drag on you, and I will settle the debt to you someday. …

/pdf/rigidity-of-the-frobenius-matlis-reflexivity-and-minimal-38feox5c0f.pdf

Rigidity of the frobenius, matlis reflexivity, and minimal flat resolutions

It is proved that a module M over a Noetherian local ring R of prime characteristic and positive dimension has finite flat dimension if $${\text {Tor}}_i^R({}^{e}\!R, M)=0$$ for $${\text {dim}}\,R$$ 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 $$e\geqslant \log _p e(R)$$, where e(R) is the multiplicity of the ring. This improves a result of Dailey et al. (J Commut Algebra), as well as generalizing a theorem due to Miller (Contemp Math 331:207–234, 2003) from finitely generated modules to arbitrary modules. We also show that if R is a complete intersection ring then the vanishing of $${\text {Tor}}_i^R({}^{e}\!R, M)$$ for single positive values of i and e is sufficient to imply M has finite flat dimension. This extends a result of Avramov and Miller (Math Res Lett 8(1–2):225–232, 2001).

/pdf/frobenius-and-homological-dimensions-of-complexes-x8yq3oh1mw.pdf

Frobenius and Homological Dimensions of Complexes

Characterizing gorenstein rings using contracting endomorphisms

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 $t\ge 0$ such that $\operatorname{Tor}_i^R(M, {}^{f^{e}}\!R)=0$ for $t\le i\le t+\dim R$ and infinitely many $e$. This extends a result of Herzog, who proved it when $M$ is finitely generated, and strengthens a result of the third author and Webb in the case $M$ is arbitrary. It is also proved that when $R$ is a Cohen-Macaulay local ring, it suffices that the Tor vanishing holds for one $e\ge \log_{p}e(R)$, where $e(R)$ is the multiplicity of $R$.

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

Let $R$ be a commutative Noetherian local ring of prime characteristic $p$ and $f:R\to R$ the Frobenius ring homomorphism. For $e\ge 1$ let $R^{(e)}$ denote the ring $R$ viewed as an $R$-module via $f^e$. Results of Peskine, Szpiro, and Herzog state that for finitely generated modules $M$, $M$ has finite projective dimension if and only if $\operatorname{Tor}_i^R(R^{(e)},M)=0$ for all $i>0$ and all (equivalently, infinitely many) $e\ge 1$. We prove this statement holds for arbitrary modules using the theory of flat covers and minimal flat resolutions.

The Acyclicity of the Frobenius Functor for Modules of Finite Flat Dimension

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

The Acyclicity of the Frobenius Functor for Modules of Finite Flat Dimension