The behavior of the Frobenius map is investigated for valuation rings of prime characteristic. We show that valuation rings are always F-pure. We introduce a generalization of the notion of strong F-regularity, which we call F-pure regularity, and show that a valuation ring is F-pure regular if and only if it is Noetherian. For valuations on function fields, we show that the Frobenius map is finite if and only if the valuation is divisorial; in this case the valuation ring is Frobenius split. For Noetherian valuation rings in function fields, we show that the valuation ring is Frobenius split if and only if Frobenius is finite, or equivalently, if and only if the valuation ring is excellent.
Let A and B be integral domains. Suppose A is Noetherian and B is a finitely generated A-algebra that contains A. Denote by A' the integral closure of A in B. We show that A' is determined by finitely many unique discrete valuation rings. Our result generalizes Rees' classical valuation theorem for ideals. We also obtain a variant of Zariski's main theorem.
Using a local monomialization result of Knaf and Kuhlmann, we prove that the valuation ring of an Abhyankar valuation of a function field over a perfect ground field of prime characteristic is Frobenius split. We show that a Frobenius splitting of a sufficiently well-behaved center lifts to a Frobenius splitting of the valuation ring. We also investigate properties of valuations centered on arbitrary Noetherian domains of prime characteristic. In contrast to [arXiv:1507.06009], this paper emphasizes the role of centers in controlling Frobenius properties of valuations rings in prime characteri…
The notion of an Ohm-Rush algebra, and its associated content map, has connections with prime characteristic algebra, polynomial extensions, and the Ananyan-Hochster proof of Stillman's conjecture. As further restrictions are placed (creating the increasingly more specialized notions of weak content, semicontent, content, and Gaussian algebras), the construction becomes more powerful. Here we settle the question in the affirmative over a Noetherian ring from our previous article of whether a faithfully flat weak content algebra is semicontent (and over an Artinian ring of whether such an algeb…
Valuation rings and perfectoid rings are examples of (usually non-noetherian) rings that behave in some sense like regular rings. We give and study an extension of the concept of regular local rings to non-noetherian rings so that it includes valuation and perfectoid rings and it is related to Grothendieck's definition of formal smoothness as in the noetherian case. For that, we have to take into account the topologies. We prove a descent theorem for regularity along flat homomorphisms (in fact for homomorphisms of finite flat dimension), extending some known results from the noetherian to the…
Let $R$ be a commutative ring. A quasi-Gorenstein $R$-module is an $R$-module such that the grade of the module and the projective dimension of the module are equal and the canonical module of the module is isomorphic to the module itself. After discussing properties of finitely generated quasi-Gorenstein modules, it is shown that this definition allows for a characterization of diagonal matrices of maximal rank over a Noetherian factorial domain $R$ extending a theorem of Frobenius and Stickelberger to modules of projective dimension 1 over a commutative Noetherian factorial domain.
In this note, we study the Cohen-Macaulayness of non-Noetherian rings. We show that Hochster's celebrated theorem that a finitely generated normal semigroup ring is Cohen-Macaulay does not extend to non-Noetherian rings. We also show that for any valuation domain $V$ of finite Krull dimension, $V[x]$ is Cohen-Macaulay in the sense of Hamilton-Marley.
For a commutative Noetherian ring $R$ of prime characteristic, denote by $^{f}R$ the ring $R$ with the left structure given by the Frobenius map. We develop Thomas Marley's work on the property of the Frobenius functor $\F(-) = - \otimes_R {^f}R$ and show the interplay between $\F$ and its dual $\widetilde{\F}(-) = \Hom_R({}^{f}R, -)$ which is introduced by Jürgen Herzog.
Let $(R,\frak m)$ be a Noetherian local ring of prime characteristic $p>0$, and $t$ an integer such that $H_{\frak m}^j(R)/0^F_{H^j_{\frak m}(R)}$ has finite length for all $j<t$. The aim of this paper is to show that there exists an uniform bound for Frobenius test exponents of ideals generated by filter regular sequences of length at most $t$.
We introduce in this work, the class of commutative rings whose lattice of ideals forms an MTL-algebra which is not necessary a BL-algebra. The so-called class of rings will be named MTL-rings. We prove that a local commutative ring with identity is an MTL-ring if and only if it is an arithmetical ring. It is shown that a noetherian commutative ring R with an identity is an MTL-ring if and only if ideals of the localization RM at a maximal ideal M are totally ordered by the set inclusion. Remarking that noetherian MTL-rings are again BL-rings, we work outside of the noetherian case by consider…
Fix any field $K$ of characteristic $p$ such that $[K:K^p]$ is finite. We discuss excellence for Noetherian domains whose fraction field is $K$, showing for example, that $R$ is excellent if and only if the Frobenius map is finite on $R$. Furthermore, we show $R$ is excellent if and only if it admits some non-zero $p^{-e}$-linear map for $R$ or equivalently, that $R$ is a solid $R$-algebra under Frobenius. In particular, this means that Frobenius split Noetherian domains that are generically $F$-finite are always excellent. We also show that non-excellent rings are abundant and easy to constru…
Let R be a local domain, v a valuation of its quotient field centred in R at its maximal ideal. We investigate the relationship between R^h, the henselisation of R as local ring, and {v}, the henselisation of the valuation v, by focussing on the recent result by de Felipe and Teissier referred to in the title. We give a new proof that simplifies the original one by using purely algebraic arguments. This proof is moreover constructive in the sense of Bishop and previous work of the authors, and allows us to obtain as a by-product a (slight) generalisation of the theorem by de Felipe and Teissie…
我们考虑的问题是，哪些半群可以出现作为半群$S_R(ν)$的秩1赋值控制的诺特局部环$R$的正值。从$P_R(n)$的上界开始，给出了$n\in\NN$的$ϕ(n)=S_R(ν)\cap (0,n)$增长的多项式型上界，其中$P_R(n)$是$R$的Hilbert函数.在Cutkosky和Teissier的论文“Semigroups of valuations on local rings，II”中，这个界被推广到任意秩赋值的一个非常一般的界，arXiv：0805.3788。这个界已经足以给出秩1良序半群的简单例子，这些半群不是控制诺特局部环的赋值的值半群$S_R(ν)$。在秩为1的情况下，有可能给出更精确的估计$ϕ(n)$，我们证明在本文中。我们还给出的例子表明，许多不同的增长率是可能的$ϕ(n)$在一个经常的本地环的尺寸2，如$n(α$的任何合理的$α$与$1\leα\le 2$，和$n{log}(n)$。
On the test properties of the Frobenius endomorphism
West Virginia University（西弗吉尼亚大学）；School of Mathematics, Institute for Research in Fundamental Sciences (IPM)（基础科学研究所数学学院）；Georgia State University（佐治亚州立大学）
A splinter is a notion of singularity that has seen numerous recent applications, especially in connection with the direct summand theorem, the mixed characteristic minimal model program, Cohen-Macaulayness of absolute integral closures and cohomology vanishing theorems. Nevertheless, many basic questions about these singularities remain elusive. One outstanding problem is whether the splinter property spreads from a point to an open neighborhood of a noetherian scheme. Our paper addresses this problem in prime characteristic, where we show that a locally noetherian scheme that has finite Frob…
Among reduced Noetherian prime characteristic commutative rings, we prove that a regular ring is precisely one where finite intersection of ideals commutes with taking bracket powers. However, reducedness is essential for this equivalence. Connections are made with Ohm-Rush content theory, intersection-flatness of the Frobenius map, and various flatness criteria.
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$_i^R({}^e R, M)=0$ 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 $e$th iteration of the Frobenius endomorphism. In the case $R$ is Cohen-Macualay, it suffices that the Tor vanishing above holds for a single $e\geq \log_p e(R)$, where $e(R)$ is the multiplicity of the ring. This improves a result of D. Dailey, S. Iyengar, and the second author, as well as genera…
In this paper we extend an epimorphism theorem of D. Wright to the case of discrete valuation rings. We will show that if $(R, t)$ is a discrete valuation ring, $n \ge 2$ is an integer not divisible by the characteristic of the residue field $R/tR$, and $g \in R[X, Y, Z]$ is a polynomial of the form $g = b(X,Y)Z^n - a(X,Y)$ such that $R[X, Y, Z]/(g)$ is a polynomial algebra in two variables, then $g$ and $Z$ form a pair of variables in $R[X, Y, Z]$. We will also show that the result holds over any Noetherian domain containing $\mathbb{Q}$.
For a reduced Noetherian ring $R$ of characteristic $p > 0$, in this paper we discuss an extension of $R$ called its perfect closure $R^\infty$. This extension contains all $p^e$-th roots of elements of $R$, and is usually non-Noetherian. We first define the generalized notions of associated primes of a module over a non-Noetherian ring. Then for any $R$-module $M$, we state a correspondence between certain generalized prime ideals of $(R^\infty \otimes_R M)/N$ over $R^\infty$, and the union of associated prime ideals of $F^e(M)/N_e$ as $e \in \mathbb{N}$ varies. Here $F$ refers to the Frobeni…
