Antoni Rangachev
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
The Japanese and universally Japanese properties for valuation rings and Prüfer domains
本文研究赋值环和Prüfer整环的Japanese与universally Japanese性质，证明二者等价，并推广Nagata经典结果至非Noetherian情形。
A well known theorem of Shuzo Izumi, strengthened by David Rees, asserts that all the divisorial valuations centered in an analytically irreducible local noetherian ring are linearly comparable to each other. In the present paper we generalize this theorem to the case of Abhyankar valuations with archimedian value semigroup. Indeed, we prove that in a certain sense linear equivalence of topologies characterizes Abhyankar valuations with archimedian semigroups, centered in analytically irreducible local noetherian rings. Then we show that some of the classical results on equivalence of topologi…
In this paper the question of which semigroups are realizable as the semigroup of values attained on a Noetherian local ring which is dominated by a valuation is considered. We give some striking examples, indicating that there may be no constraints on the semigroup beyond those known classically.
Uniformity in nonreduced rings via Noetherian operators
本文通过Noetherian微分算子证明Artin-Rees引理的微分版本，进而得到非约化环的若干均匀性结果。
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…
We develop new methods to study $\mathfrak{m}$-adic stability in an arbitrary Noetherian local ring. These techniques are used to prove results about the behavior of Hilbert-Samuel and Hilbert-Kunz multiplicities under fine $\mathfrak{m}$-adic perturbations.
Let A be a noetherian commutative ring, and let I be an ideal in A. We study questions of flatness and I-adic completeness for infinitely generated A-modules. This is done using the notions of decaying function and I-adically free A-module.
$L$-smooth factorization for Noetherian $F$-finite rings
本研究证明诺特$F$-有限环间的同态可分解为正则态射加满射，建立了有限型映射光滑-满射分解的类比，还发现此类环映射的正则性、形式光滑性等价于$L$-光滑性，是Gabber相关结论的相对版本。
A new criterion is given for a semigroup to be the semigroup of a valuation dominating an equicharacteristic local domain. The criterion is used to construct examples of well ordered subsemigroups of the positive rational numbers which are of ordinal type omega, but are not the value semigroup of a valuation on an equicharacteristic noetherian local domain. This shows that the necessary conditions on value semigroups given in Appendix 3 to Zariski and Samuel's book ``Commutative Algebra'' are not sufficient.
We consider the question of which semigroups can occur as the semigroup $S_R(ν)$ of positive values of a rank 1 valuation dominating a Noetherian local ring $R$. We give a number of bounds of polynomial type on the growth of $ϕ(n)=S_R(ν)\cap (0,n)$ for $n\in\NN$, starting with the upper bound of $P_R(n)$, where $P_R(n)$ is the Hilbert function of $R$. This bound is generalized to an extremely general bound for arbitrary rank valuations in the paper "Semigroups of valuations on local rings, II", by Cutkosky and Teissier, arXiv:0805.3788. This bound is already enough to give simple examples of r…
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 …
We examine local cohomology in the setting of valuation rings. The novelty of this investigation stems from the fact that valuation rings are usually non-Noetherian, whereas local cohomology has been extensively developed mostly in a Noetherian setting. We prove various vanishing results on local cohomology for valuation rings of finite Krull dimension. These vanishing results stem from a uniform bound on the global dimension of such rings. Our investigation reveals differences in the sheaf theoretic definition of local cohomology, and the algebraic definition in terms of a limit of certain Ex…
Valuation rings as limits of complete intersection rings
本文证明特征$p>0$的赋值环可表示为完全交环的滤过直接极限。
We extend and prove a conjecture of Benguş-Lasnier on the parametrization of valuations on a polynomial ring by certain spaces of diskoids.
A study of the relation between a noetherian local domain with a given valuation and its associated graded ring with respect to the valuation, which in some cases is an esentially toric variety, possibly of infinite embedding dimension, but of finite Krull dimension. After extension of the valuation to a suitable completion (whose existence is not established in the paper) the relation becomes much more precise, and suggests a possible way to establish local uniformization for an excellent equicharacteristic local domain with an algebraically closed residue field, by deforming a partial toric …
Structure of ind-pro completions of Noetherian rings
研究诺特环沿素理想旗标的 ind - pro 完备化结构，计算其克鲁尔维数并推导半局部性判别准则，还证明其继承基环如正规性等性质。
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…
Smooth homomorphisms admit noetherian reductions
本文证明任意光滑环同态均可由诺特环的光滑同态经基变换得到，并结合已有结果给出格罗滕迪克“光滑同态皆平坦”定理的简短初等证明。
Regular rings over valuation rings
本文探讨了估值环上本质有限生成局部环的正则性性质，研究了开放位、完美大Cohen-Macaulay代数和切线复形等经典与现代主题，并给出了诺特环上的柯代瓦消失定理的应用。
We give three proofs that valuation rings are derived splinters: a geometric proof using the absolute integral closure, a homological proof which reduces the problem to checking that valuation rings are splinters (which is done in the second author's PhD thesis and which we reprise here), and a proof by approximation which reduces the problem to Bhatt's proof of the derived direct summand conjecture. The approximation property also shows that smooth algebras over valuation rings are splinters.
In this paper, we say a ring $R$ is Nil$_{\ast}$-Noetherian provided that any nil ideal is finitely generated. First, we show that the Hilbert basis theorem holds for Nil$_{\ast}$-Noetherian rings, that is, $R$ is Nil$_{\ast}$-Noetherian if and only if $R[x]$ is Nil$_{\ast}$-Noetherian, if and only if $R[[x]]$ is Nil$_{\ast}$-Noetherian. Then we discuss some Nil$_{\ast}$-Noetherian properties on idealizations and bi-amalgamated algebras. Finally, we give the Cartan-Eilenberg-Bass Theorem for Nil$_{\ast}$-Noetherian rings in terms of Nil$_{\ast}$-injective modules and Nil$_{\ast}$-FP-injective …
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…
Two left noetherian rings $R$ and $S$ are said to be {\it singularly equivalent} if their singularity categories are equivalent as triangulated categories. The aim of this paper is to give a necessary condition for two commutative noetherian rings to be singularly equivalent. To do this, we develop the support theory for triangulated categories without tensor structure.
In this paper, we introduce a new notion, called the integral dimension, for noetherian rings. It can be regarded as the weak Briancon-Skoda numbers of rings. The point is that every noetherian local ring has finite integral dimension.
Let k be a field of characteristic zero, K an algebraic function field over k, and V a k-valuation ring of K. Zariski's theorem of local uniformization shows that there exist algebraic regular local rings R_i with quotient field K which are dominated by V, and such that the direct union of the R_i's is V. We investigate the ring T, which is the direct union of the completions of the R_i's. We give necessary and sufficient conditions for T to be a valuation ring. We then focus on the case in which the valuation has rank one.
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…
For any commutative ring $A$ we introduce a generalization of $S$-noetherian rings using a hereditary torsion theory $σ$ instead of a multiplicatively closed subset $S\subseteq{A}$. It is proved that if $A$ is a totally $σ$-noetherian ring, then $σ$ is of finite type, and that totally $σ$-noetherian is a local property.
In this paper we consider the question of when the associated graded ring along a valuation, ${\rm gr}_{ν^*}(S)$, is a finite ${\rm gr}_{ν^*}(R)$-module, where $S$ is a normal local ring which lies over a normal local ring $R$ and $ν^*$ is a valuation of the quotient field of $S$ which dominates $S$. We obtain a very general result in equicharacteristic zero in Theorem 1.5. We also obtain general results for unramified extensions of excellent local rings in Proposition 1.7.
Let $F$ be a field. We show that certain subrings contained between the polynomial ring $F[X] = F[X_1, ..., X_n]$ and the power series ring $F[X][[Y]] = F[X_1, ..., X_n][[Y]]$ have Weierstrass Factorization, which allows us to deduce both unique factorization and the Noetherian property. These intermediate subrings are obtained from elements of $F[X][[Y]]$ by bounding their total $X$-degree above by a positive real-valued monotonic up function $λ$ on their $Y$-degree. These rings arise naturally in studying $p$-adic analytic variation of zeta functions over finite fields. Future research into …
We give a necessary and sufficient condition for an extension of valuation rings containing $\bf Q$ to be a filtered direct limit of smooth algebras.
We discuss projective equivalence of ideals in Noetherian rings and the existence or failure of existence of projectively full ideals. We describe connections with the Rees valuations and Rees integers of an ideal, and consider the question of whether improvements can be made by passing to an integral extension ring.
We consider trace ideals in Noetherian rings and focus our attention to one-dimensional analytically irreducible local rings. For such rings we classify those Gorenstein rings which admit only a finite number of trace ideals.
Pure transcendental, immediate valuation ring extensions as limits of smooth algebras
本文证明包含域的纯超越、即时赋值环扩张可表示为光滑子代数的滤过并集。
There exist many characterizations of Noetherian Cohen-Macaulay rings in the literature. These characterizations do not remain equivalent if we drop the Noetherian assumption. The aim of this paper is to provide some comparisons between some of these characterizations in non Noetherian case. Toward solving a conjecture posed by Glaz, we give a generalization of the Hochster-Eagon result on Cohen-Macaulayness of invariant rings, in the context of non Noetherian rings.
Given a noetherian local domain $R$ and a valuation $ν$ of its field of fractions which is non negative on $R$, we derive some very general bounds on the growth of the number of distinct valuation ideals of $R$ corresponding to values lying in certain parts of the value group $Γ$ of $ν$. We show that this growth condition imposes restrictions on the semigroups $ν(R\setminus \{0\})$ for noetherian $R$ which are stronger that those resulting from the previous paper \cite{C2} of the first author. Given an ordered embedding $Γ\subset ({\mathbf R}^h)_{\hbox{\rm lex}}$, where $h$ is the rank of $ν$,…
Let $R$ be a ring and $S$ a multiplicative subset of $R$. Then $R$ is called a uniformly $S$-Noetherian ($u$-$S$-Noetherian for abbreviation) ring provided there exists an element $s\in S$ such that for any ideal $I$ of $R$, $sI \subseteq K$ for some finitely generated sub-ideal $K$ of $I$. We give the Eakin-Nagata-Formanek Theorem for $u$-$S$-Noetherian rings. Besides, the $u$-$S$-Noetherian properties on several ring constructions are given. The notion of $u$-$S$-injective modules is also introduced and studied. Finally, we obtain the Cartan-Eilenberg-Bass Theorem for uniformly $S$-Noetheria…
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.
Frobenius splitting of valuation rings revisited
针对特征p>0的F有限域上函数域的赋值环，证明其Frobenius分裂等价于对应赋值为Abhyankar赋值，还构造反例说明基域非F有限时该等价性不成立，方法不依赖局部一致化结果。
In the present paper, we define a notion of numerical equivalence on Chow groups or Grothendieck groups of Noetherian local rings, which is an analogue of that on smooth projective varieties. Under a mild condition, it is proved that the Chow group modulo numerical equivalence is a finite dimensional ${\Bbb Q}$-vector space, as in the case of smooth projective varieties. Numerical equivalence on local rings is deeply related to that on smooth projective varieties. For example, if Grothendieck's standard conjectures are true, then a vanishing of Chow group (of local rings) modulo numerical equi…
Let $K$ be a characteristic zero algebraic function field with a valuation $ν$. Let $L$ be a finite extension of $K$ and $ω$ be an extension of $ν$ to $L$. We establish that the valuation ring $V_ω$ of $ω$ is essentially finitely generated over the valuation ring $V_ν$ of $ν$ if and only if the initial index $ε(ω|ν)$ is equal to the ramification index $e(ω|ν)$ of the extension. This gives a positive answer, for characteristic zero algebraic function fields, to a question posed by Hagen Knaf.
In this paper we investigate a property for commutative rings with identity which is possessed by every coherent regular ring and is equivalent to Cohen-Macaulay for Noetherian rings. We study the behavior of this property in the context of ring extensions (of various types) and rings of invariants.
On S-J-Noetherian Rings
本文提出S-J-Noetherian环的概念，推广了J-Noetherian环和S-Noetherian环，并证明了Cohen型定理及S-素分解的存在性。
In a previous paper, the third author proved that finite-degree polynomial functors over infinite fields are topologically Noetherian. In this paper, we prove that the same holds for polynomial functors from free $R$-modules to finitely generated $R$-modules, for any commutative ring $R$ whose spectrum is Noetherian. As Erman-Sam-Snowden pointed out, when applying this with $R = \mathbb{Z}$ to direct sums of symmetric powers, one of their proofs of a conjecture by Stillman becomes characteristic-independent. Our paper advertises and further develops the beautiful but not so well-known machiner…
Let R be a commutative ring with identity. We investigate some ring-theoretic properties of weakly Laskerian R-modules. Our results indicate that weakly Laskerian rings behave as Noetherian ones in many respects. However, we provide some examples to illustrate the strange behavior of these rings in some other respects.
We investigate flat maps where the source or target is a Noetherian ring, giving necessary and/or sufficient conditions on a ring for such maps to exist. Along the way, we develop some general facts about flat ring maps, and exhibit many examples, including a new class of zero-dimensional local rings.
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…
We will study the Euler class group of a commutative Noetherian ring $A$ of dimension $d$, developed by S.M. Bhatwadekar and Raja Sridharan. This is my M.Phil thesis from 2001.
We will give another definition of Euler class group of a Noetherian ring.
In this note, we give the Cohen type theorem for uniformly $S$-Noetherian modules and the Eakin-Nagata type theorem for uniformly $S$-Noetherian rings. We also solve an open question proposed by Kim and Lim [5,Question 4.10].
We count the number of submodules of an arbitrary module over a countable noetherian commutative ring. We give, along the way, a structural description of meager modules, which are defined as those that do not have the square of a simple module as subquotient. We deduce in particular a characterization of uniserial modules over commutative noetherian rings.
Cohen, Levitzki, Hilbert Basis, and Lasker-Noether Theorems for Nil-S-Noetherian Rings
引入Nil-S-Noetherian环，推广S-Noetherian环和Nil*-Noetherian环，建立Cohen定理、Levitzki定理、Hilbert基定理和Lasker-Noether定理的广义版本。
On Z-Noetherian Rings
本文引入$Z$-诺特环的概念，研究其内在刻画、稳定性及相关定理，证明诺特环的部分基本性质可推广至该环类。
In this paper we study right $S$-Noetherian rings and modules, extending of notions introduced by Anderson and Dumitrescu in commutative algebra to noncommutative rings. Two characterizations of right $S$-Noetherian rings are given in terms of completely prime right ideals and point annihilator sets. We also prove an existence result for completely prime point annihilators of certain $S$-Noetherian modules with the following consequence in commutative algebra: If a module $M$ over a commutative ring is $S$-Noetherian with respect to a multiplicative set $S$ that contains no zero-divisors for $…
We show that the cosupport of a commutative noetherian ring is precisely the set of primes appearing in a minimal pure-injective resolution of the ring. As an application of this, we prove that every countable commutative noetherian ring has full cosupport. We also settle the comparison of cosupport and support of finitely generated modules over any commutative noetherian ring of finite Krull dimension. Finally, we give an example showing that the cosupport of a finitely generated module need not be a closed subset of $\operatorname{Spec}(R)$, providing a negative answer to a question of Sathe…
The purpose of this article is to prove that Gersten's conjecture for a commutative discrete valuation ring is true. Combining with the result of \cite{GL87}, we learn that Gersten's conjecture is true if the ring is a commutative regular local, smooth over a commutative discrete valuation ring.
Let $R$ be a Noetherian ring. We prove that $R$ has global dimension at most two if, and only if, every prime ideal of $R$ is of linear type. Similarly, we show that $R$ has global dimension at most three if, and only if, every prime ideal of $R$ is syzygetic. As a consequence, one derives a characterization of these rings using the André-Quillen homology.
Almost Noetherian rings and modules
本文研究了近诺特环和模的性质，提出了多个重要定理，并在特定条件下解决了相关问题。
Let $D$ be a 2-dimensional regular local ring and let $Q(D)$ denote the quadratic tree of 2-dimensional regular local overrings of $D$. We examine the Noetherian rings that are intersections of rings in $Q(D)$. To do so, we describe the desingularization of projective models over $D$ both algebraically in terms of the saturation of complete ideals and order-theoretically in terms of the quadratic tree $Q(D)$.
Let $(A,\mathfrak{m}, k=A/\mathfrak{m})$ be a noetherian local ring. Then it is equivalent $n = \dim A = \dim_k \mathfrak{m}/\mathfrak{m}^2$ and $\mathrm{Tor}^A_i(k,k) = 0$ for all $i \gg 0$. The article gives a proof with the change-of-ring spectral sequence in the derived category form.
We consider a circle of ideas involving differential algebra, local Noetherian rings, and their generic formal fibers. Connecting these ideas gives rise to what we term a "twisted" subring $R$ of a ring $S$. Each such subring $R$ arises as a pullback of a derivation taking values in an $S$-module $K$. The twisting relationship proves to be a kind of inversion of Nagata idealization: whereas idealization extends $S$ to the larger ring $S \star K$, twisting produces a subring of $S$ which behaves much like the ring $S \star K$. The rings produced in this manner exhibit pathological features, suc…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。