Miguel Angel Olalla Acosta
Let $v$ be a rank-one discrete valuation of the field $k((\X))$. We know, after \cite{Bri2}, that if $n=2$ then the dimension of $v$ is 1 and if $v$ is the usual order function over $k((\X))$ its dimension is $n-1$. In this paper we prove that, in the general case, the dimension of a rank-one discrete valuation can be any number between 1 and $n-1$.
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We study algebraic discrete valuations dominating normal local domains of dimension two. We construct a family of examples to show that the Hilbert-Samuel function of the associated graded ring of the valuation can fail to be asymptotically of the form: quasi-polynomial plus a bounded function. We also show that the associated multiplicity can be irrational, or even transcendental.
Singular loci of algebras over ramified discrete valuation rings
本文针对混合特征分歧离散赋值环上的代数，推广了经典的雅可比准则，用以刻画其奇异轨迹。
In this paper we study the rank one discrete valuations of $k((X_1,... ,X_n))$ whose center in $k\lcor\X\rcor$ is the maximal ideal $(\X)$. In sections 2 to 6 we give a construction of a system of parametric equations describing such valuations. This amounts to finding a parameter and a field of coefficients. We devote section 2 to finding an element of value 1, that is, a parameter. The field of coefficients is the residue field of the valuation, and it is given in section 5. The constructions given in these sections are not effective in the general case, because we need either to use the Zor…
Let v be a rank m discrete valuation of k[[X1,...,Xn]] with dimension n-m. We prove that there exists an inmediate extension L of K where the valuation is monomial. Therefore we compute explicitly the residue field of the valuation.
Let $D$ be an integral domain. Then $D$ is an almost valuation (AV-)domain if for $a, b\in D\setminus \{0\}$ there exists a natural number $n$ with $a^{n}\mid b^{n}$ or $b^{n}\mid a^{n}$. AV-domains are closely related to valuation domains, for example, $D$ is an AV-domain if and only if the integral closure $\bar{D}$ is a valuation domain and $D\subseteq \bar{D}$ is a root extension. In this note we explore various generalizations of DVRs (which we might call almost DVRs) such as Noetherian AV-domains, AV-domains with $\bar{D}$ a DVR, and quasilocal and local API-domains (i.e., for $\{a_α\}_{…
Let V be a rank one discrete valuation ring (DVR) on a field F and let L/F be a finite separable algebraic field extension with [L:F] = m. The integral closure of V in L is a Dedekind domain that encodes the following invariants: (i) the number of extensions of V to a valuation ring W on L, (ii) the residue degree of each W over V, and (iii) the ramification degree of each W over V. Given a finite set of DVRs on F, an m-consistent system is a family of sets enumerating what is theoretically possible for the above invariants of each V in the set. The m-consistent system is realizable if there e…
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.
We present basic properties of Gröbner bases of submodules of a free module of finite rank over a polynomial ring $R$ with coefficients in a graded truncated discrete valuations ring $A$. As an application, we give a criterion for a finitely generated $R$-module to be flat over $A$. Its non-graded version is also given.
Pontryagin Duality for Modules over Compact Discrete Valuation Rings
本文为紧离散赋值环上的模建立了庞特里亚金对偶的类似物，定义了对偶模并证明了局部紧模的双重对偶是同构同胚，同时构造了典范拓扑使函子 fully faithful，从而将拓扑结果代数化。
This is an extended introduction to discrete valuation rings and Dedekind domains. Some natural generalizations of Dedekind domains are also (briefly) discussed including "almost Dedekind domains", Prüfer domains, Krull domains, and singular integral domains of dimension 1.
Regarding non-unique factorization of integer-valued polynomials over a discrete valuation domain $(R,M)$ with finite residue field, it is known that there exist absolutely irreducible elements, that is, irreducible elements all of whose powers factor uniquely, and non-absolutely irreducible elements. We completely and constructively characterize the absolutely irreducible elements among split integer-valued polynomials. They correspond bijectively to finite sets, which we call \emph{balanced}, characterized by a combinatorial property regarding the distribution of their elements among residue…
A submodule $W$ of a p-primary module $M$ of bounded order is known to be regular if $W$ and $M$ have simultaneous bases. In this paper we derive necessary and sufficient conditions for regularity of a submodule.
Detecting and Quantifying Isolated Singularities over Discrete Valuation Rings
本文在混合特征(0,p)下发展孤立超曲面奇点理论，研究离散赋值环上的商环，引入经典Tjurina和Milnor数的类比，证明确定性定理和Mather-Yau定理的推广，并定义数值不变量以区分非可分和可分情况下的孤立奇点。
The main scope of this paper is to introduce valuation semirings in general and discrete valuation semirings in particular. In order to do that, first we define valuation maps and investigate them. Then we define valuation semirings with the help of valuation maps and prove that a multiplicatively cancellative semiring is a valuation semiring if and only if its ideals are totally ordered by inclusion. We also prove that if the unique maximal ideal of a valuation semiring is subtractive, then it is integrally closed. We end this paper by introducing discrete valuation semirings and show that a …
In this paper we study the rank one discrete valuations of the field $k((X_1,..., X_n))$ whose center in $k\lcor\X\rcor$ is the maximal ideal. In sections 2 to 6 we give a construction of a system of parametric equations describing such valuations. This amounts to finding a parameter and a field of coefficients. We devote section 2 to finding an element of value 1, that is, a parameter. The field of coefficients is the residue field of the valuation, and it is given in section 5. The constructions given in these sections are not effective in the general case, because we need either to use the …
Characterizing absolutely irreducible integer-valued polynomials over discrete valuation domains
本文针对有限剩余域离散赋值域上的不可约整值多项式，利用与多项式关联的固定因子核刻画绝对不可约性，并给出将判定归结为特定幂唯一分解的界。
Let M and N be two r x r matrices over a discrete valuation ring of characteristic zero. The orders (with respect to a uniformizing parameter) of the invariant factors of M form a partition of non-negative integers, called the invariant partition of M. Let the invariant partition of M be mu, of N be nu, and of the product MN be lambda. In this paper we construct a Littlewood-Richardson filling of the skew shape lambda/mu with content nu, and show that this filling is an invariant of the orbit of the pair (M,N) with respect to a natural group action on the pair. We relate the algebraic combinat…
For a discrete valuation domain $V$ with maximal ideal $\mathfrak{m}$ such that the residue field $V/\mathfrak{m}$ is finite, there exists a sequence of polynomials $(F_n(x))_{n \ge 0}$ defined over the quotient field $K$ of $V$ that forms a basis of the $V$-module $\text{Int}(V) = \{f \in K[x] | f(V)\subseteq V\}$. This sequence of polynomials bears many resemblances to the classical binomial polynomials $(\binom{x}{n})_{n \ge 0}$. In this paper, we introduce a generating polynomial to account for the distribution of the $V$-values of the polynomials $F_n(x)$ modulo the maximal ideal $\mathfr…
The notion of key polynomials was first introduced in 1936 by S. Maclane in the case of discrete rank 1 valuations. . Let K -> L be a field extension and ν a valuation of K. The original motivation for introducing key polynomials was the problem of describing all the extensions μ of ν to L. Take a valuation μ of L extending the valuation ν. In the case when ν is discrete of rank 1 and L is a simple algebraic extension of K Maclane introduced the notions of key polynomials for μ and augmented valuations and proved that μ is obtained as a limit of a family of augmented valuations on the polynomi…
Integer-valued polynomials on discrete valuation rings of global fields with prescribed lengths of factorizations
该研究针对全局域赋值环上整值多项式的分解问题，证明对指定的分解长度序列，存在对应整值多项式，并将结果推广到满足条件的离散赋值环，解决了相关公开问题。
Poincare Series of Non-divisorial Valuations on Two-dimensional Function Fields
本文利用双图和估值生成序列计算二维函数域上非除子估值的庞加莱级数，证明其反映双图数据并提供估值的另一种分类方法。
Classically, Groebner bases are computed by first prescribing a set monomial order. Moss Sweedler suggested an alternative and developed a framework to perform such computations by using valuation rings in place of monomial orders. We build on these ideas by providing a class of valuations on rational function fields of two variables that are suitable for this framework. For these valuations, we explicitly compute the image of an underlying polynomial ring and use this to perform computations concerning ideals in the polynomial ring. Interestingly, for these valuations, some ideals have a fini…
Kähler differentials of extensions of valuation rings and deeply ramified fields
本文研究赋值域有限Galois扩张下赋值环的Kähler微分，明确构造了素次数Kummer与Artin-Schreier无亏缺扩张的微分模，推广了离散赋值的经典结果，并应用于深分歧域的刻画。
Let ${\mathcal O}$ be a discrete valuation ring with quotient field ${\cal K}$. The affine Grassmannian ${\cal G}r$ is the set of full-rank ${\mathcal O}$-modules contained in ${\cal K}^n$. Given $Λ\in {\cal G}r$, invariant factors $inv(Λ)=λ\in {\mathbb Z}^n$ stratify ${\cal G}r$. Left-multiplication by $GL_{n}({\cal K})$ stratifies ${\cal G}r \times {\cal G}r$ where $inv(N,Λ) = μ$ if $(N,Λ)$ and $(I_{n} ,M)$ are in the same $GL_{n}({\cal K})$ orbit, and $inv(M) = μ$. We present an elementary map from ${\cal G}r \times {\cal G}r$ to hives (in the sense of Knutson and Tao) of type $(μ,ν,λ)$ whe…
This paper deals with valuations of fields of formal meromorphic functions and their residue fields. We explicitly describe the residue fields of the monomial valuations. We also classify all the discrete rank one valuations of fields of power series in two and three variables, according to their residue fields. We prove that all our cases are possible and give explicit constructions.
We consider the question of when a semigroup is the semigroup of a valuation dominating a two dimensional noetherian local domain, giving some surprising examples. We give a necessary and sufficient condition for the pair of a semigroup S and a field extension L/k to be the semigroup and residue field of a valaution dominating a regular local ring R of dimension two with residue field k, generalizing the theorem of Spivakovsky for the case when there is no residue field extension.
Dual Graphs and Generating Sequences of Non-divisorial Valuations on Two-dimensional Function Fields
本文基于Spivakovsky的双图理论，证明了二维函数域上非除数估值的最小生成序列，且定义方式与原用法不同，通过双图数据明确生成序列值。
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.
On the Equality $\sum_{j} e_jf_j=[L:K]$ \vspace{2mm} On the Equality $\sum_{j} e_jf_j=[L:K]$ for a Finite Separable Extension $L$ of $K$
本文将离散赋值扩张的基本等式推广到半离散赋值情形，并由此给出赋值环整闭包为自由模的充分必要条件。
We describe the immediate extensions of a one dimensional valuation ring $V$ which could be embedded in some separation of a ultrapower of $V$ with respect to a certain ultrafilter. For such extensions holds a kind of Artin's approximation.
$\mathbf{A}_{\text {inf}}$ has uncountable Krull dimension
本文证明了在特定条件下，O_E-Witt环的克鲁尔维数至少为连续统的基数。
The $p$-adic Galois Cohomology of Valuation Fields
本文研究预完美oid域的 $p$-进赋值域扩张，利用 Gabber-Ramero 的余切复形计算其 Galois 上同调，并推广且重新证明 Tate–Hyodo 经典定理。
The Balmer spectrum of pseudo-coherent complexes over a discrete valuation ring
基于Matsui-Takahashi前期工作，研究诺特交换环上伪凝聚复形导出范畴，证明离散赋值环情形下其Balmer谱与自然数单调序列渐近等价类的有界分配格谱空间一致，揭示其较完美复形谱更复杂及刚性与非刚性张量三角范畴谱的扩展关系。
Valuation rings of dimension one as limits of smooth algebras
本文证明在特定超越度条件下，特征p>0的一维赋值环可表示为光滑F_p-代数的滤过直接极限，并探讨了代数立即扩张的稠密性。
Let $(f\_1,\dots, f\_s) \in \mathbb{Q}\_p [X\_1,\dots, X\_n]^s$ be a sequence of homogeneous polynomials with $p$-adic coefficients. Such system may happen, for example, in arithmetic geometry. Yet, since $\mathbb{Q}\_p$ is not an effective field, classical algorithm does not apply.We provide a definition for an approximate Gr{ö}bner basis with respect to a monomial order $w.$ We design a strategy to compute such a basis, when precision is enough and under the assumption that the input sequence is regular and the ideals $\langle f\_1,\dots,f\_i \rangle$ are weakly-$w$-ideals. The conjecture of…
Flat dimension for power series over valuation rings
该研究针对秩1真稠密值群赋值环上的幂级数环，通过探究其赋值理论构造出希尔伯特合冲定理的反例，同时给出该幂级数环非凝聚环的新证明。
Suppose that $R$ is a 2 dimensional excellent local domain with quotient field $K$, $K^*$ is a finite separable extension of $K$ and $S$ is a 2 dimensional local domain with quotient field $K^*$ such that $S$ dominates $R$. Suppose that $ν^*$ is a valuation of $K^*$ such that $ν^*$ dominates $S$. Let $ν$ be the restriction of $ν^*$ to $K$. The associated graded ring ${\rm gr}_ν(R)$ was introduced by Bernard Teissier. It plays an important role in local uniformization. We show that the extension $(K,ν)\rightarrow (K^*,ν^*)$ of valued fields is without defect if and only if there exist regular l…
Classifying smashing ideals in derived categories of valuation domains
本文基于Bazzoni-Šťovíček的结果，对有限维赋值域的导出范畴的击打理想框架进行了完整分类，并构造了一类使茶杯猜想不成立的交换环，扩展了Keller的例子。
Essential dimension of cohomology classes via valuation theory
本文利用赋值理论给出严格Hensel域上上同调类基本维数的显式公式，并应用于通过Rost不变量研究例外群的基本维数。
On a factorization result of Ştefănescu -- II
本文推广了 Ştefănescu 关于离散赋值环上多项式的因式分解结果，使其覆盖更大一类多项式，该结果有助于设计多项式因式分解算法。
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