Let $F$ be a field. For each nonempty subset $X$ of the Zariski-Riemann space of valuation rings of $F$, let ${A}(X) = \bigcap_{V \in X}V$ and ${J}(X) = \bigcap_{V \in X}{\mathfrak M}_V$, where ${\mathfrak M}_V$ denotes the maximal ideal of $V$. We examine connections between topological features of $X$ and the algebraic structure of the ring ${A}(X)$. We show that if ${J}(X) \ne 0$ and $A(X)$ is a completely integrally closed local ring that is not a valuation ring of $F$, then there is a subspace $Y$ of the space of valuation rings of $F$ that is perfect in the patch topology such that ${A}(X) = {A}(Y)$. If any countable subset of points is removed from $Y$, then the resulting set remains a representation of $A(X)$. Additionally, if $F$ is a countable field, the set $Y$ can be chosen homeomorphic to the Cantor set. We apply these results to study properties of the ring ${A}(X)$ with specific focus on topological conditions that guarantee ${A}(X)$ is a Prüfer domain, a feature that is reflected in the Zariski-Riemann space when viewed as a locally ringed space. We also classify the rings ${A}(X)$ where $X$ has finitely many patch limit points, thus giving a topological generalization of the class of Krull domains, one that also includes interesting Prüfer domains. To illustrate the latter, we show how an intersection of valuation rings arising naturally in the study of local quadratic transformations of a regular local ring can be described using these techniques.
Let $D$ be an integrally closed local Noetherian domain of Krull dimension 2, and let $f$ be a nonzero element of $D$ such that $fD$ has prime radical. We consider when an integrally closed ring $H$ between $D$ and $D_f$ is determined locally by finitely many valuation overrings of $D$. We show such a local determination is equivalent to a statement about the exceptional prime divisors of normalized blow-ups of $D$, and, when $D$ is analytically normal, this property holds for $D$ if and only if it holds for the completion of $D$. This latter fact, along with MacLane's notion of key polynomial…
We consider infinite sequences {R_n} of successive local quadratic transforms of a regular local ring. Let S denote the directed union of the sequence of regular local rings R_n. We previously showed the existence of a unique limit point V of the family of order valuation rings of the sequence. In this paper, we examine asymptotic properties of this family of order valuations. We link this asymptotic behavior to ring-theoretic properties of S, namely whether S is archimedean and whether S is completely integrally closed. We construct examples of such S that are archimedean and completely integ…
We consider properties and applications of a compact, Hausdorff topology called the "ultrafilter topology" defined on an {\sl arbitrary spectral space} and we observe that this topology coincides with the constructible topology. If $K$ is a field and $A$ a subring of $K$, we show that the space Zar$(K|A)$ of all valuation domains, having $K$ as quotient field and containing $A$, (endowed with the Zariski topology) is a spectral space by giving in this general setting the explicit construction of a ring whose Zariski spectrum is homeomorphic to Zar$(K|A)$. We extend results regarding spectral t…
Suppose $F$ is a field with valuation $v$ and valuation ring $O_{v}$, $E$ is a finite field extension and $w$ is a quasi-valuation on $E$ extending $v$. We study quasi-valuations on $E$ that extend $v$; in particular, their corresponding rings and their prime spectrums. We prove that these ring extensions satisfy INC (incomparability), LO (lying over), and GD (going down) over $O_{v}$; in particular, they have the same Krull Dimension. We also prove that every such quasi-valuation is dominated by some valuation extending $v$. Under the assumption that the value monoid of the quasi-valuation is…
We consider the lattice-ordered groups Inv$(R)$ and Div$(R)$ of invertible and divisorial fractional ideals of a completely integrally closed Prüfer domain. We prove that Div$(R)$ is the completion of the group Inv$(R)$, and we show there is a faithfully flat extension $S$ of $R$ such that $S$ is a completely integrally closed Bézout domain with Div$(R) \cong $ Inv$(S)$. Among the class of completely integrally closed Prüfer domains, we focus on the one-dimensional Prüfer domains. This class includes Dedekind domains, the latter being the one-dimensional Prüfer domains whose maximal ideals are…
Let $R$ be a regular local ring of dimension at least 2. Associated to each valuation domain birationally dominating $R$, there exists a unique sequence $\{R_n\}$ of local quadratic transforms of $R$ along this valuation domain. We consider the situation where the sequence $\{ R_n \}_{n \ge 0}$ is infinite, and examine ideal-theoretic properties of the integrally closed local domain $S = \bigcup_{n \ge 0} R_n$. Among the set of valuation overrings of $R$, there exists a unique limit point $V$ for the sequence of order valuation rings of the $R_n$. We prove the existence of a unique minimal pro…
Let $F$ be a field, let $D$ be a subring of $F$ and let $Z$ be an irreducible subspace of the space of all valuation rings between $D$ and $F$ that have quotient field $F$. Then $Z$ is a locally ringed space whose ring of global sections is $A = \bigcap_{V \in Z}V$. All rings between $D$ and $F$ that are integrally closed in $F$ arise in such a way. Motivated by applications in areas such as multiplicative ideal theory and real algebraic geometry, a number of authors have formulated criteria for when $A$ is a Prüfer domain. We give geometric criteria for when $A$ is a Prüfer domain that reduce…
Lattice homology of integrally closed submodules and Artin algebras
Alfréd Rényi Institute of Mathematics（阿尔弗雷德·雷尼数学研究所）；ELTE - University of Budapest（布达佩斯罗兰大学）；HUN-REN Alfréd Rényi Institute of Mathematics（匈牙利研究与创新网络阿尔弗雷德·雷尼数学研究所）
This paper gives a survey on a valuation theoretical approach to local uniformization in positive characteristic, the model theory of valued fields in positive characteristic, and their connection with the valuation theoretical phenomenon of defect.
This paper is inspired by Michael Artin's paper "On The Join of Hensel Rings". In his paper, Artin proves that in an absolutely integrally closed ring the sum of two prime ideals is either prime or the whole ring. A more elementary proof of the previous statement is given in M. Hochster and C. Huneke's "Infinite Integral Extensions and Big Cohen-Macaulay Algebras" for the larger class of quadratically closed rings. Motivated by Hochster and Huneke's proof, we give an elementary proof of the fact that an absolutely integrally closed ring has no polygons. We prove this for the larger class of 2n…
We study two variants of the following question: "Given two finitely generated subalgebras R_1, R_2 of C[x_1, \ldots, x_n], is their intersection also finitely generated?" We show that the smallest value of $n$ for which there is a counterexample is 2 in the general case, and 3 in the case that R_1 and R_2 are integrally closed. We also explain the relation of this question to the problem of constructing algebraic compactifications of C^n and to the moment problem on semialgebraic subsets of R^n. The counterexample for the general case is a simple modification of a construction of Neena Gupta,…
We give a valuation theoretic characterization for a real closed field to be recursively saturated. Our result extends the characterization of Harnik and Ressayre \cite{hr} for a divisible ordered abelian group to be recursively saturated.
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…
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…
Let $D$ be an integral domain. A nonzero nonunit $a$ of $D$ is called a valuation element if there is a valuation overring $V$ of $D$ such that $aV\cap D=aD$. We say that $D$ is a valuation factorization domain (VFD) if each nonzero nonunit of $D$ can be written as a finite product of valuation elements. In this paper, we study some ring-theoretic properties of VFDs. Among other things, we show that (i) a VFD $D$ is Schreier, and hence ${\rm Cl}_t(D)=\{0\}$, (ii) if $D$ is a P$v$MD, then $D$ is a VFD if and only if $D$ is a weakly Matlis GCD-domain, if and only if $D[X]$, the polynomial ring o…
We give model theoretic criteria for $\exists \forall$ and $\forall \exists$- formulas in the ring language to define uniformly the valuation rings $\mathcal{O}$ of models $(K, \mathcal{O})$ of an elementary theory $Σ$ of henselian valued fields. As one of the applications we obtain the existence of an $\exists \forall$-formula defining uniformly the valuation rings $\mathcal{O}$ of valued henselian fields $(K, \mathcal{O})$ whose residue class field $k$ is finite, pseudo-finite, or hilbertian. We also obtain $\forall \exists$-formulas $φ_2$ and $φ_4$ such that $φ_2$ defines uniformly $k[[t]]$…
Let $D$ be a domain with fraction field $K$, and let $M_n(D)$ be the ring of $n \times n$ matrices with entries in $D$. The ring of integer-valued polynomials on the matrix ring $M_n(D)$, denoted ${\rm Int}_K(M_n(D))$, consists of those polynomials in $K[x]$ that map matrices in $M_n(D)$ back to $M_n(D)$ under evaluation. It has been known for some time that ${\rm Int}_{\mathbb{Q}}(M_n(\mathbb{Z}))$ is not integrally closed. However, it was only recently that an example of a polynomial in the integral closure of ${\rm Int}_{\mathbb{Q}}(M_n(\mathbb{Z}))$ but not in the ring itself appeared in t…
