Valuation rings as limits of complete intersection rings
Simion Stoilow Institute of Mathematics of the Romanian Academy（罗马尼亚科学院西奥多·斯托伊洛数学研究所）；University of Bucharest, Faculty of Mathematics and Computer Science（布加勒斯特大学数学与计算机科学系）
We show that a valuation ring containing its residue field of characteristic $p>0$ is a filtered direct limit of complete intersection ${\bf F}_p$-algebras.
Algebraic valuation ring extensions as limits of complete intersection algebras
Simion Stoilow Institute of Mathematics of the Romanian Academy（罗马尼亚科学院西蒙·斯托伊洛数学研究所）；University of Bucharest, Faculty of Mathematics and Computer Science（布加勒斯特大学数学与计算机科学系）
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.
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…
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.
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 $K$ 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 consider birational properties of ramification in excellent local rings. We extend earlier results of the author with Olivier Piltant to show that strong local monomialization is true along a valuation dominating a defectless extension of two dimensional excellent local rings. We also obtain general results on the structure of the extension of associated graded rings along a valuation, and show that the invariants alpha and beta of stable forms of two dimensional extensions in characteristic p of the author and Olivier Piltant are not eventually constant.
Let $(K,v)$ be a valued field. We review some results of MacLane and Vaquié on extensions of $v$ to valuations on the polynomial ring $K[x]$. We introduce certain MacLane-Vaquié chains of residually transcendental valuations, and we prove that every valuation $μ$ on $K[x]$ is a limit of a finite or countably infinite MacLane-Vaquié chain. This chain underlying $μ$ is essentially unique and contains arithmetic data yielding an explicit description of the graded algebra of $μ$ as an algebra over the graded algebra of $v$.
Suppose that R\rightarrow S is an extension of local domains and ν^* is a valuation dominating S. We consider the natural extension of associated graded rings along the valuation gr_{ν^*}(R)\rightarrow gr_{ν^*}(S). We give examples showing that in general, this extension does not share good properties of the extension $\rightarrow S, but after enough blow ups above the valuations, good properties of the extension R\rightarrow S are reflected in the extension of associated graded rings. Stable properties of this extension (after blowing up) are much better in characteristic zero than in positiv…
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.
Extension of valuation rings as limits of complete intersection algebras
Simion Stoilow Institute of Mathematics of the Romanian Academy（罗马尼亚科学院西奥多·斯托伊洛数学研究所）；University of Bucharest, Faculty of Mathematics and Computer Science（布加勒斯特大学数学与计算机科学系）
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 extend Greenberg's strong approximation theorem to schemes of finite presentation over valuation rings with arbitrary value group, using the ultraproduct method of Becker, Denef, Lipshitz and van den Dries. As an application, we prove a closed image theorem (in the strong topology on rational points) for proper morphisms of varieties over valued fields.
Valuation rings of mixed characteristic as limits of complete intersection rings
Simion Stoilow Institute of Mathematics of the Romanian Academy（罗马尼亚科学院西蒙·斯托伊洛数学研究所）；University of Bucharest, Faculty of Mathematics and Computer Science（布加勒斯特大学数学与计算机科学系）
In this paper we provide necessary and sufficient conditions for $ R=A\propto E $ to be a valuation ring where $E$ is a non-torsion or finitely generated $A-$module. Also, we investigate the $ (n,d) $ property of the valuation ring.
An intersection of sets $A = \bigcap_{i \in I}B_i$ is irredundant if no $B_i$ can be omitted from this intersection. We develop a topological approach to irredundance by introducing a notion of a spectral representation, a spectral space whose members are sets that intersect to a given set $A$ and whose topology encodes set membership. We define a notion of a minimal representation and show that for such representations, irredundance is a topological property. We apply this approach to intersections of valuation rings and ideals. In the former case we focus on Krull-like domains and Prüfer $v$…
Suppose that $k$ is an arbitrary field. Consider the field $k((x_1,...,x_n))$, which is the quotient field of the ring $k[[x_1,...,x_n]]$ of formal power series in the variables $x_1,...,x_n$, with coefficients in $k$. Suppose that $σ$ is a formal power series in $x_1,...,x_n$ with coefficints in the algebraic closure of $k$. We give a very simple necessary and sufficient condition for $σ$ to be algebraic over $k((x_1,...,x_n))$. As an application of our methods, we give a characterization of valuation rings $V$ which dominate an excellent, Noetherian local domain $R$ of dimension two, and suc…
In this paper, we study the structure of the graded ring associated to a limit key polynomial $Q_n$ in terms of the key polynomials that define $Q_n$. In order to do that, we use direct limits. In general, we describe the direct limit of a family of graded rings associated to a totally ordered set of valuations. As an example, we describe the graded ring associated to a valuation-algebraic valuation as a direct limit of graded rings associated to residue-transcendental valuations.
We show that if $R$ is a local domain which is dominated by a valuation $ν$, then there does not always exist a regular local ring $R'$ which birationally dominates R and is dominated by $ν$ and an extension of $ν$ to the Henselization $(R')^h$ of $R'$ such that the associated graded rings of $R'$ and $(R')^h$ along the valuations are equal. We also show that there does not always exist $R'$, a prime ideal $p$ of the completion $\widehat{R'}$ such that $p\cap R'=(0)$ and an extension of $ν$ to $\widehat{R'}$ such that the associated graded rings of $R'$ and $\widehat{R'}/p$ along the valuation…
Étant donné un anneau de valuation $V$, de corps résiduel $F$ et de groupe des valeurs $Γ$, on donne une condition suffisante pour qu'un anneau local dominant $V$ soit un anneau de valuation de groupe $Γ$. Lorsque $V$ contient un corps $k$, ce résultat est appliqué à la construction d'un anneau de valuation contenant $V$ et une extension donnée $k'$ de $k$, de groupe $Γ$ et de corps résiduel engendré par $k'$ et $F$. Cela s'avère possible, notamment, lorsque $k'$ ou $F$ est séparable sur $k$. Given a valuation ring $V$, with residue field $F$ and value group $Γ$, we give a sufficient condition…
In this paper we prove that the Hermite ring conjecture holds for valuation rings $V$, and the special liner group $SL_n(V[x])$ coincides with the group generated by elementary matrices $E_n(V[x])$ for $n\geq3$. For any arithmetical ring $R$ and $n\geq3$, we show $SL_n(R[x]) = SL_n(R)\cdot E_n(R[x])$.
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…
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 …
In his work of 1969, Merle E. Manis introduced valuations on commutative rings. Recently, the class of totally quasi-ordered rings was developped by the second author. In the present paper, we establish the notion of compatibility between valuations and quasi-orders on rings, leading to a definition of the rank of a quasi-ordered ring. Moreover, we prove a Baer-Krull Theorem for quasi-ordered rings: fixing a Manis valuation v on R, we characterize all v-compatible quasi-orders of R by lifting the quasi-orders from the residue class ring to R itself.
In this article, we give a full description of the topology of the one dimensional affine analytic space $\mathbb{A}_R^1$ over a complete valuation ring $R$ (i.e. a valuation ring with "real valued valuation" which is complete under the induced metric), when its field of fractions $K$ is algebraically closed. In particular, we show that $\mathbb{A}_R^1$ is both connected and locally path connected. Furthermore, $\mathbb{A}_R^1$ is the completion of $K\times (1,\infty)$ under a canonical uniform structure. As an application, we describe the Berkovich spectrum $\mathfrak{M}(\mathbb{Z}_p[G])$ of …
In this paper we discuss stable forms of extensions of algebraic local rings along a valuation in all dimensions over a field k of characteristic zero, and generalize a formula of Ghezzi, Hà and Kashcheyeva describing the extension of associated graded rings along the valuation for stable extensions of regular algebraic local rings of dimension two to arbitrary ground fields k of characteristic zero. We discuss the failure of this result in positive characteristic.
Let $F$ be a field, and let Zar$(F)$ be the space of valuation rings of $F$ with respect to the Zariski topology. We prove that if $X$ is a quasicompact set of rank one valuation rings in Zar$(F)$ whose maximal ideals do not intersect to $0$, then the intersection of the rings in $X$ is an integral domain with quotient field $F$ such that every finitely generated ideal is a principal ideal. To prove this result, we develop a duality between (a) quasicompact sets of rank one valuation rings whose maximal ideals do not intersect to $0$, and (b) one-dimensional Prüfer domains with nonzero Jacobso…
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.
We give a characterization, in terms of the residue field, of those henselian valuation rings and those henselian valuation ideals that are diophantine. This characterization gives a common generalization of all the positive and negative results on diophantine henselian valuation rings and diophantine valuation ideals in the literature. We also treat questions of uniformity and we apply the results to show that a given field can carry at most one diophantine nontrivial equicharacteristic henselian valuation ring or valuation ideal.
This is a continuation of a previous paper by the same authors. In the former paper, it was proved that in order to obtain local uniformization for valuations centered on local domains, it is enough to prove it for rank one valuations. In this paper, we extend this result to the case of valuations centered on rings which are not necessarily integral domains and may even contain nilpotents.
The aim of this paper is to prove that every non-empty set of valuations centered at a two-dimensional regular domain has an infimum. We also generalize some results related to a non-metric tree.
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 …
Let K be a field with a valuation $ν$ and let L = K(x) be a transcendental extension of K, then any valuation $μ$ of L which extends $ν$ is determined by its restriction to the polynomial ring K[x]. We know how to associate to this valuation $μ$ a family of valuations A = ($μ$i)i$\in$I of K[x], called the associated admise family, which converges in a certain sense towards the valuation $μ$. Although the definition of this family, as well as the notion of convergence, essentially imply the structure of the polynomial ring, in particular the degree of polynomials, we show in this note that the …
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…
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…
The main goal of this paper is to study some properties of an extension of valuations from classical invariants. More specifically, we consider a valued field $(K,ν)$ and an extension $ω$ of $ν$ to a finite extension $L$ of $K$. Then we study when the valuation ring of $ω$ is essentially finitely generated over the valuation ring of $ν$. We present a necessary condition in terms of classic invariants of the extension by Hagen Knaf and show that in some particular cases, this condition is also sufficient. We also study when the corresponding extension of graded algebras is finitely generated. F…
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 show how several results about p-adic lattices generalize easily to lattices over valuation ring of arbitrary rank having only the Henselian property for quadratic polynomial. If 2 is invertible we obtain the uniqueness of the Jordan decomposition and the Witt Cancelation Theorem. We show that the isomorphism classes of indecomposable rank 2 lattices over such a ring in which 2 is not invertible are characterized by two invariants, provided that the lattices contain a primitive norm divisible by 2 of maximal valuation.
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…
Given a well-ordered semi-group $Γ$ with a minimal system of generators of ordinal type at most $ωn$ and of rational rank $r$, which satisfies a positivity and increasing condition, we construct a zero-dimensional valuation centered on the ring of polynomials with $r$ variables such that the semi-group of the values of the polynomial ring is equal to $Γ$. The construction uses a generalization of Favre and Jonsson's version of MacLane's sequence of key-polynomials.
Using the notion of cyclically pure injective modules, a characterization for rings which are locally valuation is established. As applications, new characterizations for Prufer domains and pure semi-simple rings are provided. Namely, we show that a domain R is Prufer if and only if two of the three classes of pure injective, cyclically pure injective and RD-injective modules are equal. Also, we prove that a commutative ring R is pure semi-simple if and only if every R-module is cyclically pure injective.
We discuss definability of henselian valuation rings in the Macintyre language $\mathcal{L}_{\rm Mac}$, the language of rings expanded by n-th power predicates. In particular, we show that henselian valuation rings with finite or Hilbertian residue field are uniformly $\exists$-$\emptyset$-definable in $\mathcal{L}_{\rm Mac}$, and henselian valuation rings with value group $\mathbb{Z}$ are uniformly $\exists\forall$-$\emptyset$-definable in the ring language, but not uniformly $\exists$-$\emptyset$-definable in $\mathcal{L}_{\rm Mac}$. We apply these results to local fields $\mathbb{Q}_p$ and …
Let $F$ be a field, let $D$ be a subring of $F$, and let ${\mathfrak{X}}$ be the Zariski-Riemann space of valuation rings containing $D$ and having quotient field $F$. We consider the Zariski, inverse and patch topologies on ${\mathfrak{X}}$ when viewed as a projective limit of projective integral schemes having function field contained in $F$, and we characterize the locally ringed subspaces of ${\mathfrak{X}}$ that are affine schemes.
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.
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.
We prove that any complete regular local ring with a valuation of rank 1 can be embedded, as a valued ring, in a ring of generalized Puiseux expansions.
