A criterion for rings which are locally valuation

紧离散赋值环上模的庞特里亚金对偶

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。

On the geometry of Prüfer intersections of valuation rings

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…

Non-compact subsets of the Zariski space of an integral domain

Let $V$ be a minimal valuation overring of an integral domain $D$ and let $\mathrm{Zar}(D)$ be the Zariski space of the valuation overrings of $D$. Starting from a result in the theory of semistar operations, we prove a criterion under which the set $\mathrm{Zar}(D)\setminus\{V\}$ is not compact. We then use it to prove that, in many cases, $\mathrm{Zar}(D)$ is not a Noetherian space, and apply it to the study of the spaces of Kronecker function rings and of Noetherian overrings.

On the topology of valuation-theoretic representations of integrally closed domains

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}(…

The topology on Berkovich affine lines over complete valuation rings

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 …

On a Theorem of Dedekind

Let $(K,ν)$ be an arbitrary valued field with valuation ring $R_ν$ and $L=K(α)$, where $α$ is a root of a monic irreducible polynomial $f\in R_ν[x]$. In this paper, we characterize the integral closedness of $R_ν[α]$ in such a way that extend Dedekind's criterion. Without the assumption of separability of the extension $L/K$, we show that Dedekind's theorem and its converse hold.

非诺特环中的上同调Cohen--Macaulay性

Cohomological Cohen--Macaulayness in Non-Noetherian Rings

本文研究非诺特环的上同调Cohen--Macaulay性，证明局部有限维CCM环必为局部HMCM环，给出满足条件的环类实例，并建立特征p>0诺特环与其完备化的相关等价关系。

Erratic birational behavior of mappings in positive characteristic

Birational properites of generically finite morphisms $X\rightarrow Y$ of algebraic varieties can be understood locally by a valuation of the function field of $X$. In finite extensions of algebraic local rings in characteristic zero algebraic function fields which are dominated by a valuation there are nice monomial forms of the mapping after blowing up enough, which reflect classical invariants of the valuation. Further, these forms are stable upon suitable further blowing up. In positive characteristic algebraic function fields it is not always possible to find a monomial form after blowing…