Some closure operations in Zariski-Riemann spaces of valuation domains: a survey

In this survey we present several results concerning various topologies that were introduced in recent years on spaces of valuation domains

On quasi-Pr\"{u}fer and UMtt domains

In this note we show that an integral domain $D$ of finite $w$-dimension is a quasi-Pr\"{u}fer domain if and only if each overring of $D$ is a $w$-Jaffard domain. Similar characterizations of quasi-Pr\"{u}fer domains are given by replacing $w$-Jaffard domain by $w$-stably strong S-domain, and $w$-strong S-domain. We also give new characterizations of UM$t$ domains.Comment: 6 Page

An historical overview of Kronecker function rings, Nagata rings, and related star and semistar operations

An historical overview of Kronecker function rings, Nagata rings, and related star and semistar operationsComment: "Multiplicative Ideal Theory in Commutative Algebra: A tribute to the work of Robert Gilmer", Jim Brewer, Sarah Glaz, William Heinzer, and Bruce Olberding Editors, Springer (to appear

Equivariant class group. I. Finite generation of the Picard and the class groups of an invariant subring

The purpose of this paper is to define equivariant class group of a locally Krull scheme (that is, a scheme which is locally a prime spectrum of a Krull domain) with an action of a flat group scheme, study its basic properties, and apply it to prove the finite generation of the class group of an invariant subring. In particular, we prove the following. Let $k$ be a field, $G$ a smooth $k$-group scheme of finite type, and $X$ a quasi-compact quasi-separated locally Krull $G$-scheme. Assume that there is a $k$-scheme $Z$ of finite type and a dominating $k$-morphism $Z\rightarrow X$. Let $\varphi:X\rightarrow Y$ be a $G$-invariant morphism such that $\mathcal O_Y\rightarrow (\varphi_*\mathcal O_X)^G$ is an isomorphism. Then $Y$ is locally Krull. If, moreover, \Cl(X) is finitely generated, then \Cl(G,X) and \Cl(Y) are also finitely generated, where \Cl(G,X) is the equivariant class group. In fact, \Cl(Y) is a subquotient of \Cl(G,X). For actions of connected group schemes on affine schemes, there are similar results of Magid and Waterhouse, but our result also holds for disconnected $G$. The proof depends on a similar result on (equivariant) Picard groups.Comment: 39 page

Elementary geometric local-global principles for fields

We define and investigate a family of local-global principles for fields involving both orderings and p-valuations. This family contains the PAC, PRC and PpC fields and exhausts the class of pseudo classically closed fields. We show that the fields satisfying such a local-global principle form an elementary class, admit diophantine definitions of holomorphy domains, and their orderings satisfy the strong approximation property.Comment: final version published in Annals of Pure and Applied Logic, Volume 164, Issue 10, October 2013, Pages 989-100

Totally ordered sets and the prime spectra of rings

Let $T$ be a totally ordered set and let $D(T)$ denote the set of all cuts of $T$. We prove the existence of a discrete valuation domain $O_{v}$ such that $T$ is order isomorphic to two special subsets of Spec$(O_{v})$. We prove that if $A$ is a ring (not necessarily commutative) whose prime spectrum is totally ordered and satisfies (K2), then there exists a totally ordered set $U \subseteq \text{Spec}(A)$ such that the prime spectrum of $A$ is order isomorphic to $D(U)$. We also present equivalent conditions for a totally ordered set to be a Dedekind totally ordered set. At the end, we present an algebraic geometry point of viewComment: 13 page

Ultrafilter and Constructible topologies on spaces of valuation domains

Let $K$ be a field and let $A$ be a subring of $K$. We consider properties and applications of a compact, Hausdorff topology called the "ultrafilter topology" defined on the space Zar$(K|A)$ of all valuation domains having $K$ as quotient field and containing $A$. We show that the ultrafilter topology coincides with the constructible topology on the abstract Riemann-Zariski surface Zar$(K|A)$. We extend results regarding distinguished spectral topologies on spaces of valuation domains.Comment: Comm. Algebra (accepted for publication

Characterization of Completions of Noncatenary Local Domains and Noncatenary Local UFDs

We find necessary and sufficient conditions for a complete local ring to be the completion of a noncatenary local (Noetherian) domain, as well as necessary and sufficient conditions for it to be the completion of a noncatenary local (Noetherian) unique factorization domain. We use our first result to demonstrate a large class of quasi-excellent domains that are not excellent, as well as a large class of catenary domains that are not universally catenary. We use our second result to find a larger class of noncatenary local UFDs than was previously known, and we show that there is no bound on how noncatenary a UFD can be.Comment: 18 page

Cohomology of locally-closed semi-algebraic subsets

Let k be a non archimedean field. If X is a k-algebraic variety and U a locally closed semi-algebraic subset of X^{an} -- the Berkovich space associated to X -- we show that for l \neq char(\tilde{k}), the cohomology groups H^i_c (\bar{U}, Q_l) behave like H^i_c(\bar{X}, Q_l), where \bar{U} = U \otimes \hat{\bar{k}}. In particular, they are finite-dimensional vector spaces. This result has been used by E. Hrushovski and F. Loeser. Moreover, we prove analogous finiteness properties concerning rigid semi-analytic subsets of compact Berkovich spaces (resp. adic spaces associated to quasi-compact quasi-separated k-rigid spaces) when char(\tilde{k}) \neq 0 (resp in any characteristic).Comment: We obtain a more general result using a recent cohomological finiteness result for affinoid spaces proved by Vladimir Berkovic

Mathieu Subspaces of Associative Algebras

Motivated by the Mathieu conjecture [Ma], the image conjecture [Z3] and the well-known Jacobian conjecture [K] (see also [BCW] and [E]), the notion of Mathieu subspaces as a natural generalization of the notion of ideals has been introduced recently in [Z4] for associative algebras. In this paper, we first study algebraic elements in the radicals of Mathieu subspaces of associative algebras over fields and prove some properties and characterizations of Mathieu subspaces with algebraic radicals. We then give some characterizations or classifications for strongly simple algebras (the algebras with no non-trivial Mathieu subspaces) over arbitrary commutative rings, and for quasi-stable algebras (the algebras all of whose subspaces that do not contain the identity element of the algebra are Mathieu spaces) over arbitrary fields. Furthermore, co-dimension one Mathieu subspaces and the minimal non-trivial Mathieu subspaces of the matrix algebras over fields are also completely determined.Comment: A new case of Mathieu subspaces has been added; some mistakes and misprints have been corrected. Latex, 42 page