Related papers: Valuation rings as limits of complete intersection…

We show that an algebraic immediate valuation ring extension of characteristic $p>0$ is a filtered union of complete intersection algebras of finite type.

We show that a mixed characteristic valuation ring with a value group $\Gamma$, $\val$ its valuation and a residue field of characteristic $p>0$, is a filtered colimit of complete intersection $\bf Z$-algebras if $\Gamma/{\bf Z}\val(p)$ has…

As in Zariski's Uniformization Theorem we show that a valuation ring $V$ of characteristic $p>0$ of dimension one is a filtered direct limit of smooth ${\bf F}_p$-algebras under some conditions of transcendence degree. Under mild…

This is mainly a small exposition on extensions of valuation rings

We associate to any given finite set of valuations on the polynomial ring in two variables over an algebraically closed field a numerical invariant whose positivity characterizes the case when the intersection of their valuation rings has…

We give a necessary and sufficient condition for an extension of valuation rings containing $\bf Q$ to be a filtered direct limit of smooth algebras.

We characterize those valued fields for which the image of the valuation ring under every polynomial in several variables contains an element of maximal value, or zero.

We show that a pure transcendental, immediate extension of valuation rings $V\subset V'$ containing a field is a filtered union of smooth $V$-subalgebras of $V'$.

Let $K$ be a complete discrete valued field of characteristic $p$ with residue $k$ which is not necessarily perfect. We prove the Conjecture in \cite{cs} that a $p$-algebra over $K$ contains a totally ramified cyclic maximal subfield if it…

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…

We prove that the theory of a Henselian valued field of characteristic zero, with finite ramification, and whose value group is a $Z$-group, is model-complete in the language of rings if the theory of its residue field is model-complete in…

We extend the characterization of extremal valued fields given in \cite{[AKP]} to the missing case of valued fields of mixed characteristic with perfect residue field. This leads to a complete characterization of the tame valued fields that…

We consider complete intersection ideals in a polynomial ring over a field of characteristic zero that are stable under the action of the symmetric group permuting the variables. We determine the possible representation types for these…

We prove that an expansion of an algebraically closed field by $n$ arbitrary valuation rings is NTP${}_2$, and in fact has finite burden. It fails to be NIP, however, unless the valuation rings form a chain. Moreover, the incomplete theory…

Let $K$ be a characteristic zero algebraic function field with a valuation $\nu$. Let $L$ be a finite extension of $K$ and $\omega$ be an extension of $\nu$ to $L$. We establish that the valuation ring $V_{\omega}$ of $\omega$ is…

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…

We previously obtained a generalization and refinement of results about the ramification theory of Artin-Schreier extensions of discretely valued fields in characteristic $p$ with perfect residue fields to the case of fields with more…

Local fields, and fields complete with respect to a discrete valuation, are essential objects in commutative algebra, with applications to number theory and algebraic geometry. We formalize in Lean the basic theory of discretely valued…

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…

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…