Extension of valuation rings as limits of complete intersection algebras
We give sufficient conditions for an extension of valuation rings to be a filtered colimit of complete intersection algebras.
math.AC↗
arXiv subjects
Dorin Popescu
Publications and source records attributed to Dorin Popescu.
Valuation rings as limits of complete intersection rings
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.
math.AC↗
A special form of Zariski's Uniformization Theorem in positive characteristic
This is mainly a small exposition on extensions of valuation rings as a filtered union of smooth algebras.
math.AC↗
Pure transcendental, immediate valuation ring extensions as limits of smooth algebras
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 $V'$.
math.AC↗
Filtered colimits of complete intersection algebras
This is mainly a small exposition on extensions of valuation rings
math.AC↗
Algebraic valuation ring extensions as limits of complete intersection algebras
We show that an algebraic immediate valuation ring extension of characteristic $p>0$ is a filtered union of complete intersection algebras of finite type.
math.AC↗
Extensions of valuation rings containing $\bf Q$ as limits of smooth algebras
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.
math.AC↗
Valuation rings of dimension one as limits of smooth algebras
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 conditions, the algebraic immediate extensions of valuation rings are dense if they are filtered direct limit of smooth morphisms.
math.AC↗
Valuation rings of mixed characteristic as limits of complete intersection rings
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 no $p$-torsion and $V$ is Henselian.
math.AC↗
Immediate extensions of valuation rings and ultrapowers
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.
math.AC↗
Néron desingularization of extensions of valuation rings with an Appendix by Kęstutis Česnavičius
Zariski's local uniformization, a weak form of resolution of singularities, implies that every valuation ring containing $\bf Q$ is a filtered direct limit of smooth $\bf Q$-algebras. Given an immediate extension of valuation rings $V\subset V'$ containing $\bf Q$ we show that $V'$ is a filtered direct limit of smooth $V$-algebras. This corrects a paper of us \cite{Po1} where we thought that we may reduce to the case when the value groups are finitely generated. For this correction we use an infinite tower of ultrapowers construction that rests on results from model theory. .
math.AC↗
On the Bass-Quillen Conjecture and Swan's question
We present a question which implies a complete positive answer for the Bass-Quillen Conjecture.
math.AC↗
On a question of Swan. With an Appendix by Kęstutis Česnavičius
We show that a regular local ring is a filtered inductive limit of regular local rings, essentially of finite type over $\bf Z$. As an application the cohomological purity conjecture is reduced to the complete case.
math.AC↗
Simple General Neron Desingularization in local ${\bf Q}$-algebras
In this form will appear in Communications in Algebra.
math.AC↗
Remarks on Artin approximation with constraints
We study various approximation results of solutions of equations $f(x,Y)=0$ where $f(x,Y)\in\mathbb K[[x]][Y]^r$ and $x$ and $Y$ are two sets of variables, and where some components of the solutions $y(x)\in\mathbb K[[x]]^m$ do not depend on all the variables $x_j$. These problems have been highlighted by M. Artin.
math.AC↗
Linear nested Artin approximation theorem for algebraic power series
We give a new and elementary proof of the nested Artin approximation Theorem for linear equations with algebraic power series coefficients. Moreover, for any Noetherian local subring of the ring of formal power series, we clarify the relationship between this theorem and the problem of the com-mutation of two operations for ideals: the operation of replacing an ideal by its completion and the operation of replacing an ideal by one of its elimination ideals.
math.AC↗
Constructive Néron Desingularization of algebras with big smooth locus
An algorithmic proof of the General Néron Desingularization theorem and its uniform version is given for morphisms with big smooth locus. This generalizes the results for the one-dimensional case.
math.AC↗
An uniform General Neron Desingularization in dimension one
We give an uniform General Neron Desingularization for one dimensional local rings with respect to morphisms which coincide modulo a high power of the maximal ideal. The result has interesting applications in the case of Cohen-Macaulay rings.
math.AC↗