Chains of differential subvarieties in an algebraic variety.
type: Item
instance of: scholarly article
title: Chains of differential subvarieties in an algebraic variety.
zbMATH Open document ID: 1079.12004
author: Wai Yan Pong
published in: Journal of Algebra
publication date: 2002-01-01T00:00:00Z
review text: A \(\delta\)-ring is a commutative ring \(A\) with \(1\) containing \(\mathbb{Q}\) together with a derivation \(\delta\) of \(A\). In the first part of his paper, the author shows that the lying over theorem, the going-up theorem, and the going-down theorem for \(\delta\)-integral domains still hold, if only \(\delta\)-invariant prime ideals are considered instead of arbitrary prime ideals. These results are applied to the theory of increasing chains of \(\delta\)-subvarieties of a \(\delta\)-variety \(X\) over some differentially closed field \(F\) of characteristic \(0\). The author calls such a chain a long chain, if it has at least \(d\) disjoint subchains of infinitely many subvarieties, where \(d\) is the \(\delta\)-transcendence degree of \(F\langle X\rangle\) over \(F\). He proves the so-called Kolchin catenary problem: Given any \(x\in X\), there is a long chain in \(X\) starting at \(x\).
Mathematics Subject Classification ID: 12H05
Mathematics Subject Classification ID: 13N15
Mathematics Subject Classification ID: 13B24
Mathematics Subject Classification ID: 14F10
Mathematics Subject Classification ID: 16S32
zbMATH DE Number: 1749650
zbMATH Keywords: Kolchin catenary problem
zbMATH Keywords: \(\delta\)-ring
zbMATH Keywords: \(\delta\)-variety
zbMATH Keywords: derivation
zbMATH Keywords: going-up theorem
zbMATH Keywords: going-down theorem
reviewed by: Hartmut Wiebe
MaRDI profile type: Publication
full work available at URL: jabr.2001.9065
OpenAlex ID: W2015301826
cites work: Q5691061
cites work: A notion of Krull dimension for differential rings
cites work: Q5680222
cites work: Specializations in Differential Algebra
DOI: 10.1006/JABR.2001.9065
description: scientific article; zbMATH DE number 1749650