Completeness in partial differential algebraic geometry
type: Item
instance of: scholarly article
title: Completeness in partial differential algebraic geometry
zbMATH Open document ID: 1308.14005
author: James Freitag
published in: Journal of Algebra
publication date: 2014-10-01T00:00:00Z
review text: This paper studies various aspects of complete differential algebraic varieties. The setting is the following, introduced by Kolchin. If \(F\) is a field of characteristic \(0\) equipped with \(m\) commuting derivations, then one has the notion of \textit{differential algebraic varieties} over \(F\). For simplicity, we fix a \textit{universal domain} \(\mathcal U \supset F\) and we will identify a differential variety with its \(\mathcal U\)-rational points. In this setting, a universal domain \(\mathcal U\) has to be a model of \(\text{DCF}_{0,m}\), i.e., it should be \textit{differentially closed} -- the appropriate replacement for algebraically closed in the presence of the derivations. The Kolchin topology on \(\mathbb A^n = \mathcal U^n\) is the differential analogue of the Zariski topology: closed sets are zero sets of differential polynomials with coefficients in \(F\); in a similar way, one also defines the Kolchin topology on \(\mathbb P^n\). The present paper considers the notion of completeness of quasi-projective differential varieties in this setting, which is defined analogously to the non-differential setting. It generalizes results from \textit{W. Y. Pong} [J. Algebra 224, No. 2, 454--466 (2000; Zbl 1037.12009)] in the \(m=1\) case. The main results are the following: \(\bullet\) Every complete projective differential variety is isomorphic to a closed differential subvariety of \(\mathbb A^1\) (Corollary~4.9). (This is in strong contrast to the non-differential setting; a key ingredient is that \(\mathbb P^1\) is \textit{not} differentially complete.) \(\bullet\) A valuative criterion for completeness is given (Theorem~5.9), similar to the one in the non-differential setting (which says that a variety \(V\) is complete if for every valuation ring \(R\) with quotient field \(K\), the map \(V(R) \to V(K)\) is a bijection). The paper also contains some results about embedability of arbitrary (i.e., not necessarily projective) differential varieties into…
reviewed by: Immanuel Halupczok
Mathematics Subject Classification ID: 14A10
Mathematics Subject Classification ID: 12H05
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 12L12
zbMATH DE Number: 6350924
zbMATH Keywords: differential algebraic varieties
zbMATH Keywords: completeness
zbMATH Keywords: differentially closed fields
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2014.07.025
OpenAlex ID: W2083831403
Wikidata QID: Q115350906
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DOI: 10.1016/J.JALGEBRA.2014.07.025
description: scientific article; zbMATH DE number 6350924