Continuous derivations
type: Item
instance of: scholarly article
title: Continuous derivations
zbMATH Open document ID: 0644.12012
DOI: 10.1016/0021-8693(87)90058-5
author: Sally D. Morrison
published in: Journal of Algebra
publication date: 1987-01-01T00:00:00Z
review text: In many and very diverse problems of differential algebra a fundamental role is played implicitly or explicitly by a valuation related to the derivation. Let L be a field, v a rank one non-archimedean valuation of L trivial on the prime field of L. Then \((L,v)\) is called a valued field. Let K be a subfield of L and \(D: K\to L\) a derivation. Let \(S\subset K\). Then D is bounded below on S if there exists \(\gamma\) in the value group of v such that v(Dx/x)\(\geq \gamma\) for all \(x\in S\). It is shown (proposition 2.2) that if \(v\neq 0\), D is bounded below on K if and only if D is continuous on K when L is normed via \(\| x\| =e^{-v(x)}\), \(x\in L\). This paper is the first to investigate in a general way the properties of continuous derivations, thus coordinating many results scattered through the literature of the applications (most of which impose further restrictions on D). Many of the theorems generalize to all finite ranks. Let K/F be a finitely generated field extension. If K/F has transcendence degree one, a classic result assures that every F-derivation of K is continuous in the topology of every valuation of K/F. In the general case one asks first whether for a given valuation v of K over F, there exists a non-zero F-derivation of K continuous with respect to the topology of v; and second, whether, given an F-derivation D of K there exists a valuation v of K over F such that D is continuous in the corresponding topology. These questions, somewhat generalized, are answered in many cases. It is shown that the answer to the first question is affirmative in important cases but not always. Information is obtained (theorem 5.2 and corollary 5.3) on the dimension of the space of continuous derivations. The answer to the second question is affirmative if K/F is finitely generated and of characteristic 0, but there is a counter-example if K/F is not finitely generated. Pursuing a related but somewhat different question, theorems 4.1 and 7.1 b generalize ear…
reviewed by: Richard M. Cohn
Mathematics Subject Classification ID: 12H05
Mathematics Subject Classification ID: 12J10
Mathematics Subject Classification ID: 13A18
zbMATH DE Number: 4049751
zbMATH Keywords: non-archimedean valuation
zbMATH Keywords: continuous derivations
zbMATH Keywords: differential specialization
zbMATH Keywords: power series
MaRDI profile type: Publication
full work available at URL: 0021-8693(87)90058-5
OpenAlex ID: W4213103992
cites work: Q5810622
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description: scientific article; zbMATH DE number 4049751