On some properties of locally nilpotent derivations
type: Item
instance of: scholarly article
title: On some properties of locally nilpotent derivations
zbMATH Open document ID: 0885.13003
DOI: 10.1016/0022-4049(95)00173-5
published in: Journal of Pure and Applied Algebra
publication date: 1998-04-20T00:00:00Z
review text: Let \(K\) be a field of characteristic 0, \(B\) a \(K\)-domain. Sections 1 and 2 of this paper study derivations \(D: B\to B\), particularly those which are locally nilpotent; that is which are such that for each \(b\in B\) there exists \(n\in\mathbb{N}\) such that \(D^n b=0\). Section 3 is devoted to theorems on almost smooth morphisms which are needed for the concluding results of section 2. The culminating result, corollary 2.5, of the work on locally nilpotent derivations is a formula for the locally nilpotent derivations in an important special case: Let \(B =K[X_1, \dots, X_n]\), the \(X_i\) indeterminates over \(K\). Let \(D:B\to B\) be locally nilpotent and non-zero. From an earlier result of the paper it is known that \(\text{Ker} D\) is of transcendence degree \(n-1\) over \(K\). Suppose \(\text{Ker} D\) is a purely transcendental extension of \(K\), so that \(\text{Ker} D= K[f_1, \dots, f_{n-1}]\), the \(f_i\) algebraically independent over \(K\). Then \(D= a\Delta_f\), where \(a\in \text{Ker} D\) and \(\Delta_f\) is given by the Jacobian matrix \[ \Delta_f(g)= \text{determinant of } {\partial (f_1, \dots, f_{n-1},g) \over \partial (X_1, \dots, X_n)} \text{(for each } g\in B). \] Furthermore, \(\Delta_f\) is an irreducible derivation; that is, \(\Delta_f\) is not of the form \(b\Delta\), \(\Delta\): \(B\to B\) a derivation and \(b\) a non-unit of \(B\).
Mathematics Subject Classification ID: 13B10
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 13G05
Mathematics Subject Classification ID: 12H05
zbMATH DE Number: 986700
zbMATH Keywords: almost smooth morphisms
zbMATH Keywords: locally nilpotent derivations
author: Daniel Daigle
reviewed by: Richard M. Cohn
MaRDI profile type: Publication
full work available at URL: 0022-4049(95)00173-5
OpenAlex ID: W2040696824
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description: scientific article; zbMATH DE number 986700