On some properties of locally nilpotent derivations
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 \(B\).
scientific article; zbMATH DE number 986700
scientific article