Lifting of the additive group scheme actions
The authors deal with the question of lifting a locally nilpotent derivation \(\delta\) on an affine domain \(A/\mathbb C\) to a finite integral extension \(B\) of \(A\). Geometrically, a derivation corresponds to a vector field on Spec\,\(A\)-Sing (Spec\,\(A\)), and locally nilpotent derivation corresponds to an action of the additive group \((\mathbb C,+)\) on Spec\,\(A\). Then the map Spec\,\(B\to \text{Spec}\,A\) is étale outside a codimension 2 subvariety, the vector field corresponding to \(\delta\) lifts to a vector field on Spec\,\(B\)-Sing (Spec\,\(B\)). This then gives a lift of \(\delta\) to Spec\,\(B\), at least when \(B\) is normal. An old result of A. Seidenberg says that any derivation of \(A/\mathbb C\) extends to its integral closure in its quotient field. On the other hand, a result of W. Vasconcelos says that if the locally nilpotent derivation \(\delta\) extends as a derivation on B then the extension is also locally nilpotent. The authors prove a slightly weaker result than Vasconcelos and so extension of \(\delta\) to \(B\) when the ramification is in codimension 2. The proofs are purely algebraic. These results are illustrated by working out in detail some examples. {\parindent5mm \begin{itemize}\item[1.] Action of the symmetric group \(S_n\) on \(B =\mathbb C[X_1,\dots, X_n]\) and \(A = B^{S_n}\). \item[2.] Dihedral group \(G\) acting on \(B=\mathbb C[X,Y],\;A = B^G\). \item[3.] A characterization of \(\mathbb C^2/G\), with \(G\) a finite cyclic group. \end{itemize}}
scientific article; zbMATH DE number 5608375
scientific article