Derivations of \(R[X,Y,Z]\) with a slice. (English) Zbl 1220.13018
The author considers locally nilpotent derivations \(D\) on \(R[X,Y,Z]\) having a slice (which is an element \(s\) such that \(D(s)=1\)). He considers the case where \(R\) is a polynomial ring over a field of characteristic zero. The central question is: is the kernel \(A\) of \(D\) isomorphic to \(R^{[2]}\)?
It is shown that \(A\) is an \(\mathbb A^2\)-fibration over \(R\).
The author gives a family of interesting examples: define \(R:=k[a,b]\cong k^{[2]}\), \(T_n:=a^nY+bZ+X^2\), \(v_n:=bX+a^nT_n\), \(t_n:=Z+bv_nY-2v_nXT_n-bv_n^2T_n^2\) and \(\varphi_n\) a locally nilpotent derivation on \(R[X,Y,Z]\) by \(\varphi_n(X)=-a^n\), \(\varphi_n(Y)=2(X+bv_nT_n)\), \(\varphi_n(Z)=1-2a^nv_nT_n\).
It is shown that \(A\) is an \(\mathbb A^2\)-fibration over \(R\).
The author gives a family of interesting examples: define \(R:=k[a,b]\cong k^{[2]}\), \(T_n:=a^nY+bZ+X^2\), \(v_n:=bX+a^nT_n\), \(t_n:=Z+bv_nY-2v_nXT_n-bv_n^2T_n^2\) and \(\varphi_n\) a locally nilpotent derivation on \(R[X,Y,Z]\) by \(\varphi_n(X)=-a^n\), \(\varphi_n(Y)=2(X+bv_nT_n)\), \(\varphi_n(Z)=1-2a^nv_nT_n\).
Reviewer: Stefan Maubach (Bremen)
MSC:
| 13N15 | Derivations and commutative rings |
| 14R25 | Affine fibrations |
| 14R10 | Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem) |
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