Roots and logarithms of automorphisms of complete local rings
Let R be a complete (Noetherian) local ring with maximal ideal m and coefficient field \({\mathbb{C}}\), i.e. R/m\(\cong {\mathbb{C}}R\). Then R is the inverse limit of the \(R_ 1:=R/m^{\ell +1}\). Furthermore let \({\mathcal A}\) be the group of \({\mathbb{C}}\)-automorphisms of R and \({\mathcal D}\) the Lie algebra of \({\mathbb{C}}\)-derivations from R to m. \(F\in {\mathcal A}\) is called to have a logarithm if there is some \(D\in {\mathcal D}\) such that \(F=\exp D\) and \(G\in {\mathcal A}\) is called an n-th root of F in G \(n=F\). Using this notions then among others it is shown that for every \(F\in {\mathcal A}\) there is some power F h which has a logarithm. This generalizes an old result of \textit{D. C. Lewis} jun. [Duke Math. J. 5, 794-805 (1939; Zbl 0022.32703)] who proved this in the case of a regular ring R. Furthermore for a fixed set of generators of R special automorphisms and derivations called x-normal are considered and it is shown that \(F\in {\mathcal A}\) has an x-normal logarithm if and only if F has x-normal roots of all orders.
scientific article; zbMATH DE number 4054213