Abstract
Let k be a valued field of characteristic zero. We consider analytic k-algebras, i.e. finite algebras over rings of convergent power series over k, and their k-derivations. The following theorem is proved: Let k be algebraically closed and A a normal analytic k-algebra of dimension 2. If there is a k-derivation on A not acting nilpotently, then A is homogeneous, i.e. a residue class ring of a power series ring by a (weighted) homogeneous ideal.
The basic tool is the Chevalley decomposition of k-derivations of analytic algebras. We also use some general lemmata concerning extensions and restrictions of k-derivations which describe homogeneity.
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Scheja, G., Wiebe, H. Zur Chevalley-Zerlegung von Derivationen. Manuscripta Math 33, 159–176 (1980). https://doi.org/10.1007/BF01316974
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DOI: https://doi.org/10.1007/BF01316974