On the group of automorphisms of a power series ring
Let k be a commutative \({\mathbb{Q}}\)-algebra and \(k^{n}\) the ring of (formal) power series in n variables (n\(\geq 2)\) \(X_ 1,...,X_ n\) over k. This paper concerns the structure of the group \(GA_ n((k))\) of continuous k-algebra automorphisms of \(k^{n}\), where \(k^{n}\) is considered with the topology defined by the powers of the ideal ((X)) generated by \(X_ 1,...,X_ n\). If \(GA^ 0_ n((k))\) is the subgroup of \(GA_ n((k))\) consisting of automorphisms which preserve the ideal ((X)), it is proved that every element of \(GA_ n((k))\) is the product of an element of \(GA^ 0_ n((k))\) and a translation. The author shows that \(GA^ 0_ n((k))\) is the semidirect product of \(GA^ 1_ n((k))\) and \(GL_ n(k)\), where \(GA^ 1_ n((k))\) is the torsion-free normal subgroup consisting of all automorphisms F such that F(f)\(\equiv f (mod ((X))^{e+1})\) for all f in \(((X))^ e\). She also proves that, if \(n\geq 3\), every closed subgroup of \(GA^ 0_ n((k))\), which is normalized by the subgroup \(E_ n(k)\) of \(SL_ n(k)\) consisting of elementary matrices, is factorized with respect to the above semidirect decomposition of the group \(GA^ 0_ n((k))\). The analogous problem for the Lie algebra of k-derivations of the ring of polynomials in n variables over k was considered by the author in a previous paper [J. Pure Appl. Algebra 36, 299-314 (1985; Zbl 0565.20022)].
scientific article; zbMATH DE number 3993862
scientific article