Group action on polynomial and power series rings
Let \(S\) be a finitely generated graded integral domain over a field \(k\), with finite dimensional homogeneous components, such as a polynomial ring. Let \(G\) be a finite group of grading preserving automorphisms of \(S\). The paper concerns the structure of \(S\) as a \(kG\)-module. The main result is that \(S\) contains a free \(kG\)-module summand, which is large in an appropriate sense, made precise in terms of the Hilbert-Serre theory relating the growth of the dimension of the homogeneous components of \(S\). Versions of this theorem (with restrictions on \(k\) or \(S\)) were proved by \textit{R. Howe} [J. Algebra 122, 384-379 (1989; Zbl 0693.20005)] and by \textit{R. M. Bryant} [J. Algebra 154, 416-436 (1993; Zbl 0828.20002)]. A similar result holds for filtered algebras, such as power series rings.
scientific article; zbMATH DE number 1537849