Relation modules of prounipotent groups
The object is to develop for prounipotent groups a concept analogous to that one of relation modules from discrete group theory. Let G be a prounipotent group over the algebraically closed field k of characteristic zero. G has a presentation as quotient of a free prounipotent group F by a normal prounipotent subgroup R. Denote by \(\bar R\) the quotient of R by the Zariski-closure (R,R) of its derived group. G acts induced by conjugation on \(\bar R.\) The first part of the paper consists of the construction of a suitable resolution of k, in which \(\bar R\) occurs as the second syzygy. Assume that F is free on \(d<\infty\) generators, however allow G to be infinitely related. Then the resolution has the form \[ k\ll G\gg^ r\to k\ll G\gg^ d\to k\ll G\gg \to k\to 0. \] The terms of the resolution are direct products of the complete group algebra (the dual of the coordinate ring of G) \(k\ll G\gg\), the cardinal r can be infinite. So \(k\ll G\gg\) is the analogue to the group ring in the discrete case. In the second part the author analyzes the structure of \(\bar R\) as \(k\ll G\gg\)-module and the structure of the group F/(R,R). In particular the Magnus embedding of F/(R,R) is constructed. Finally it is shown that higher relation modules (the other quotients of the closed lower central series of R) are also submodules of direct products of copies of \(k\ll G\gg.\) All constructions depend on the properties of prounipotent groups developed by \textit{A. Lubotzky} and the author [J. Algebra 74, 76-95 (1982; Zbl 0473.20031), ibid. 80, 323-349 (1983; Zbl 0503.20008), Am. J. Math. 107, 531-553 (1985; Zbl 0565.20023), J. Algebra 95, 46-58 (1985; Zbl 0596.20033)]. In this sense the article may be regarded as a continuation of this work.
scientific article; zbMATH DE number 4029830