Exercise 2, § 5

Let A be a principal ring, E a free A-module of finite rank l, and G a finite subgroup of $\mathbf{GL}(E)$. Assume the following:

(i) If $q = \mathrm{Card}(G)$, the element $q.1$ of A is invertible. (ii) G is generated by pseudo-reflections in E (i.e. by elements s such that $(s-1)(E)$ is a monogenic submodule of E, cf. § 2, Exerc. 1.)

⁸ As E. Vinberg has shown, the group W corresponding to this last graph is not a subgroup of G of "arithmetic" type. The same situation arises for various other graphs of hyperbolic type (non-compact, this time), notably for the last four in Exerc. 16.

Let $S(E)$ be the symmetric algebra of $E$, and let $S(E)^G$ be the subalgebra of $S(E)$ consisting of the elements invariant under $G$. Show that, for any homomorphism from $A$ into a field $k$, $S(E)^G \otimes k$ can be identified with $S(E \otimes k)^G$. Deduce, by applying Th. 4, that $S(E)^G$ is a graded polynomial algebra over $A$.