Exercise 7, § 5

*Let R be a regular local ring (Commutative Algebra, Chap. VIII, § 5, no. 1), with maximal ideal m and residue field k. Let G be a finite group of automorphisms of R, and let $R' = R^G$ be the subring of R consisting of the elements invariant under G. This is a local ring with maximal ideal $m' = m \cap R'$. Assume the following:

(i) $R'$ is noetherian and R is an $R'$-module of finite type.

(ii) The composite $R' \to R \to k$ is surjective.

Put $V = m/m^2$; this is a $k$-vector space. The action of G on R defines a homomorphism $\varepsilon : G \to \mathbf{GL}(V)$.

a) Let $\mathfrak{p}$ be a prime ideal of R of height 1 (Commutative Algebra, Chap. VII, § 1, no. 6) and let $s \in G$ be such that $s(\mathfrak{p}) = \mathfrak{p}$ and that s acts trivially on $R/\mathfrak{p}$. Show that $\varepsilon(s)$ is a pseudo-reflection of V. (Remark that the image of $\mathfrak{p}$ in $m/m^2$ is of dimension 0 or 1.)

b) Show that, if $R'$ is regular, the subgroup $\varepsilon(G)$ of $\mathbf{GL}(V)$ is generated by pseudo-reflections, (Let H be the subgroup of G generated by elements whose image under $\varepsilon$ is a pseudo-reflection, and let $R^H$ be the subring of R consisting of elements invariant under H. Show, using a), that no prime ideal of $R'$ of height 1 is ramified in $R^H$; using the fact that $R^H$ is integrally closed, deduce by means of the Purity Theorem$^{11}$ that $R' = R^H$, so $H = G$.)

c) Assume now that the order of G is coprime to the characteristic of $k$. Show that $\varepsilon$ is injective.

Let $(m'_n)$ be the filtration of $R'$ induced by the m-adic filtration $(m^n)$ of R, and let

be the canonical homomorphism of the associated graded algebra of $R'$ to that of R (Commutative Algebra, Chap. III, § 2). Show that i is injective, and that its image is the subring $\operatorname{gr}(R)^G$ of $\operatorname{gr}(R)$ consisting of the elements invariant under G.

d) Retain the assumptions and notation of c), and assume further that $\varepsilon(G)$ is generated by pseudo-reflections. If $l = \dim V$, let $P_1, \ldots, P_l$ be algebraically independent homogeneous generators of the $k$-algebra $\operatorname{gr}(R)^G$ (such elements exist by Th. 4 and the fact that $\operatorname{gr}(R)$ can be identified with the symmetric algebra of V); let $p_1, \ldots, p_l$ be their degrees. By c), we can find $x_i \in m'_{p_i}$ such that $\operatorname{gr}(x_i) = P_i$. Show that the $x_i$ generate the ideal $m'$ and deduce that $R'$ is regular.*