Exercise 10, § 2

(a) Let $A'$ be the polynomial ring $\mathbf{Z}[X, Y]$ in two indeterminates and let $\sigma$ be the automorphism of $A'$ leaving $Y$ invariant and such that $\sigma(X) = -X$; $\sigma$ and the identity automorphism form a group $\mathscr{G}$ of automorphisms of $A'$ and the ring of invariants of $\mathscr{G}$ is $A = \mathbf{Z}[X^2, Y]$. Let $p'$ be the prime ideal $A'(Y - 2X)$, $p = A \cap p'$; the decomposition group $\mathscr{G}^{\mathbf{Z}}(p')$ is reduced to the identity element. Show that $A/p$ and $A'/p'$ are not isomorphic (consider the tensor products of these rings with $2/22$).

(b) Let $K$ be a field of characteristic $\neq 2$, $A'$ the polynomial ring $K[X, Y]$ and $\mathscr{G}$ the automorphism group of $A'$ consisting of the identity and the $K[Y]$-automorphism $\sigma$ of $A'$ such that $\sigma(X) = -X$; the ring of invariants $A$ of $\mathscr{G}$ is $K[X^2, Y]$. Let $p'$ be the prime ideal $A'(X^3 - Y)$ of $A'$ and $p = A \cap p'$; the decomposition group $\mathscr{G}^{\mathbf{Z}}(p')$ is reduced to the identity element but $A/p$ and $A'/p'$ are not isomorphic.