Exercise 28, § 1

Let k be a field and A, the polynomial ring $k[X_n, Y_n]_{n \in \mathbf{Z}}$ in two infinite families of indeterminates. Let G be an infinite monogenous group (therefore isomorphic to $\mathbf{Z}$) and let $\mathfrak{s}$ be a generator of $\mathscr{G}$. G is defined as operating on A, by the conditions $\sigma(Y_n) = Y_n$, and $\sigma(X_n) = X_{n+1}$ for all $n \in \mathbf{Z}$. Let 3 be the ideal of A, generated by the elements $Y_n(X_n - X_{n+1})$ for all $n \in \mathbf{Z}$ and let A be the quotient ring $A_0/\mathfrak{s}$; as $\sigma(\mathfrak{s}) \subset 3$, the group $\mathscr{G}$ also operates on A by passing to the quotient. Let S be the multiplicative subset of A generated by the canonical images of the Y, in A, so that S consists of the elements invariant under G. Show that $S^{-1}(A^{\mathscr{G}}) \neq (S^{-1}A)^{\mathscr{G}}$.