Lemma 22.1.9476252E7717AC

Proof. It suffices to argue that $I$ can be expanded as in (22.1.95). To see this, assume that $a\in I$. We can write $a=a_0+a_1t+\cdots +a_ mt^ m$ with $a_ i\in R$. Then our assumption implies that $\sum _ i a_ i \rho ^ i t^ i\in I$ as well for all $\rho \in S^1$. So by the Lagrange interpolation formula, $a_ it^ i\in I$ for all $i$. Therefore, we can write $I$ as $I_0+I_1t+I_2t^2+\cdots $ for some ideals $I_0\subseteq I_1\subseteq \cdots $ in $R$. But as $R$ is noetherian, there is $m\geq 0$ so that $I_{m'}=I_ m$ for $m'>m$. (22.1.95) follows. $\square$