Lemma 10.126.8. Let $R$ be a ring. Let $I \subset R$ be a nilpotent ideal. Let $S$ be an $R$-algebra such that $R/I \to S/IS$ is of finite type. Then $R \to S$ is of finite type.
Proof. Choose $s_1, \ldots , s_ n \in S$ whose images in $S/IS$ generate $S/IS$ as an algebra over $R/I$. By Lemma 10.20.1 part (11) we see that the $R$-algebra map $R[x_1, \ldots , x_ n \to S$, $x_ i \mapsto s_ i$ is surjective and we conclude. $\square$
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