Lemma 88.8.6. Let $A_1 \to A_2$ be a map of Noetherian rings. Let $I_ i \subset A_ i$ be an ideal such that $V(I_1A_2) = V(I_2)$. Let $B_1$ be in (88.2.0.2) for $(A_1, I_1)$. Let $B_2$ be the base change of $B_1$ as in Remark 88.2.3. If $B_1$ is rig-étale over $(A_1, I_1)$, then $B_2$ is rig-étale over $(A_2, I_2)$.

**Proof.**
Follows from Lemma 88.8.5 and Definition 88.8.1 and the fact that $I_2^ c \subset I_1A_2$ for some $c \geq 0$ as $A_2$ is Noetherian.
$\square$

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