Lemma 88.4.5. 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-smooth over $(A_1, I_1)$, then $B_2$ is rig-smooth over $(A_2, I_2)$.
Proof. Follows from Lemma 88.4.4 and Definition 88.4.1 and the fact that $I_2^ c$ is contained in $I_1A_2$ for some $c \geq 0$ as $A_2$ is Noetherian. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$
). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)