The Stacks project

Lemma 87.12.2. Consider the property $P$ on arrows of $\textit{WAdm}^{count}$ defined in Lemma 87.12.1. Then $P$ is stable under base change (Formal Spaces, Situation 86.21.6).

Proof. The statement makes sense by Lemma 87.12.1. To see that it is true assume we have morphisms $B \to A$ and $B \to C$ in $\textit{WAdm}^{count}$ such that $B/\mathfrak b \to A/\mathfrak a$ is of finite type where $\mathfrak b \subset B$ and $\mathfrak a \subset A$ are the ideals of topologically nilpotent elements. Since $A$ and $B$ are weakly admissible, the ideals $\mathfrak a$ and $\mathfrak b$ are open. Let $\mathfrak c \subset C$ be the (open) ideal of topologically nilpotent elements. Then we find a surjection $A \widehat{\otimes }_ B C \to A/\mathfrak a \otimes _{B/\mathfrak b} C/\mathfrak c$ whose kernel is a weak ideal of definition and hence consists of topologically nilpotent elements (please compare with the proof of Formal Spaces, Lemma 86.4.12). Since already $C/\mathfrak c \to A/\mathfrak a \otimes _{B/\mathfrak b} C/\mathfrak c$ is of finite type as a base change of $B/\mathfrak b \to A/\mathfrak a$ we conclude. $\square$

Comments (0)

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Your email address will not be published.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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0GC3. Beware of the difference between the letter 'O' and the digit '0'.