The Stacks project

Lemma 87.30.2. Notation and assumptions as in Lemma 87.30.1. Let $f : X' \to \mathop{\mathrm{Spec}}(A)$ correspond to $g : Y' \to \mathop{\mathrm{Spec}}(B)$ via the equivalence. Then $f$ is quasi-compact, quasi-separated, separated, proper, finite, and add more here if and only if $g$ is so.

Proof. You can deduce this for the statements quasi-compact, quasi-separated, separated, and proper by using Lemmas 87.28.1 87.28.2, 87.28.3, 87.28.2, and 87.28.4 to translate the corresponding property into a property of the formal completion and using the argument of the proof of Lemma 87.30.1. However, there is a direct argument using fpqc descent as follows. First, you can reduce to proving the lemma for $A \to A^\wedge $ and $B \to B^\wedge $ since $A^\wedge \to B^\wedge $ is an isomorphism. Then note that $\{ U \to \mathop{\mathrm{Spec}}(A), \mathop{\mathrm{Spec}}(A^\wedge ) \to \mathop{\mathrm{Spec}}(A)\} $ is an fpqc covering with $U = \mathop{\mathrm{Spec}}(A) \setminus V(I)$ as before. The base change of $f$ by $U \to \mathop{\mathrm{Spec}}(A)$ is $\text{id}_ U$ by definition of our category ( Let $P$ be a property of morphisms of algebraic spaces which is fpqc local on the base (Descent on Spaces, Definition 73.10.1) such that $P$ holds for identity morphisms. Then we see that $P$ holds for $f$ if and only if $P$ holds for $g$. This applies to $P$ equal to quasi-compact, quasi-separated, separated, proper, and finite by Descent on Spaces, Lemmas 73.11.1, 73.11.2, 73.11.18, 73.11.19, and 73.11.23. $\square$

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Your email address will not be published.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 0BH5. Beware of the difference between the letter 'O' and the digit '0'.