The Stacks project

Lemma 67.6.6. Let $S$ be a scheme. Let $X \to Y$ be an étale morphism of algebraic spaces over $S$. If $Y$ is decent, resp. reasonable, then so is $X$.

Proof. Let $U$ be an affine scheme and $U \to X$ an étale morphism. Set $R = U \times _ X U$ and $R' = U \times _ Y U$. Note that $R \to R'$ is a monomorphism.

Let $x \in |X|$. To show that $X$ is decent, we have to show that the fibres of $|U| \to |X|$ and $|R| \to |X|$ over $x$ are finite. But if $Y$ is decent, then the fibres of $|U| \to |Y|$ and $|R'| \to |Y|$ are finite. Hence the result for “decent”.

To show that $X$ is reasonable, we have to show that the fibres of $U \to X$ are universally bounded. However, if $Y$ is reasonable, then the fibres of $U \to Y$ are universally bounded, which immediately implies the same thing for the fibres of $U \to X$. Hence the result for “reasonable”. $\square$

Comments (0)

There are also:

  • 2 comment(s) on Section 67.6: Reasonable and decent algebraic spaces

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Your email address will not be published.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