The Stacks project

Lemma 101.33.7. Let $X \to Y$ be a smooth morphism of algebraic spaces. Let $G$ be a group algebraic space over $Y$ which is flat and locally of finite presentation over $Y$. Let $G$ act on $X$ over $Y$. Then the quotient stack $[X/G]$ is smooth over $Y$.

Proof. The quotient $[X/G]$ is an algebraic stack by Criteria for Representability, Theorem 97.17.2. The smoothness of $[X/G]$ over $Y$ follows from the fact that smoothness descends under fppf coverings: Choose a surjective smooth morphism $U \to [X/G]$ where $U$ is a scheme. Smoothness of $[X/G]$ over $Y$ is equivalent to smoothness of $U$ over $Y$. Observe that $U \times _{[X/G]} X$ is smooth over $X$ and hence smooth over $Y$ (because compositions of smooth morphisms are smooth). On the other hand, $U \times _{[X/G]} X \to U$ is locally of finite presentation, flat, and surjective (because it is the base change of $X \to [X/G]$ which has those properties for example by Criteria for Representability, Lemma 97.17.1). Therefore we may apply Descent on Spaces, Lemma 74.8.4. $\square$


Comments (0)

There are also:

  • 2 comment(s) on Section 101.33: Smooth morphisms

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Your email address will not be published.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