Lemma 99.33.7. 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 95.17.2. 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 95.17.1). Therefore we may apply Descent on Spaces, Lemma 72.7.4.
$\square$

## 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Your email address will not be published.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).

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.

## Comments (0)