Remark 97.19.3. Let $P$ be a property of algebraic spaces over fields which is invariant under ground field extensions. Given an algebraic stack $\mathcal{X}$ and $x \in |\mathcal{X}|$, we say the automorphism group of $\mathcal{X}$ at $x$ has $P$ if the equivalent conditions of Lemma 97.19.2 are satisfied. For example, we say *the automorphism group of $\mathcal{X}$ at $x$ is finite*, if $G_ x \to \mathop{\mathrm{Spec}}(k)$ is finite whenever $x : \mathop{\mathrm{Spec}}(k) \to \mathcal{X}$ is a representative of $x$. Similarly for smooth, proper, etc. (There is clearly an abuse of language going on here, but we believe it will not cause confusion or imprecision.)

## 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Your email address will not be published.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)

There are also: