The Stacks project

Remark 101.19.3. 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 101.19.2 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.)