Every algebraic stack has a largest open substack which is a Deligne-Mumford stack; this is more or less clear but we also write out the proof below. Of course this substack may be empty, for example if $X = [\mathop{\mathrm{Spec}}(\mathbf{Z})/\mathbf{G}_{m, \mathbf{Z}}]$. Below we will characterize the points of the DM locus.
Lemma 101.22.1. Let $\mathcal{X}$ be an algebraic stack. There exist open substacks such that $\mathcal{X}''$ is DM, $\mathcal{X}'$ is quasi-DM, and such that these are the largest open substacks with these properties.
\[ \mathcal{X}'' \subset \mathcal{X}' \subset \mathcal{X} \]
Proof. All we are really saying here is that if $\mathcal{U} \subset \mathcal{X}$ and $\mathcal{V} \subset \mathcal{X}$ are open substacks which are DM, then the open substack $\mathcal{W} \subset \mathcal{X}$ with $|\mathcal{W}| = |\mathcal{U}| \cup |\mathcal{V}|$ is DM as well. (Similarly for quasi-DM.) Although this is a cheat, let us use Theorem 101.21.6 to prove this. By that theorem we can choose schemes $U$ and $V$ and surjective étale morphisms $U \to \mathcal{U}$ and $V \to \mathcal{V}$. Then of course $U \amalg V \to \mathcal{W}$ is surjective and étale. The quasi-DM case is proven by exactly the same method using Theorem 101.21.3. $\square$
Lemma 101.22.2. Let $\mathcal{X}$ be an algebraic stack. Let $x \in |\mathcal{X}|$ correspond to $x : \mathop{\mathrm{Spec}}(k) \to \mathcal{X}$. Let $G_ x/k$ be the automorphism group algebraic space of $x$. Then
$x$ is in the DM locus of $\mathcal{X}$ if and only if $G_ x \to \mathop{\mathrm{Spec}}(k)$ is unramified, and
$x$ is in the quasi-DM locus of $\mathcal{X}$ if and only if $G_ x \to \mathop{\mathrm{Spec}}(k)$ is locally quasi-finite.
Proof. Proof of (2). Choose a scheme $U$ and a surjective smooth morphism $U \to \mathcal{X}$. Consider the fibre product
Recall that $G$ is the automorphism group algebraic space of $U \to \mathcal{X}$. By Groupoids in Spaces, Lemma 78.6.3 there is a maximal open subscheme $U' \subset U$ such that $G_{U'} \to U'$ is locally quasi-finite. Moreover, formation of $U'$ commutes with arbitrary base change. In particular the two inverse images of $U'$ in $R = U \times _\mathcal {X} U$ are the same open subspace of $R$ (since after all the two maps $R \to \mathcal{X}$ are isomorphic and hence have isomorphic automorphism group spaces). Hence $U'$ is the inverse image of an open substack $\mathcal{X}' \subset \mathcal{X}$ by Properties of Stacks, Lemma 100.9.11 and we have a cartesian diagram
Thus the morphism $\mathcal{I}_{\mathcal{X}'} \to \mathcal{X}'$ is locally quasi-finite and we conclude that $\mathcal{X}'$ is quasi-DM by Lemma 101.6.1 part (5). On the other hand, if $\mathcal{W} \subset \mathcal{X}$ is an open substack which is quasi-DM, then the inverse image $W \subset U$ of $\mathcal{W}$ must be contained in $U'$ by our construction of $U'$ since $\mathcal{I}_\mathcal {W} = \mathcal{W} \times _\mathcal {X} \mathcal{I}_\mathcal {X}$ is locally quasi-finite over $\mathcal{W}$. Thus $\mathcal{X}'$ is the quasi-DM locus. Finally, choose a field extension $K/k$ and a $2$-commutative diagram
Then we find an isomorphism $G_ x \times _{\mathop{\mathrm{Spec}}(k)} \mathop{\mathrm{Spec}}(K) \cong G \times _ U \mathop{\mathrm{Spec}}(K)$ of group algebraic spaces over $K$. Hence $G_ x$ is locally quasi-finite over $k$ if and only if $\mathop{\mathrm{Spec}}(K) \to U$ maps into $U'$ (use the commutation of formation of $U'$ and Groupoids in Spaces, Lemma 78.6.3 applied to $\mathop{\mathrm{Spec}}(K) \to \mathop{\mathrm{Spec}}(k)$ and $G_ x$ to see this). This finishes the proof of (2). The proof of (1) is exactly the same. $\square$
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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)