The Stacks project

Lemma 101.32.3. Let $\mathcal{X}$ be an algebraic stack. Assume $\mathcal{X}$ is quasi-DM with separated diagonal (equivalently $\mathcal{I}_\mathcal {X} \to \mathcal{X}$ is locally quasi-finite and separated). Let $x \in |\mathcal{X}|$. Then there exists a morphism of algebraic stacks

\[ \mathcal{U} \longrightarrow \mathcal{X} \]

with the following properties

  1. there exists a point $u \in |\mathcal{U}|$ mapping to $x$,

  2. $\mathcal{U} \to \mathcal{X}$ is representable by algebraic spaces and étale,

  3. $\mathcal{U} = [U/R]$ where $(U, R, s, t, c)$ is a groupoid scheme with $U$, $R$ affine, and $s, t$ finite, flat, and locally of finite presentation.

Proof. (The parenthetical statement follows from the equivalences in Lemma 101.6.1). Choose an affine scheme $U$ and a flat, locally finitely presented, locally quasi-finite morphism $U \to \mathcal{X}$ such that $x$ is the image of some point $u \in U$. This is possible by Theorem 101.21.3 and the assumption that $\mathcal{X}$ is quasi-DM. Let $(U, R, s, t, c)$ be the groupoid in algebraic spaces we obtain by setting $R = U \times _\mathcal {X} U$, see Algebraic Stacks, Lemma 94.16.1. Let $\mathcal{X}' \subset \mathcal{X}$ be the open substack corresponding to the open image of $|U| \to |\mathcal{X}|$ (Properties of Stacks, Lemmas 100.4.7 and 100.9.12). Clearly, we may replace $\mathcal{X}$ by the open substack $\mathcal{X}'$. Thus we may assume $U \to \mathcal{X}$ is surjective and then Algebraic Stacks, Remark 94.16.3 gives $\mathcal{X} = [U/R]$. Observe that $s, t : R \to U$ are flat, locally of finite presentation, and locally quasi-finite. Since $R = U \times U \times _{(\mathcal{X} \times \mathcal{X})} \mathcal{X}$ and since the diagonal of $\mathcal{X}$ is separated, we find that $R$ is separated. Hence $s, t : R \to U$ are separated. It follows that $R$ is a scheme by Morphisms of Spaces, Proposition 67.50.2 applied to $s : R \to U$.

Above we have verified all the assumptions of More on Groupoids in Spaces, Lemma 79.15.13 are satisfied for $(U, R, s, t, c)$ and $u$. Hence we can find an elementary étale neighbourhood $(U', u') \to (U, u)$ such that the restriction $R'$ of $R$ to $U'$ is quasi-split over $u$. Note that $R' = U' \times _\mathcal {X} U'$ (small detail omitted; hint: transitivity of fibre products). Replacing $(U, R, s, t, c)$ by $(U', R', s', t', c')$ and shrinking $\mathcal{X}$ as above, we may assume that $(U, R, s, t, c)$ has a quasi-splitting over $u$ (the point $u$ is irrelevant from now on as can be seen from the footnote in More on Groupoids in Spaces, Definition 79.15.1). Let $P \subset R$ be a quasi-splitting of $R$ over $u$. Apply Lemma 101.32.2 to see that

\[ \mathcal{U} = [U/P] \longrightarrow [U/R] = \mathcal{X} \]

has all the desired properties. $\square$

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