The Stacks project

Lemma 101.32.5. 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 $x \in |\mathcal{X}|$. Assume $x$ can be represented by a quasi-compact morphism $\mathop{\mathrm{Spec}}(k) \to \mathcal{X}$. Then there exists a morphism of algebraic stacks

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

with the following properties

  1. there exists a point $u \in |\mathcal{U}|$ mapping to $x$ and $g$ induces an isomorphism between the residual gerbes at $u$ and $x$,

  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 first part of the proof is exactly the same as the first part of the proof of Lemma 101.32.3. Thus we may assume $\mathcal{X} = [U/R]$ where $(U, R, s, t, c)$ and $u \in U$ mapping to $x$ satisfy all the assumptions of More on Groupoids in Spaces, Lemma 79.15.13. Observe that $u = \mathop{\mathrm{Spec}}(\kappa (u)) \to \mathcal{X}$ is quasi-compact, see Properties of Stacks, Lemma 100.14.1. Consider the cartesian diagram

\[ \xymatrix{ F \ar[d] \ar[r] & U \ar[d] \\ u \ar[r]^ u & \mathcal{X} } \]

Since $U$ is an affine scheme and $F \to U$ is quasi-compact, we see that $F$ is quasi-compact. Since $U \to \mathcal{X}$ is locally quasi-finite, we see that $F \to u$ is locally quasi-finite. Hence $F \to u$ is quasi-finite and $F$ is an affine scheme whose underlying topological space is finite discrete (Spaces over Fields, Lemma 72.10.8). Observe that we have a monomorphism $u \times _\mathcal {X} u \to F$. In particular the set $\{ r \in R : s(r) = u, t(r) = u\} $ which is the image of $|u \times _\mathcal {X} u| \to |R|$ is finite. we conclude that all the assumptions of More on Groupoids in Spaces, Lemma 79.15.11 hold.

Thus we can find an elementary étale neighbourhood $(U', u') \to (U, u)$ such that the restriction $R'$ of $R$ to $U'$ is strongly 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 strong splitting over $u$. Let $P \subset R$ be a strong splitting of $R$ over $u$. Apply Lemma 101.32.2 to see that

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

is representable by algebraic spaces and étale. Since $P \subset R$ is open and contains $\{ r \in R : s(r) = u, t(r) = u\} $ by construction we see that $u \times _\mathcal {U} u \to u \times _\mathcal {X} u$ is an isomorphism. The statement on residual gerbes then follows from Properties of Stacks, Lemma 100.11.14 (we observe that the residual gerbes in question exist by Lemma 101.31.2). $\square$

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