101.32 Étale local structure

In this section we start discussing what we can say about the étale local structure of an algebraic stack.

Lemma 101.32.1. Let $Y$ be an algebraic space. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $Y$. Assume $U \to Y$ is flat and locally of finite presentation and $R \to U \times _ Y U$ an open immersion. Then $X = [U/R] = U/R$ is an algebraic space and $X \to Y$ is étale.

Proof. The quotient stack $[U/R]$ is an algebraic stacks by Criteria for Representability, Theorem 97.17.2. On the other hand, since $R \to U \times U$ is a monomorphism, it is an algebraic space (by our abuse of language and Algebraic Stacks, Proposition 94.13.3) and of course it is equal to the algebraic space $U/R$ (of Bootstrap, Theorem 80.10.1). Since $U \to X$ is surjective, flat, and locally of finite presenation (Bootstrap, Lemma 80.11.6) we conclude that $X \to Y$ is flat and locally of finite presentation by Morphisms of Spaces, Lemma 67.31.5 and Descent on Spaces, Lemma 74.8.2. Finally, consider the cartesian diagram

$\xymatrix{ R \ar[d] \ar[r] & U \times _ Y U \ar[d] \\ X \ar[r] & X \times _ Y X }$

Since the right vertical arrow is surjective, flat, and locally of finite presentation (small detail omitted), we find that $X \to X \times _ Y X$ is an open immersion as the top horizonal arrow has this property by assumption (use Properties of Stacks, Lemma 100.3.3). Thus $X \to Y$ is unramified by Morphisms of Spaces, Lemma 67.38.9. We conclude by Morphisms of Spaces, Lemma 67.39.12. $\square$

Lemma 101.32.2. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $S$. Assume $s, t$ are flat and locally of finite presentation. Let $P \subset R$ be an open subspace such that $(U, P, s|_ P, t|_ P, c|_{P \times _{s, U, t} P})$ is a groupoid in algebraic spaces over $S$. Then

$[U/P] \longrightarrow [U/R]$

is a morphism of algebraic stacks which is representable by algebraic spaces, surjective, and étale.

Proof. Since $P \subset R$ is open, we see that $s|_ P$ and $t|_ P$ are flat and locally of finite presentation. Thus $[U/R]$ and $[U/P]$ are algebraic stacks by Criteria for Representability, Theorem 97.17.2. To see that the morphism is representable by algebraic spaces, it suffices to show that $[U/P] \to [U/R]$ is faithful on fibre categories, see Algebraic Stacks, Lemma 94.15.2. This follows immediately from the fact that $P \to R$ is a monomorphism and the explicit description of quotient stacks given in Groupoids in Spaces, Lemma 78.24.1. Having said this, we know what it means for $[U/P] \to [U/R]$ to be surjective and étale by Algebraic Stacks, Definition 94.10.1. Surjectivity follows for example from Criteria for Representability, Lemma 97.7.3 and the description of objects of quotient stacks (see lemma cited above) over spectra of fields. It remains to prove that our morphism is étale.

To do this it suffices to show that $U \times _{[U/R]} [U/P] \to U$ is étale, see Properties of Stacks, Lemma 100.3.3. By Groupoids in Spaces, Lemma 78.21.2 the fibre product is equal to $[R/P \times _{s, U, t} R]$ with morphism to $U$ induced by $s : R \to U$. The maps $s', t' : P \times _{s, U, t} R \to R$ are given by $s' : (p, r) \mapsto r$ and $t' : (p, r) \mapsto c(p, r)$. Since $P \subset R$ is open we conclude that $(t', s') : P \times _{s, U, t} R \to R \times _{s, U, s} R$ is an open immersion. Thus we may apply Lemma 101.32.1 to conclude. $\square$

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. $\square$

Lemma 101.32.4. 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 the automorphism group of $\mathcal{X}$ at $x$ is finite (Remark 101.19.3). 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 automorphism groups at $u$ and $x$ (Remark 101.19.5),

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. Observe that $G_ x$ is a group scheme by Lemma 101.19.1. 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. Our assumption on $G_ x$ implies that $G_ u$ is finite over $u$. Hence all the assumptions of More on Groupoids in Spaces, Lemma 79.15.12 are satisfied. Hence we can find an elementary étale neighbourhood $(U', u') \to (U, u)$ such that the restriction $R'$ of $R$ to $U'$ is 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 splitting over $u$. Let $P \subset R$ be a 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. By construction $G_ u$ is contained in $P$, hence this morphism defines an isomorphism on automorphism groups at $u$ as desired. $\square$

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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