The Stacks project

100.28 Gerbes

An important type of algebraic stack are the stacks of the form $[B/G]$ where $B$ is an algebraic space and $G$ is a flat and locally finitely presented group algebraic space over $B$ (acting trivially on $B$), see Criteria for Representability, Lemma 96.18.3. It turns out that an algebraic stack is a gerbe when it locally in the fppf topology is of this form, see Lemma 100.28.7. In this section we briefly discuss this notion and the corresponding relative notion.

Definition 100.28.1. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. We say $\mathcal{X}$ is a gerbe over $\mathcal{Y}$ if $\mathcal{X}$ is a gerbe over $\mathcal{Y}$ as stacks in groupoids over $(\mathit{Sch}/S)_{fppf}$, see Stacks, Definition 8.11.4. We say an algebraic stack $\mathcal{X}$ is a gerbe if there exists a morphism $\mathcal{X} \to X$ where $X$ is an algebraic space which turns $\mathcal{X}$ into a gerbe over $X$.

The condition that $\mathcal{X}$ be a gerbe over $\mathcal{Y}$ is defined purely in terms of the topology and category theory underlying the given algebraic stacks; but as we will see later this condition has geometric consequences. For example it implies that $\mathcal{X} \to \mathcal{Y}$ is surjective, flat, and locally of finite presentation, see Lemma 100.28.8. The absolute notion is trickier to parse, because it may not be at first clear that $X$ is well determined. Actually, it is.

Lemma 100.28.2. Let $\mathcal{X}$ be an algebraic stack. If $\mathcal{X}$ is a gerbe, then the sheafification of the presheaf

\[ (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}, \quad U \mapsto \mathop{\mathrm{Ob}}\nolimits (\mathcal{X}_ U)/\! \! \cong \]

is an algebraic space and $\mathcal{X}$ is a gerbe over it.

Proof. (In this proof the abuse of language introduced in Section 100.2 really pays off.) Choose a morphism $\pi : \mathcal{X} \to X$ where $X$ is an algebraic space which turns $\mathcal{X}$ into a gerbe over $X$. It suffices to prove that $X$ is the sheafification of the presheaf $\mathcal{F}$ displayed in the lemma. It is clear that there is a map $c : \mathcal{F} \to X$. We will use Stacks, Lemma 8.11.3 properties (2)(a) and (2)(b) to see that the map $c^\# : \mathcal{F}^\# \to X$ is surjective and injective, hence an isomorphism, see Sites, Lemma 7.11.2. Surjective: Let $T$ be a scheme and let $f : T \to X$. By property (2)(a) there exists an fppf covering $\{ h_ i : T_ i \to T\} $ and morphisms $x_ i : T_ i \to \mathcal{X}$ such that $f \circ h_ i$ corresponds to $\pi \circ x_ i$. Hence we see that $f|_{T_ i}$ is in the image of $c$. Injective: Let $T$ be a scheme and let $x, y : T \to \mathcal{X}$ be morphisms such that $c \circ x = c \circ y$. By (2)(b) we can find a covering $\{ T_ i \to T\} $ and morphisms $x|_{T_ i} \to y|_{T_ i}$ in the fibre category $\mathcal{X}_{T_ i}$. Hence the restrictions $x|_{T_ i}, y|_{T_ i}$ are equal in $\mathcal{F}(T_ i)$. This proves that $x, y$ give the same section of $\mathcal{F}^\# $ over $T$ as desired. $\square$

Lemma 100.28.3. Let

\[ \xymatrix{ \mathcal{X}' \ar[r] \ar[d] & \mathcal{X} \ar[d] \\ \mathcal{Y}' \ar[r] & \mathcal{Y} } \]

be a fibre product of algebraic stacks. If $\mathcal{X}$ is a gerbe over $\mathcal{Y}$, then $\mathcal{X}'$ is a gerbe over $\mathcal{Y}'$.

Proof. Immediate from the definitions and Stacks, Lemma 8.11.5. $\square$

Lemma 100.28.4. Let $\mathcal{X} \to \mathcal{Y}$ and $\mathcal{Y} \to \mathcal{Z}$ be morphisms of algebraic stacks. If $\mathcal{X}$ is a gerbe over $\mathcal{Y}$ and $\mathcal{Y}$ is a gerbe over $\mathcal{Z}$, then $\mathcal{X}$ is a gerbe over $\mathcal{Z}$.

Proof. Immediate from Stacks, Lemma 8.11.6. $\square$

Lemma 100.28.5. Let

\[ \xymatrix{ \mathcal{X}' \ar[r] \ar[d] & \mathcal{X} \ar[d] \\ \mathcal{Y}' \ar[r] & \mathcal{Y} } \]

be a fibre product of algebraic stacks. If $\mathcal{Y}' \to \mathcal{Y}$ is surjective, flat, and locally of finite presentation and $\mathcal{X}'$ is a gerbe over $\mathcal{Y}'$, then $\mathcal{X}$ is a gerbe over $\mathcal{Y}$.

Proof. Follows immediately from Lemma 100.27.13 and Stacks, Lemma 8.11.7. $\square$

Lemma 100.28.6. Let $\pi : \mathcal{X} \to U$ be a morphism from an algebraic stack to an algebraic space and let $x : U \to \mathcal{X}$ be a section of $\pi $. Set $G = \mathit{Isom}_\mathcal {X}(x, x)$, see Definition 100.5.3. If $\mathcal{X}$ is a gerbe over $U$, then

  1. there is a canonical equivalence of stacks in groupoids

    \[ x_{can} : [U/G] \longrightarrow \mathcal{X}. \]

    where $[U/G]$ is the quotient stack for the trivial action of $G$ on $U$,

  2. $G \to U$ is flat and locally of finite presentation, and

  3. $U \to \mathcal{X}$ is surjective, flat, and locally of finite presentation.

Proof. Set $R = U \times _{x, \mathcal{X}, x} U$. The morphism $R \to U \times U$ factors through the diagonal $\Delta _ U : U \to U \times U$ as it factors through $U \times _ U U = U$. Hence $R = G$ because

\begin{align*} G & = \mathit{Isom}_\mathcal {X}(x, x) \\ & = U \times _{x, \mathcal{X}} \mathcal{I}_\mathcal {X} \\ & = U \times _{x, \mathcal{X}} (\mathcal{X} \times _{\Delta , \mathcal{X} \times _ S \mathcal{X}, \Delta } \mathcal{X}) \\ & = (U \times _{x, \mathcal{X}, x} U) \times _{U \times U, \Delta _ U} U \\ & = R \times _{U \times U, \Delta _ U} U \\ & = R \end{align*}

for the fourth equality use Categories, Lemma 4.31.12. Let $t, s : R \to U$ be the projections. The composition law $c : R \times _{s, U, t} R \to R$ constructed on $R$ in Algebraic Stacks, Lemma 93.16.1 agrees with the group law on $G$ (proof omitted). Thus Algebraic Stacks, Lemma 93.16.1 shows we obtain a canonical fully faithful $1$-morphism

\[ x_{can} : [U/G] \longrightarrow \mathcal{X} \]

of stacks in groupoids over $(\mathit{Sch}/S)_{fppf}$. To see that it is an equivalence it suffices to show that it is essentially surjective. To do this it suffices to show that any object of $\mathcal{X}$ over a scheme $T$ comes fppf locally from $x$ via a morphism $T \to U$, see Stacks, Lemma 8.4.8. However, this follows the condition that $\pi $ turns $\mathcal{X}$ into a gerbe over $U$, see property (2)(a) of Stacks, Lemma 8.11.3.

By Criteria for Representability, Lemma 96.18.3 we conclude that $G \to U$ is flat and locally of finite presentation. Finally, $U \to \mathcal{X}$ is surjective, flat, and locally of finite presentation by Criteria for Representability, Lemma 96.17.1. $\square$

Lemma 100.28.7. Let $\pi : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. The following are equivalent

  1. $\mathcal{X}$ is a gerbe over $\mathcal{Y}$, and

  2. there exists an algebraic space $U$, a group algebraic space $G$ flat and locally of finite presentation over $U$, and a surjective, flat, and locally finitely presented morphism $U \to \mathcal{Y}$ such that $\mathcal{X} \times _\mathcal {Y} U \cong [U/G]$ over $U$.

Proof. Assume (2). By Lemma 100.28.5 to prove (1) it suffices to show that $[U/G]$ is a gerbe over $U$. This is immediate from Groupoids in Spaces, Lemma 77.27.2.

Assume (1). Any base change of $\pi $ is a gerbe, see Lemma 100.28.3. As a first step we choose a scheme $V$ and a surjective smooth morphism $V \to \mathcal{Y}$. Thus we may assume that $\pi : \mathcal{X} \to V$ is a gerbe over a scheme. This means that there exists an fppf covering $\{ V_ i \to V\} $ such that the fibre category $\mathcal{X}_{V_ i}$ is nonempty, see Stacks, Lemma 8.11.3 (2)(a). Note that $U = \coprod V_ i \to V$ is surjective, flat, and locally of finite presentation. Hence we may replace $V$ by $U$ and assume that $\pi : \mathcal{X} \to U$ is a gerbe over a scheme $U$ and that there exists an object $x$ of $\mathcal{X}$ over $U$. By Lemma 100.28.6 we see that $\mathcal{X} = [U/G]$ over $U$ for some flat and locally finitely presented group algebraic space $G$ over $U$. $\square$

Lemma 100.28.8. Let $\pi : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. If $\mathcal{X}$ is a gerbe over $\mathcal{Y}$, then $\pi $ is surjective, flat, and locally of finite presentation.

Proof. By Properties of Stacks, Lemma 99.5.4 and Lemmas 100.25.4 and 100.27.11 it suffices to prove to the lemma after replacing $\pi $ by a base change with a surjective, flat, locally finitely presented morphism $\mathcal{Y}' \to \mathcal{Y}$. By Lemma 100.28.7 we may assume $\mathcal{Y} = U$ is an algebraic space and $\mathcal{X} = [U/G]$ over $U$. Then $U \to [U/G]$ is surjective, flat, and locally of finite presentation, see Lemma 100.28.6. This implies that $\pi $ is surjective, flat, and locally of finite presentation by Properties of Stacks, Lemma 99.5.5 and Lemmas 100.25.5 and 100.27.12. $\square$

Proposition 100.28.9. Let $\mathcal{X}$ be an algebraic stack. The following are equivalent

  1. $\mathcal{X}$ is a gerbe, and

  2. $\mathcal{I}_\mathcal {X} \to \mathcal{X}$ is flat and locally of finite presentation.

Proof. Assume (1). Choose a morphism $\mathcal{X} \to X$ into an algebraic space $X$ which turns $\mathcal{X}$ into a gerbe over $X$. Let $X' \to X$ be a surjective, flat, locally finitely presented morphism and set $\mathcal{X}' = X' \times _ X \mathcal{X}$. Note that $\mathcal{X}'$ is a gerbe over $X'$ by Lemma 100.28.3. Then both squares in

\[ \xymatrix{ \mathcal{I}_{\mathcal{X}'} \ar[r] \ar[d] & \mathcal{X}' \ar[r] \ar[d] & X' \ar[d] \\ \mathcal{I}_\mathcal {X} \ar[r] & \mathcal{X} \ar[r] & X } \]

are fibre product squares, see Lemma 100.5.5. Hence to prove $\mathcal{I}_\mathcal {X} \to \mathcal{X}$ is flat and locally of finite presentation it suffices to do so after such a base change by Lemmas 100.25.4 and 100.27.11. Thus we can apply Lemma 100.28.7 to assume that $\mathcal{X} = [U/G]$. By Lemma 100.28.6 we see $G$ is flat and locally of finite presentation over $U$ and that $x : U \to [U/G]$ is surjective, flat, and locally of finite presentation. Moreover, the pullback of $\mathcal{I}_\mathcal {X}$ by $x$ is $G$ and we conclude that (2) holds by descent again, i.e., by Lemmas 100.25.4 and 100.27.11.

Conversely, assume (2). Choose a smooth presentation $\mathcal{X} = [U/R]$, see Algebraic Stacks, Section 93.16. Denote $G \to U$ the stabilizer group algebraic space of the groupoid $(U, R, s, t, c, e, i)$, see Groupoids in Spaces, Definition 77.16.2. By Lemma 100.5.7 we see that $G \to U$ is flat and locally of finite presentation as a base change of $\mathcal{I}_\mathcal {X} \to \mathcal{X}$, see Lemmas 100.25.3 and 100.27.3. Consider the following action

\[ a : G \times _{U, t} R \to R, \quad (g, r) \mapsto c(g, r) \]

of $G$ on $R$. This action is free on $T$-valued points for any scheme $T$ as $R$ is a groupoid. Hence $R' = R/G$ is an algebraic space and the quotient morphism $\pi : R \to R'$ is surjective, flat, and locally of finite presentation by Bootstrap, Lemma 79.11.7. The projections $s, t : R \to U$ are $G$-invariant, hence we obtain morphisms $s' , t' : R' \to U$ such that $s = s' \circ \pi $ and $t = t' \circ \pi $. Since $s, t : R \to U$ are flat and locally of finite presentation we conclude that $s', t'$ are flat and locally of finite presentation, see Morphisms of Spaces, Lemmas 66.31.5 and Descent on Spaces, Lemma 73.16.1. Consider the morphism

\[ j' = (t', s') : R' \longrightarrow U \times U. \]

We claim this is a monomorphism. Namely, suppose that $T$ is a scheme and that $a, b : T \to R'$ are morphisms which have the same image in $U \times U$. By definition of the quotient $R' = R/G$ there exists an fppf covering $\{ h_ j : T_ j \to T\} $ such that $a \circ h_ j = \pi \circ a_ j$ and $b \circ h_ j = \pi \circ b_ j$ for some morphisms $a_ j, b_ j : T_ j \to R$. Since $a_ j, b_ j$ have the same image in $U \times U$ we see that $g_ j = c(a_ j, i(b_ j))$ is a $T_ j$-valued point of $G$ such that $c(g_ j, b_ j) = a_ j$. In other words, $a_ j$ and $b_ j$ have the same image in $R'$ and the claim is proved. Since $j : R \to U \times U$ is a pre-equivalence relation (see Groupoids in Spaces, Lemma 77.11.2) and $R \to R'$ is surjective (as a map of sheaves) we see that $j' : R' \to U \times U$ is an equivalence relation. Hence Bootstrap, Theorem 79.10.1 shows that $X = U/R'$ is an algebraic space. Finally, we claim that the morphism

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

turns $\mathcal{X}$ into a gerbe over $X$. This follows from Groupoids in Spaces, Lemma 77.27.1 as $R \to R'$ is surjective, flat, and locally of finite presentation (if needed use Bootstrap, Lemma 79.4.6 to see this implies the required hypothesis). $\square$

Lemma 100.28.10. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks which makes $\mathcal{X}$ a gerbe over $\mathcal{Y}$. Then

  1. $\mathcal{I}_{\mathcal{X}/\mathcal{Y}} \to \mathcal{X}$ is flat and locally of finite presentation,

  2. $\mathcal{X} \to \mathcal{X} \times _\mathcal {Y} \mathcal{X}$ is surjective, flat, and locally of finite presentation,

  3. given algebraic spaces $T_ i$, $i = 1, 2$ and morphisms $x_ i : T_ i \to \mathcal{X}$, with $y_ i = f \circ x_ i$ the morphism

    \[ T_1 \times _{x_1, \mathcal{X}, x_2} T_2 \longrightarrow T_1 \times _{y_1, \mathcal{Y}, y_2} T_2 \]

    is surjective, flat, and locally of finite presentation,

  4. given an algebraic space $T$ and morphisms $x_ i : T \to \mathcal{X}$, $i = 1, 2$, with $y_ i = f \circ x_ i$ the morphism

    \[ \mathit{Isom}_\mathcal {X}(x_1, x_2) \longrightarrow \mathit{Isom}_\mathcal {Y}(y_1, y_2) \]

    is surjective, flat, and locally of finite presentation.

Proof. Proof of (1). Choose a scheme $Y$ and a surjective smooth morphism $Y \to \mathcal{Y}$. Set $\mathcal{X}' = \mathcal{X} \times _\mathcal {Y} Y$. By Lemma 100.5.5 we obtain cartesian squares

\[ \xymatrix{ \mathcal{I}_{\mathcal{X}'} \ar[r] \ar[d] & \mathcal{X}' \ar[r] \ar[d] & Y \ar[d] \\ \mathcal{I}_{\mathcal{X}/\mathcal{Y}} \ar[r] & \mathcal{X} \ar[r] & \mathcal{Y} } \]

By Lemmas 100.25.4 and 100.27.11 it suffices to prove that $\mathcal{I}_{\mathcal{X}'} \to \mathcal{X}'$ is flat and locally of finite presentation. This follows from Proposition 100.28.9 (because $\mathcal{X}'$ is a gerbe over $Y$ by Lemma 100.28.3).

Proof of (2). With notation as above, note that we may assume that $\mathcal{X}' = [Y/G]$ for some group algebraic space $G$ flat and locally of finite presentation over $Y$, see Lemma 100.28.7. The base change of the morphism $\Delta : \mathcal{X} \to \mathcal{X} \times _\mathcal {Y} \mathcal{X}$ over $\mathcal{Y}$ by the morphism $Y \to \mathcal{Y}$ is the morphism $\Delta ' : \mathcal{X}' \to \mathcal{X}' \times _ Y \mathcal{X}'$. Hence it suffices to show that $\Delta '$ is surjective, flat, and locally of finite presentation (see Lemmas 100.25.4 and 100.27.11). In other words, we have to show that

\[ [Y/G] \longrightarrow [Y/G \times _ Y G] \]

is surjective, flat, and locally of finite presentation. This is true because the base change by the surjective, flat, locally finitely presented morphism $Y \to [Y/G \times _ Y G]$ is the morphism $G \to Y$.

Proof of (3). Observe that the diagram

\[ \xymatrix{ T_1 \times _{x_1, \mathcal{X}, x_2} T_2 \ar[d] \ar[r] & T_1 \times _{y_1, \mathcal{Y}, y_2} T_2 \ar[d] \\ \mathcal{X} \ar[r] & \mathcal{X} \times _\mathcal {Y} \mathcal{X} } \]

is cartesian. Hence (3) follows from (2).

Proof of (4). This is true because

\[ \mathit{Isom}_\mathcal {X}(x_1, x_2) = (T \times _{x_1, \mathcal{X}, x_2} T) \times _{T \times T, \Delta _ T} T \]

hence the morphism in (4) is a base change of the morphism in (3). $\square$

Proposition 100.28.11. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. The following are equivalent

  1. $\mathcal{X}$ is a gerbe over $\mathcal{Y}$, and

  2. $f : \mathcal{X} \to \mathcal{Y}$ and $\Delta : \mathcal{X} \to \mathcal{X} \times _\mathcal {Y} \mathcal{X}$ are surjective, flat, and locally of finite presentation.

Proof. The implication (1) $\Rightarrow $ (2) follows from Lemmas 100.28.8 and 100.28.10.

Assume (2). It suffices to prove (1) for the base change of $f$ by a surjective, flat, and locally finitely presented morphism $\mathcal{Y}' \to \mathcal{Y}$, see Lemma 100.28.5 (note that the base change of the diagonal of $f$ is the diagonal of the base change). Thus we may assume $\mathcal{Y}$ is a scheme $Y$. In this case $\mathcal{I}_\mathcal {X} \to \mathcal{X}$ is a base change of $\Delta $ and we conclude that $\mathcal{X}$ is a gerbe by Proposition 100.28.9. We still have to show that $\mathcal{X}$ is a gerbe over $Y$. Let $\mathcal{X} \to X$ be the morphism of Lemma 100.28.2 turning $\mathcal{X}$ into a gerbe over the algebraic space $X$ classifying isomorphism classes of objects of $\mathcal{X}$. It is clear that $f : \mathcal{X} \to Y$ factors as $\mathcal{X} \to X \to Y$. Since $f$ is surjective, flat, and locally of finite presentation, we conclude that $X \to Y$ is surjective as a map of fppf sheaves (for example use Lemma 100.27.13). On the other hand, $X \to Y$ is injective too: for any scheme $T$ and any two $T$-valued points $x_1, x_2$ of $X$ which map to the same point of $Y$, we can first fppf locally on $T$ lift $x_1, x_2$ to objects $\xi _1, \xi _2$ of $\mathcal{X}$ over $T$ and second deduce that $\xi _1$ and $\xi _2$ are fppf locally isomorphic by our assumption that $\Delta : \mathcal{X} \to \mathcal{X} \times _ Y \mathcal{X}$ is surjective, flat, and locally of finite presentation. Whence $x_1 = x_2$ by construction of $X$. Thus $X = Y$ and the proof is complete. $\square$

At this point we have developed enough machinery to prove that residual gerbes (when they exist) are gerbes.

Lemma 100.28.12. Let $\mathcal{Z}$ be a reduced, locally Noetherian algebraic stack such that $|\mathcal{Z}|$ is a singleton. Then $\mathcal{Z}$ is a gerbe over a reduced, locally Noetherian algebraic space $Z$ with $|Z|$ a singleton.

Proof. By Properties of Stacks, Lemma 99.11.3 there exists a surjective, flat, locally finitely presented morphism $\mathop{\mathrm{Spec}}(k) \to \mathcal{Z}$ where $k$ is a field. Then $\mathcal{I}_ Z \times _\mathcal {Z} \mathop{\mathrm{Spec}}(k) \to \mathop{\mathrm{Spec}}(k)$ is representable by algebraic spaces and locally of finite type (as a base change of $\mathcal{I}_\mathcal {Z} \to \mathcal{Z}$, see Lemmas 100.5.1 and 100.17.3). Therefore it is locally of finite presentation, see Morphisms of Spaces, Lemma 66.28.7. Of course it is also flat as $k$ is a field. Hence we may apply Lemmas 100.25.4 and 100.27.11 to see that $\mathcal{I}_\mathcal {Z} \to \mathcal{Z}$ is flat and locally of finite presentation. We conclude that $\mathcal{Z}$ is a gerbe by Proposition 100.28.9. Let $\pi : \mathcal{Z} \to Z$ be a morphism to an algebraic space such that $\mathcal{Z}$ is a gerbe over $Z$. Then $\pi $ is surjective, flat, and locally of finite presentation by Lemma 100.28.8. Hence $\mathop{\mathrm{Spec}}(k) \to Z$ is surjective, flat, and locally of finite presentation as a composition, see Properties of Stacks, Lemma 99.5.2 and Lemmas 100.25.2 and 100.27.2. Hence by Properties of Stacks, Lemma 99.11.3 we see that $|Z|$ is a singleton and that $Z$ is locally Noetherian and reduced. $\square$

Lemma 100.28.13. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. If $\mathcal{X}$ is a gerbe over $\mathcal{Y}$ then $f$ is a universal homeomorphism.

Proof. By Lemma 100.28.3 the assumption on $f$ is preserved under base change. Hence it suffices to show that the map $|\mathcal{X}| \to |\mathcal{Y}|$ is a homeomorphism of topological spaces. Let $k$ be a field and let $y$ be an object of $\mathcal{Y}$ over $\mathop{\mathrm{Spec}}(k)$. By Stacks, Lemma 8.11.3 property (2)(a) there exists an fppf covering $\{ T_ i \to \mathop{\mathrm{Spec}}(k)\} $ and objects $x_ i$ of $\mathcal{X}$ over $T_ i$ with $f(x_ i) \cong y|_{T_ i}$. Choose an $i$ such that $T_ i \not= \emptyset $. Choose a morphism $\mathop{\mathrm{Spec}}(K) \to T_ i$ for some field $K$. Then $k \subset K$ and $x_ i|_ K$ is an object of $\mathcal{X}$ lying over $y|_ K$. Thus we see that $|\mathcal{Y}| \to |\mathcal{X}|$. is surjective. The map $|\mathcal{Y}| \to |\mathcal{X}|$ is also injective. Namely, if $x, x'$ are objects of $\mathcal{X}$ over $\mathop{\mathrm{Spec}}(k)$ whose images $f(x), f(x')$ become isomorphic (over an extension) in $\mathcal{Y}$, then Stacks, Lemma 8.11.3 property (2)(b) guarantees the existence of an extension of $k$ over which $x$ and $x'$ become isomorphic (details omitted). Hence $|\mathcal{X}| \to |\mathcal{Y}|$ is continuous and bijective and it suffices to show that it is also open. This follows from Lemmas 100.28.8 and 100.27.15. $\square$

Lemma 100.28.14. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks such that $\mathcal{X}$ is a gerbe over $\mathcal{Y}$. If $\Delta _\mathcal {X}$ is quasi-compact, so is $\Delta _\mathcal {Y}$.

Proof. Consider the diagram

\[ \xymatrix{ \mathcal{X} \ar[r] & \mathcal{X} \times _\mathcal {Y} \mathcal{X} \ar[r] \ar[d] & \mathcal{X} \times \mathcal{X} \ar[d] \\ & \mathcal{Y} \ar[r] & \mathcal{Y} \times \mathcal{Y} } \]

By Proposition 100.28.11 we find that the arrow on the top left is surjective. Since the composition of the top horizontal arrows is quasi-compact, we conclude that the top right arrow is quasi-compact by Lemma 100.7.6. The square is cartesian and the right vertical arrow is surjective, flat, and locally of finite presentation. Thus we conclude by Lemma 100.27.16. $\square$

The following lemma tells us that residual gerbes exist for all points on any algebraic stack which is a gerbe.

Lemma 100.28.15. Let $\mathcal{X}$ be an algebraic stack. If $\mathcal{X}$ is a gerbe then for every $x \in |\mathcal{X}|$ the residual gerbe of $\mathcal{X}$ at $x$ exists.

Proof. Let $\pi : \mathcal{X} \to X$ be a morphism from $\mathcal{X}$ into an algebraic space $X$ which turns $\mathcal{X}$ into a gerbe over $X$. Let $Z_ x \to X$ be the residual space of $X$ at $x$, see Decent Spaces, Definition 67.13.6. Let $\mathcal{Z} = \mathcal{X} \times _ X Z_ x$. By Lemma 100.28.3 the algebraic stack $\mathcal{Z}$ is a gerbe over $Z_ x$. Hence $|\mathcal{Z}| = |Z_ x|$ (Lemma 100.28.13) is a singleton. Since $\mathcal{Z} \to Z_ x$ is locally of finite presentation as a base change of $\pi $ (see Lemmas 100.28.8 and 100.27.3) we see that $\mathcal{Z}$ is locally Noetherian, see Lemma 100.17.5. Thus the residual gerbe $\mathcal{Z}_ x$ of $\mathcal{X}$ at $x$ exists and is equal to $\mathcal{Z}_ x = \mathcal{Z}_{red}$ the reduction of the algebraic stack $\mathcal{Z}$. Namely, we have seen above that $|\mathcal{Z}_{red}|$ is a singleton mapping to $x \in |\mathcal{X}|$, it is reduced by construction, and it is locally Noetherian (as the reduction of a locally Noetherian algebraic stack is locally Noetherian, details omitted). $\square$


Comments (2)

Comment #3224 by typo on

The reference to Lemma 92.27.8 in th the fifth line of the first paragraph seems wrong.


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