The Stacks project

77.28 Quotient stacks and change of big site

We suggest skipping this section on a first reading. Pullbacks of stacks are defined in Stacks, Section 8.12.

Lemma 77.28.1. Suppose given big sites $\mathit{Sch}_{fppf}$ and $\mathit{Sch}'_{fppf}$. Assume that $\mathit{Sch}_{fppf}$ is contained in $\mathit{Sch}'_{fppf}$, see Topologies, Section 34.12. Let $S \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_{fppf})$. Let $B, U, R \in \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{fppf})$ be algebraic spaces, and let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. Let $f : (\mathit{Sch}'/S)_{fppf} \to (\mathit{Sch}/S)_{fppf}$ the morphism of sites corresponding to the inclusion functor $u : \mathit{Sch}_{fppf} \to \mathit{Sch}'_{fppf}$. Then we have a canonical equivalence

\[ [f^{-1}U/f^{-1}R] \longrightarrow f^{-1}[U/R] \]

of stacks in groupoids over $(\mathit{Sch}'/S)_{fppf}$.

Proof. Note that $f^{-1}B, f^{-1}U, f^{-1}R \in \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}'/S)_{fppf})$ are algebraic spaces by Spaces, Lemma 64.15.1 and hence $(f^{-1}U, f^{-1}R, f^{-1}s, f^{-1}t, f^{-1}c)$ is a groupoid in algebraic spaces over $f^{-1}B$. Thus the statement makes sense.

The category $u_ p[U/_{\! p}R]$ is the localization of the category $u_{pp}[U/_{\! p}R]$ at right multiplicative system $I$ of morphisms. An object of $u_{pp}[U/_{\! p}R]$ is a triple

\[ (T', \phi : T' \to T, x) \]

where $T' \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}'/S)_{fppf})$, $T \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$, $\phi $ is a morphism of schemes over $S$, and $x : T \to U$ is a morphism of sheaves on $(\mathit{Sch}/S)_{fppf}$. Note that the morphism of schemes $\phi : T' \to T$ is the same thing as a morphism $\phi : T' \to u(T)$, and since $u(T)$ represents $f^{-1}T$ it is the same thing as a morphism $T' \to f^{-1}T$. Moreover, as $f^{-1}$ on algebraic spaces is fully faithful, see Spaces, Lemma 64.15.2, we may think of $x$ as a morphism $x : f^{-1}T \to f^{-1}U$ as well. From now on we will make such identifications without further mention. A morphism

\[ (a, a', \alpha ) : (T'_1, \phi _1 : T'_1 \to T_1, x_1) \longrightarrow (T'_2, \phi _2 : T'_2 \to T_2, x_2) \]

of $u_{pp}[U/_{\! p}R]$ is a commutative diagram

\[ \xymatrix{ & & U \\ T'_1 \ar[d]_{a'} \ar[r]_{\phi _1} & T_1 \ar[d]_ a \ar[ru]^{x_1} \ar[r]_\alpha & R \ar[d]^ t \ar[u]_ s \\ T'_2 \ar[r]^{\phi _2} & T_2 \ar[r]^{x_2} & U } \]

and such a morphism is an element of $I$ if and only if $T'_1 = T'_2$ and $a' = \text{id}$. We define a functor

\[ u_{pp}[U/_{\! p}R] \longrightarrow [f^{-1}U/_{\! p}f^{-1}R] \]

by the rules

\[ (T', \phi : T' \to T, x) \longmapsto (x \circ \phi : T' \to f^{-1}U) \]

on objects and

\[ (a, a', \alpha ) \longmapsto (\alpha \circ \phi _1 : T'_1 \to f^{-1}R) \]

on morphisms as above. It is clear that elements of $I$ are transformed into isomorphisms as $(f^{-1}U, f^{-1}R, f^{-1}s, f^{-1}t, f^{-1}c)$ is a groupoid in algebraic spaces over $f^{-1}B$. Hence this functor factors in a canonical way through a functor

\[ u_ p[U/_{\! p}R] \longrightarrow [f^{-1}U/_{\! p}f^{-1}R] \]

Applying stackification we obtain a functor of stacks

\[ f^{-1}[U/R] \longrightarrow [f^{-1}U/f^{-1}R] \]

over $(\mathit{Sch}'/S)_{fppf}$, as by Stacks, Lemma 8.12.11 the stack $f^{-1}[U/R]$ is the stackification of $u_ p[U/_{\! p}R]$.

At this point we have a morphism of stacks, and to verify that it is an equivalence it suffices to show that it is fully faithful and that objects are locally in the essential image, see Stacks, Lemmas 8.4.7 and 8.4.8. The statement on objects holds as $f^{-1}R$ admits a surjective étale morphism $f^{-1}W \to f^{-1}R$ for some object $W$ of $(\mathit{Sch}/S)_{fppf}$. To show that the functor is “full”, it suffices to show that morphisms are locally in the image of the functor which holds as $f^{-1}U$ admits a surjective étale morphism $f^{-1}W \to f^{-1}U$ for some object $W$ of $(\mathit{Sch}/S)_{fppf}$. We omit the proof that the functor is faithful. $\square$

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