The Stacks project

78.28 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 78.28.1. 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 65.15.1 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 65.15.2, 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. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 04WW. Beware of the difference between the letter 'O' and the digit '0'.