The Stacks project

Lemma 96.10.1. Let $S$ be a scheme. Let $\mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids. Assume $\mathcal{X}$ is representable by an algebraic space $F$. Then there exists a continuous and cocontinuous functor $ F_{\acute{e}tale}\to \mathcal{X}_{\acute{e}tale}$ which induces a morphism of ringed sites

\[ \pi _ F : (\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X}) \longrightarrow (F_{\acute{e}tale}, \mathcal{O}_ F) \]

and a morphism of ringed topoi

\[ i_ F : (\mathop{\mathit{Sh}}\nolimits (F_{\acute{e}tale}), \mathcal{O}_ F) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{\acute{e}tale}), \mathcal{O}_\mathcal {X}) \]

such that $\pi _ F \circ i_ F = \text{id}$. Moreover $\pi _{F, *} = i_ F^{-1}$.

Proof. Choose an equivalence $j : \mathcal{S}_ F \to \mathcal{X}$, see Algebraic Stacks, Sections 94.7 and 94.8. An object of $F_{\acute{e}tale}$ is a scheme $U$ together with an étale morphism $\varphi : U \to F$. Then $\varphi $ is an object of $\mathcal{S}_ F$ over $U$. Hence $j(\varphi )$ is an object of $\mathcal{X}$ over $U$. In this way $j$ induces a functor $u : F_{\acute{e}tale}\to \mathcal{X}$. It is clear that $u$ is continuous and cocontinuous for the étale topology on $\mathcal{X}$. Since $j$ is an equivalence, the functor $u$ is fully faithful. Also, fibre products and equalizers exist in $F_{\acute{e}tale}$ and $u$ commutes with them because these are computed on the level of underlying schemes in $F_{\acute{e}tale}$. Thus Sites, Lemmas 7.21.5, 7.21.6, and 7.21.7 apply. In particular $u$ defines a morphism of topoi $i_ F : \mathop{\mathit{Sh}}\nolimits (F_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{\acute{e}tale})$ and there exists a left adjoint $i_{F, !}$ of $i_ F^{-1}$ which commutes with fibre products and equalizers.

We claim that $i_{F, !}$ is exact. If this is true, then we can define $\pi _ F$ by the rules $\pi _ F^{-1} = i_{F, !}$ and $\pi _{F, *} = i_ F^{-1}$ and everything is clear. To prove the claim, note that we already know that $i_{F, !}$ is right exact and preserves fibre products. Hence it suffices to show that $i_{F, !}* = *$ where $*$ indicates the final object in the category of sheaves of sets. Let $U$ be a scheme and let $\varphi : U \to F$ be surjective and étale. Set $R = U \times _ F U$. Then

\[ \xymatrix{ h_ R \ar@<1ex>[r] \ar@<-1ex>[r] & h_ U \ar[r] & {*} } \]

is a coequalizer diagram in $\mathop{\mathit{Sh}}\nolimits (F_{\acute{e}tale})$. Using the right exactness of $i_{F, !}$, using $i_{F, !} = (u_ p\ )^\# $, and using Sites, Lemma 7.5.6 we see that

\[ \xymatrix{ h_{u(R)} \ar@<1ex>[r] \ar@<-1ex>[r] & h_{u(U)} \ar[r] & i_{F, !}{*} } \]

is a coequalizer diagram in $\mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{\acute{e}tale})$. Using that $j$ is an equivalence and that $F = U/R$ it follows that the coequalizer in $\mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{\acute{e}tale})$ of the two maps $h_{u(R)} \to h_{u(U)}$ is $*$. We omit the proof that these morphisms are compatible with structure sheaves. $\square$


Comments (3)

Comment #27 by David Zureick-Brown on

Typo: representably.

Comment #8566 by ZL on

A minor typo: The third line from bottom "is a coequalizer diagram in " should be "is a coequalizer diagram in "

Also a small question: The conclusion does it include that the composition of structure ring maps is isomorphic to identity?

Comment #9145 by on

Thanks for the type; it is fixed here. The answer to your final question is yes (it's also true in the corresponding schemes case as you can more easily verify by tracing through the definitions).

There are also:

  • 4 comment(s) on Section 96.10: Restriction to algebraic spaces

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