Section 35.39 (02W4): Descent data in terms of sheaves

35.39 Descent data in terms of sheaves

Here is another way to think about descent data in case of a covering on a site.

Lemma 35.39.1. Let $\tau \in \{ Zariski, fppf, {\acute{e}tale}, smooth, syntomic\} $¹. Let $\mathit{Sch}_\tau $ be a big $\tau $-site. Let $S \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_\tau )$. Let $\{ S_ i \to S\} _{i \in I}$ be a covering in the site $(\mathit{Sch}/S)_\tau $. There is an equivalence of categories

\[ \left\{ \begin{matrix} \text{descent data }(X_ i, \varphi _{ii'})\text{ such that} \\ \text{each }X_ i \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_\tau ) \end{matrix} \right\} \leftrightarrow \left\{ \begin{matrix} \text{sheaves }F\text{ on }(\mathit{Sch}/S)_\tau \text{ such that} \\ \text{each }h_{S_ i} \times F\text{ is representable} \end{matrix} \right\} . \]

Moreover,

the objects representing $h_{S_ i} \times F$ on the right hand side correspond to the schemes $X_ i$ on the left hand side, and
the sheaf $F$ is representable if and only if the corresponding descent datum $(X_ i, \varphi _{ii'})$ is effective.

Proof. We have seen in Section 35.13 that representable presheaves are sheaves on the site $(\mathit{Sch}/S)_\tau $. Moreover, the Yoneda lemma (Categories, Lemma 4.3.5) guarantees that maps between representable sheaves correspond one to one with maps between the representing objects. We will use these remarks without further mention during the proof.

Let us construct the functor from right to left. Let $F$ be a sheaf on $(\mathit{Sch}/S)_\tau $ such that each $h_{S_ i} \times F$ is representable. In this case let $X_ i$ be a representing object in $(\mathit{Sch}/S)_\tau $. It comes equipped with a morphism $X_ i \to S_ i$. Then both $X_ i \times _ S S_{i'}$ and $S_ i \times _ S X_{i'}$ represent the sheaf $h_{S_ i} \times F \times h_{S_{i'}}$ and hence we obtain an isomorphism

\[ \varphi _{ii'} : X_ i \times _ S S_{i'} \to S_ i \times _ S X_{i'} \]

It is straightforward to see that the maps $\varphi _{ii'}$ are morphisms over $S_ i \times _ S S_{i'}$ and satisfy the cocycle condition. The functor from right to left is given by this construction $F \mapsto (X_ i, \varphi _{ii'})$.

Let us construct a functor from left to right. For each $i$ denote $F_ i$ the sheaf $h_{X_ i}$. The isomorphisms $\varphi _{ii'}$ give isomorphisms

\[ \varphi _{ii'} : F_ i \times h_{S_{i'}} \longrightarrow h_{S_ i} \times F_{i'} \]

over $h_{S_ i} \times h_{S_{i'}}$. Set $F$ equal to the coequalizer in the following diagram

\[ \xymatrix{ \coprod _{i, i'} F_ i \times h_{S_{i'}} \ar@<1ex>[rr]^-{\text{pr}_0} \ar@<-1ex>[rr]_-{\text{pr}_1 \circ \varphi _{ii'}} & & \coprod _ i F_ i \ar[r] & F } \]

The cocycle condition guarantees that $h_{S_ i} \times F$ is isomorphic to $F_ i$ and hence representable. The functor from left to right is given by this construction $(X_ i, \varphi _{ii'}) \mapsto F$.

We omit the verification that these constructions are mutually quasi-inverse functors. The final statements (1) and (2) follow from the constructions. $\square$

Remark 35.39.2. In the statement of Lemma 35.39.1 the condition that $h_{S_ i} \times F$ is representable is equivalent to the condition that the restriction of $F$ to $(\mathit{Sch}/S_ i)_\tau $ is representable.

[1] The fact that fpqc is missing is not a typo. See discussion in Topologies, Section 34.9.

Comments (7)

Comment #493 by Mark on March 25, 2014 at 12:25

This lemma isn't displaying properly for me; it looks okay in the PDF version. Maybe it has to do with the footnote; or maybe it's just me?

Comment #494 by Johan on March 25, 2014 at 18:47

Thanks for pointing this out. I think you are right that it is the footnote. Let me try to fix it.

Comment #574 by Keenan Kidwell on May 15, 2014 at 20:15

This is perhaps a dumb question, but it seems to me that the statement (2) in 02W5 hinges on the fact that, if $X/S$ yields a descent to $S$ of $\{(X_i,\varphi_{ii^\prime})\}$ , then the universal families $\eta_i\in F(X_i)$ (suppressing the morphisms $\pi_i:X_i\to X$ which are part of the isomorphism $h_{X_i}\simeq F\times h_{S_i}$ ) have to glue to $\eta\in F(X)$ (then we get a morphism $h_X\to F$ which pulls back to an isomorphism on the covering by the $S_i$ , and hence is an isomorphism of sheaves). More precisely, we have the $\tau$ -covering $(X_i\to X)$ by base change of $(S_i\to S)$ along $X\to S$ , and since $F$ is a $\tau$ -sheaf, we get $F(X)\to \prod_i F(X_i)\rightrightarrow\prod_{i,j}F(X_i\times_XX_j)$ , and we need to prove that $(\eta_i)$ is sent to the same thing by the two arrows to $\prod_{i,j}F(X_i\times_XX_j)$ . Is this clear from the compatibility between the descent data on the $X_i$ with the canonical descent datum on the $X\times_SS_i$ ?

Comment #575 by Keenan Kidwell on May 15, 2014 at 20:18

The $\pi_i$ in my comment above should go from $X_i$ to $S_i$ , and the messed up LaTeX should be the sequence $F(X)\to \prod_i F(X_i)\rightrightarrows\prod_{i,j}F(X_i\times_XX_j)$ . I just realized the eye icon allows me to check my LaTeX. Sorry about that.

Comment #591 by Johan on May 23, 2014 at 19:48

Hi! I agree with what you wrote. Another way to make your point might be that $h_X$ is the coequalizer of the diagram $\prod h_{X_i \times_S S_j} \rightrightarrows \prod h_{X_i}$ since after all that is the way we go from the left hand side to the right hand side and this just follows from the fact that $X_i \times_S S_j = X_i \times_X X_j$ and that $\{X_i \to X\}$ is a covering.

But maybe you are talking about the converse, i.e., we assume that $X$ represents $F$ and now we want to show that $\{X_i, \varphi_{ij}\}$ is effective?

Anyway, I will gladly take any edits you suggest for cleaning up the proof.

Comment #6324 by fpqcquestions on July 09, 2021 at 01:27

Can I ask what specifically we are afraid of about letting \tau = fpqc? Is it simply that due to set theoretic issues we can't define the fpqc site? Is it because we are uncertain if the right hand side would be a "big" category? There are still interesting applications of (2) in that context which would be worth having here (checking representability of an fpqc sheaf by quasi-affines fpqc locally, etc.) unless something goes deeply wrong in the proof.

Comment #6326 by Johan on July 09, 2021 at 11:32

@#6324. We're not afraid! The statement as given does not make sense for $\tau = fpqc$ with notation and definitions as in the Stacks project. The other questions are too wide ranging to answer in a comment. If you have a precise and specific alternative statement of this lemma for $\tau = fpqc$ you would like to see added here, then please let us know.

The Stacks project

35.39 Descent data in terms of sheaves

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