Lemma 34.4.20. Let $S$ be a scheme contained in a big étale site $\mathit{Sch}_{\acute{e}tale}$. A sheaf $\mathcal{F}$ on the big étale site $(\mathit{Sch}/S)_{\acute{e}tale}$ is given by the following data:
for every $T/S \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{\acute{e}tale})$ a sheaf $\mathcal{F}_ T$ on $T_{\acute{e}tale}$,
for every $f : T' \to T$ in $(\mathit{Sch}/S)_{\acute{e}tale}$ a map $c_ f : f_{small}^{-1}\mathcal{F}_ T \to \mathcal{F}_{T'}$.
These data are subject to the following conditions:
given any $f : T' \to T$ and $g : T'' \to T'$ in $(\mathit{Sch}/S)_{\acute{e}tale}$ the composition $c_ g \circ g_{small}^{-1}c_ f$ is equal to $c_{f \circ g}$, and
if $f : T' \to T$ in $(\mathit{Sch}/S)_{\acute{e}tale}$ is étale then $c_ f$ is an isomorphism.
Proof.
This lemma follows from a purely sheaf theoretic statement discussed in Sites, Remark 7.26.7. We also give a direct proof in this case.
Given a sheaf $\mathcal{F}$ on $\mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{\acute{e}tale})$ we set $\mathcal{F}_ T = i_ p^{-1}\mathcal{F}$ where $p : T \to S$ is the structure morphism. Note that $\mathcal{F}_ T(U) = \mathcal{F}(U/S)$ for any $U \to T$ in $T_{\acute{e}tale}$ see Lemma 34.4.13. Hence given $f : T' \to T$ over $S$ and $U \to T$ we get a canonical map $\mathcal{F}_ T(U) = \mathcal{F}(U/S) \to \mathcal{F}(U \times _ T T'/S) = \mathcal{F}_{T'}(U \times _ T T')$ where the middle is the restriction map of $\mathcal{F}$ with respect to the morphism $U \times _ T T' \to U$ over $S$. The collection of these maps are compatible with restrictions, and hence define a map $c'_ f : \mathcal{F}_ T \to f_{small, *}\mathcal{F}_{T'}$ where $u : T_{\acute{e}tale}\to T'_{\acute{e}tale}$ is the base change functor associated to $f$. By adjunction of $f_{small, *}$ (see Sites, Section 7.13) with $f_{small}^{-1}$ this is the same as a map $c_ f : f_{small}^{-1}\mathcal{F}_ T \to \mathcal{F}_{T'}$. It is clear that $c'_{f \circ g}$ is the composition of $c'_ f$ and $f_{small, *}c'_ g$, since composition of restriction maps of $\mathcal{F}$ gives restriction maps, and this gives the desired relationship among $c_ f$, $c_ g$ and $c_{f \circ g}$.
Conversely, given a system $(\mathcal{F}_ T, c_ f)$ as in the lemma we may define a presheaf $\mathcal{F}$ on $\mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{\acute{e}tale})$ by simply setting $\mathcal{F}(T/S) = \mathcal{F}_ T(T)$. As restriction mapping, given $f : T' \to T$ we set for $s \in \mathcal{F}(T)$ the pullback $f^*(s)$ equal to $c_ f(s)$ where we think of $c_ f$ as a map $\mathcal{F}_ T \to f_{small, *}\mathcal{F}_{T'}$ again. The condition on the $c_ f$ guarantees that pullbacks satisfy the required functoriality property. We omit the verification that this is a sheaf. It is clear that the constructions so defined are mutually inverse.
$\square$
Comments (2)
Comment #1237 by Antoine Chambert-Loir on
Comment #1250 by Johan on