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