The Stacks project

Lemma 66.18.14. With $S$, $\varphi : U \to X$, and $(U, R, s, t, c, e, i)$ as above. For any sheaf $\mathcal{F}$ on $X_{\acute{e}tale}$ the sheaf1 $\mathcal{G} = \varphi ^{-1}\mathcal{F}$ comes equipped with a canonical isomorphism

\[ \alpha : t^{-1}\mathcal{G} \longrightarrow s^{-1}\mathcal{G} \]

such that the diagram

\[ \xymatrix{ & \text{pr}_1^{-1}t^{-1}\mathcal{G} \ar[r]_-{\text{pr}_1^{-1}\alpha } & \text{pr}_1^{-1}s^{-1}\mathcal{G} \ar@{=}[rd] & \\ \text{pr}_0^{-1}s^{-1}\mathcal{G} \ar@{=}[ru] & & & c^{-1}s^{-1}\mathcal{G} \\ & \text{pr}_0^{-1}t^{-1}\mathcal{G} \ar[lu]^{\text{pr}_0^{-1}\alpha } \ar@{=}[r] & c^{-1}t^{-1}\mathcal{G} \ar[ru]_{c^{-1}\alpha } } \]

is a commutative. The functor $\mathcal{F} \mapsto (\mathcal{G}, \alpha )$ defines an equivalence of categories between sheaves on $X_{\acute{e}tale}$ and pairs $(\mathcal{G}, \alpha )$ as above.

First proof of Lemma 66.18.14. Let $\mathcal{C} = X_{spaces, {\acute{e}tale}}$. By Lemma 66.18.11 and its proof we have $U_{spaces, {\acute{e}tale}} = \mathcal{C}/U$ and the pullback functor $\varphi ^{-1}$ is just the restriction functor. Moreover, $\{ U \to X\} $ is a covering of the site $\mathcal{C}$ and $R = U \times _ X U$. The isomorphism $\alpha $ is just the canonical identification

\[ \left(\mathcal{F}|_{\mathcal{C}/U}\right)|_{\mathcal{C}/U \times _ X U} = \left(\mathcal{F}|_{\mathcal{C}/U}\right)|_{\mathcal{C}/U \times _ X U} \]

and the commutativity of the diagram is the cocycle condition for glueing data. Hence this lemma is a special case of glueing of sheaves, see Sites, Section 7.26. $\square$

Second proof of Lemma 66.18.14. The existence of $\alpha $ comes from the fact that $\varphi \circ t = \varphi \circ s$ and that pullback is functorial in the morphism, see Lemma 66.18.8. In exactly the same way, i.e., by functoriality of pullback, we see that the isomorphism $\alpha $ fits into the commutative diagram. The construction $\mathcal{F} \mapsto (\varphi ^{-1}\mathcal{F}, \alpha )$ is clearly functorial in the sheaf $\mathcal{F}$. Hence we obtain the functor.

Conversely, suppose that $(\mathcal{G}, \alpha )$ is a pair. Let $V \to X$ be an object of $X_{\acute{e}tale}$. In this case the morphism $V' = U \times _ X V \to V$ is a surjective étale morphism of schemes, and hence $\{ V' \to V\} $ is an étale covering of $V$. Set $\mathcal{G}' = (V' \to V)^{-1}\mathcal{G}$. Since $R = U \times _ X U$ with $t = \text{pr}_0$ and $s = \text{pr}_0$ we see that $V' \times _ V V' = R \times _ X V$ with projection maps $s', t' : V' \times _ V V' \to V'$ equal to the pullbacks of $t$ and $s$. Hence $\alpha $ pulls back to an isomorphism $\alpha ' : (t')^{-1}\mathcal{G}' \to (s')^{-1}\mathcal{G}'$. Having said this we simply define

\[ \xymatrix{ \mathcal{F}(V) \ar@{=}[r] & \text{Equalizer}(\mathcal{G}(V') \ar@<1ex>[r] \ar@<-1ex>[r] & \mathcal{G}(V' \times _ V V'). } \]

We omit the verification that this defines a sheaf. To see that $\mathcal{G}(V) = \mathcal{F}(V)$ if there exists a morphism $V \to U$ note that in this case the equalizer is $H^0(\{ V' \to V\} , \mathcal{G}) = \mathcal{G}(V)$. $\square$

[1] In this lemma and its proof we write simply $\varphi ^{-1}$ instead of $\varphi _{small}^{-1}$ and similarly for all the other pullbacks.

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