The Stacks project

Lemma 66.18.10. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a sheaf of sets on $X_{\acute{e}tale}$ and let $\mathcal{G}$ be a sheaf of sets on $Y_{\acute{e}tale}$. There are canonical bijections between the following three sets:

  1. The set of maps $\mathcal{G} \to f_{small, *}\mathcal{F}$.

  2. The set of maps $f_{small}^{-1}\mathcal{G} \to \mathcal{F}$.

  3. The set of $f$-maps $\varphi : \mathcal{G} \to \mathcal{F}$.

Proof. Note that (1) and (2) are the same because the functors $f_{small, *}$ and $f_{small}^{-1}$ are a pair of adjoint functors. Suppose that $\alpha : f_{small}^{-1}\mathcal{G} \to \mathcal{F}$ is a map of sheaves on $Y_{\acute{e}tale}$. Let a diagram

\[ \xymatrix{ U \ar[d]_ g \ar[r]_{j_ U} & X \ar[d]^ f \\ V \ar[r]^{j_ V} & Y } \]

as in Definition 66.18.9 be given. By the commutativity of the diagram we also get a map $g_{small}^{-1}(j_ V)^{-1}\mathcal{G} \to (j_ U)^{-1}\mathcal{F}$ (compare Sites, Section 7.25 for the description of the localization functors). Hence we certainly get a map $\varphi _{(V, U, g)} : \mathcal{G}(V) = (j_ V)^{-1}\mathcal{G}(V) \to (j_ U)^{-1}\mathcal{F}(U) = \mathcal{F}(U)$. We omit the verification that this rule is compatible with further restrictions and defines an $f$-map from $\mathcal{G}$ to $\mathcal{F}$.

Conversely, suppose that we are given an $f$-map $\varphi = (\varphi _{(U, V, g)})$. Let $\mathcal{G}'$ (resp. $\mathcal{F}'$) denote the extension of $\mathcal{G}$ (resp. $\mathcal{F}$) to $Y_{spaces, {\acute{e}tale}}$ (resp. $X_{spaces, {\acute{e}tale}}$), see Lemma 66.18.3. Then we have to construct a map of sheaves

\[ \mathcal{G}' \longrightarrow (f_{spaces, {\acute{e}tale}})_*\mathcal{F}' \]

To do this, let $V \to Y$ be an étale morphism of algebraic spaces. We have to construct a map of sets

\[ \mathcal{G}'(V) \to \mathcal{F}'(X \times _ Y V) \]

Choose an étale surjective morphism $V' \to V$ with $V'$ a scheme, and after that choose an étale surjective morphism $U' \to X \times _ U V'$ with $U'$ a scheme. We get a morphism of schemes $g' : U' \to V'$ and also a morphism of schemes

\[ g'' : U' \times _{X \times _ Y V} U' \longrightarrow V' \times _ V V' \]

Consider the following diagram

\[ \xymatrix{ \mathcal{F}'(X \times _ Y V) \ar[r] & \mathcal{F}(U') \ar@<1ex>[r] \ar@<-1ex>[r] & \mathcal{F}(U' \times _{X \times _ Y V} U') \\ \mathcal{G}'(X \times _ Y V) \ar[r] \ar@{..>}[u] & \mathcal{G}(V') \ar@<1ex>[r] \ar@<-1ex>[r] \ar[u]_{\varphi _{(U', V', g')}} & \mathcal{G}(V' \times _ V V') \ar[u]_{\varphi _{(U'', V'', g'')}} } \]

The compatibility of the maps $\varphi _{...}$ with restriction shows that the two right squares commute. The definition of coverings in $X_{spaces, {\acute{e}tale}}$ shows that the horizontal rows are equalizer diagrams. Hence we get the dotted arrow. We leave it to the reader to show that these arrows are compatible with the restriction mappings. $\square$

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