## 65.18 The étale site of an algebraic space

In this section we define the small étale site of an algebraic space. This is the analogue of the small étale site $S_{\acute{e}tale}$ of a scheme. Lemma 65.16.1 implies that in the definition below any morphism between objects of the étale site of $X$ is étale, and that any scheme étale over an object of $X_{\acute{e}tale}$ is also an object of $X_{\acute{e}tale}$.

Definition 65.18.1. Let $S$ be a scheme. Let $\mathit{Sch}_{fppf}$ be a big fppf site containing $S$, and let $\mathit{Sch}_{\acute{e}tale}$ be the corresponding big étale site (i.e., having the same underlying category). Let $X$ be an algebraic space over $S$. The *small étale site $X_{\acute{e}tale}$* of $X$ is defined as follows:

An object of $X_{\acute{e}tale}$ is a morphism $\varphi : U \to X$ where $U \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{\acute{e}tale})$ is a scheme and $\varphi $ is an étale morphism,

a morphism $(\varphi : U \to X) \to (\varphi ' : U' \to X)$ is given by a morphism of schemes $\chi : U \to U'$ such that $\varphi = \varphi ' \circ \chi $, and

a family of morphisms $\{ (U_ i \to X) \to (U \to X)\} _{i \in I}$ of $X_{\acute{e}tale}$ is a covering if and only if $\{ U_ i \to U\} _{i \in I}$ is a covering of $(\mathit{Sch}/S)_{\acute{e}tale}$.

A consequence of our choice is that the étale site of an algebraic space in general does not have a final object! On the other hand, if $X$ happens to be a scheme, then the definition above agrees with Topologies, Definition 34.4.8.

The above is our default site, but there are a couple of variants which we will also use. Namely, we can consider all *algebraic spaces* $U$ which are étale over $X$ and this produces the site $X_{spaces, {\acute{e}tale}}$ we define below or we can consider all *affine schemes* $U$ which are étale over $X$ and this produces the site $X_{affine, {\acute{e}tale}}$ we define below. The first of these two notions is used when discussing functoriality of the small étale site, see Lemma 65.18.8.

Definition 65.18.2. Let $S$ be a scheme. Let $\mathit{Sch}_{fppf}$ be a big fppf site containing $S$, and let $\mathit{Sch}_{\acute{e}tale}$ be the corresponding big étale site (i.e., having the same underlying category). Let $X$ be an algebraic space over $S$. The site *$X_{spaces, {\acute{e}tale}}$* of $X$ is defined as follows:

An object of $X_{spaces, {\acute{e}tale}}$ is a morphism $\varphi : U \to X$ where $U$ is an algebraic space over $S$ and $\varphi $ is an étale morphism of algebraic spaces over $S$,

a morphism $(\varphi : U \to X) \to (\varphi ' : U' \to X)$ of $X_{spaces, {\acute{e}tale}}$ is given by a morphism of algebraic spaces $\chi : U \to U'$ such that $\varphi = \varphi ' \circ \chi $, and

a family of morphisms $\{ \varphi _ i : (U_ i \to X) \to (U \to X)\} _{i \in I}$ of $X_{spaces, {\acute{e}tale}}$ is a covering if and only if $|U| = \bigcup \varphi _ i(|U_ i|)$.

As usual we choose a set of coverings of this type, including at least the coverings in $X_{\acute{e}tale}$, as in Sets, Lemma 3.11.1 to turn $X_{spaces, {\acute{e}tale}}$ into a site.

Since the identity morphism of $X$ is étale it is clear that $X_{spaces, {\acute{e}tale}}$ does have a final object. Let us show right away that the corresponding topos equals the small étale topos of $X$.

Lemma 65.18.3. The functor

\[ X_{\acute{e}tale}\longrightarrow X_{spaces, {\acute{e}tale}}, \quad U/X \longmapsto U/X \]

is a special cocontinuous functor (Sites, Definition 7.29.2) and hence induces an equivalence of topoi $\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (X_{spaces, {\acute{e}tale}})$.

**Proof.**
We have to show that the functor satisfies the assumptions (1) – (5) of Sites, Lemma 7.29.1. It is clear that the functor is continuous and cocontinuous, which proves assumptions (1) and (2). Assumptions (3) and (4) hold simply because the functor is fully faithful. Assumption (5) holds, because an algebraic space by definition has a covering by a scheme.
$\square$

Definition 65.18.5. Let $S$ be a scheme. Let $\mathit{Sch}_{fppf}$ be a big fppf site containing $S$, and let $\mathit{Sch}_{\acute{e}tale}$ be the corresponding big étale site (i.e., having the same underlying category). Let $X$ be an algebraic space over $S$. The site *$X_{affine, {\acute{e}tale}}$* of $X$ is defined as follows:

An object of $X_{affine, {\acute{e}tale}}$ is a morphism $\varphi : U \to X$ where $U \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{\acute{e}tale})$ is an affine scheme and $\varphi $ is an étale morphism,

a morphism $(\varphi : U \to X) \to (\varphi ' : U' \to X)$ of $X_{affine, {\acute{e}tale}}$ is given by a morphism of schemes $\chi : U \to U'$ such that $\varphi = \varphi ' \circ \chi $, and

a family of morphisms $\{ \varphi _ i : (U_ i \to X) \to (U \to X)\} _{i \in I}$ of $X_{affine, {\acute{e}tale}}$ is a covering if and only if $\{ U_ i \to U\} $ is a standard étale covering, see Topologies, Definition 34.4.5.

As usual we choose a set of coverings of this type, as in Sets, Lemma 3.11.1 to turn $X_{affine, {\acute{e}tale}}$ into a site.

Lemma 65.18.6. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The functor $X_{affine, {\acute{e}tale}} \to X_{\acute{e}tale}$ is special cocontinuous and induces an equivalence of topoi from $\mathop{\mathit{Sh}}\nolimits (X_{affine, {\acute{e}tale}})$ to $\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$.

**Proof.**
Omitted. Hint: compare with the proof of Topologies, Lemma 34.4.11.
$\square$

Definition 65.18.7. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The *étale topos* of $X$, or more precisely the *small étale topos* of $X$ is the category $\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$ of sheaves of sets on $X_{\acute{e}tale}$.

By Lemma 65.18.3 we have $\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) = \mathop{\mathit{Sh}}\nolimits (X_{spaces, {\acute{e}tale}})$, so we can also think of this as the category of sheaves of sets on $X_{spaces, {\acute{e}tale}}$. Similarly, by Lemma 65.18.6 we see that $\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) = \mathop{\mathit{Sh}}\nolimits (X_{affine, {\acute{e}tale}})$. It turns out that the topos is functorial with respect to morphisms of algebraic spaces. Here is a precise statement.

Lemma 65.18.8. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$.

The continuous functor

\[ Y_{spaces, {\acute{e}tale}} \longrightarrow X_{spaces, {\acute{e}tale}}, \quad V \longmapsto X \times _ Y V \]

induces a morphism of sites

\[ f_{spaces, {\acute{e}tale}} : X_{spaces, {\acute{e}tale}} \to Y_{spaces, {\acute{e}tale}}. \]

The rule $f \mapsto f_{spaces, {\acute{e}tale}}$ is compatible with compositions, in other words $(f \circ g)_{spaces, {\acute{e}tale}} = f_{spaces, {\acute{e}tale}} \circ g_{spaces, {\acute{e}tale}}$ (see Sites, Definition 7.14.5).

The morphism of topoi associated to $f_{spaces, {\acute{e}tale}}$ induces, via Lemma 65.18.3, a morphism of topoi $f_{small} : \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale})$ whose construction is compatible with compositions.

If $f$ is a representable morphism of algebraic spaces, then $f_{small}$ comes from a morphism of sites $X_{\acute{e}tale}\to Y_{\acute{e}tale}$, corresponding to the continuous functor $V \mapsto X \times _ Y V$.

**Proof.**
Let us show that the functor described in (1) satisfies the assumptions of Sites, Proposition 7.14.7. Thus we have to show that $Y_{spaces, {\acute{e}tale}}$ has a final object (namely $Y$) and that the functor transforms this into a final object in $X_{spaces, {\acute{e}tale}}$ (namely $X$). This is clear as $X \times _ Y Y = X$ in any category. Next, we have to show that $Y_{spaces, {\acute{e}tale}}$ has fibre products. This is true since the category of algebraic spaces has fibre products, and since $V \times _ Y V'$ is étale over $Y$ if $V$ and $V'$ are étale over $Y$ (see Lemmas 65.16.4 and 65.16.5 above). OK, so the proposition applies and we see that we get a morphism of sites as described in (1).

Part (2) you get by unwinding the definitions. Part (3) is clear by using the equivalences for $X$ and $Y$ from Lemma 65.18.3 above. Part (4) follows, because if $f$ is representable, then the functors above fit into a commutative diagram

\[ \xymatrix{ X_{\acute{e}tale}\ar[r] & X_{spaces, {\acute{e}tale}} \\ Y_{\acute{e}tale}\ar[r] \ar[u] & Y_{spaces, {\acute{e}tale}} \ar[u] } \]

of categories.
$\square$

We can do a little bit better than the lemma above in describing the relationship between sheaves on $X$ and sheaves on $Y$. Namely, we can formulate this in turns of $f$-maps, compare Sheaves, Definition 6.21.7, as follows.

Definition 65.18.9. 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 $Y_{\acute{e}tale}$. An *$f$-map $\varphi : \mathcal{G} \to \mathcal{F}$* is a collection of maps $\varphi _{(U, V, g)} : \mathcal{G}(V) \to \mathcal{F}(U)$ indexed by commutative diagrams

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

where $U \in X_{\acute{e}tale}$, $V \in Y_{\acute{e}tale}$ such that whenever given an extended diagram

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

with $V' \to V$ and $U' \to U$ étale morphisms of schemes the diagram

\[ \xymatrix{ \mathcal{G}(V) \ar[rr]_{\varphi _{(U, V, g)}} \ar[d]_{\text{restriction of }\mathcal{G}} & & \mathcal{F}(U) \ar[d]^{\text{restriction of }\mathcal{F}} \\ \mathcal{G}(V') \ar[rr]^{\varphi _{(U', V', g')}} & & \mathcal{F}(U') } \]

commutes.

Lemma 65.18.10. 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:

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

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

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 65.18.9 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 65.18.3. 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.
$\square$

If the morphism of algebraic spaces $X \to Y$ is étale, then the morphism of topoi $\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale})$ is a localization. Here is a statement.

Lemma 65.18.11. Let $S$ be a scheme, and let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is étale. In this case there is a functor

\[ j : X_{\acute{e}tale}\to Y_{\acute{e}tale}, \quad (\varphi : U \to X) \mapsto (f \circ \varphi : U \to Y) \]

which is cocontinuous. The morphism of topoi $f_{small}$ is the morphism of topoi associated to $j$, see Sites, Lemma 7.21.1. Moreover, $j$ is continuous as well, hence Sites, Lemma 7.21.5 applies. In particular $f_{small}^{-1}\mathcal{G}(U) = \mathcal{G}(jU)$ for all sheaves $\mathcal{G}$ on $Y_{\acute{e}tale}$.

**Proof.**
Note that by our very definition of an étale morphism of algebraic spaces (Definition 65.16.2) it is indeed the case that the rule given defines a functor $j$ as indicated. It is clear that $j$ is cocontinuous and continuous, simply because a covering $\{ U_ i \to U\} $ of $j(\varphi : U \to X)$ in $Y_{\acute{e}tale}$ is the same thing as a covering of $(\varphi : U \to X)$ in $X_{\acute{e}tale}$. It remains to show that $j$ induces the same morphism of topoi as $f_{small}$. To see this we consider the diagram

\[ \xymatrix{ X_{\acute{e}tale}\ar[r] \ar[d]^ j & X_{spaces, {\acute{e}tale}} \ar@/_/[d]_{j_{spaces}} \\ Y_{\acute{e}tale}\ar[r] & Y_{spaces, {\acute{e}tale}} \ar@/_/[u]_{v : V \mapsto X \times _ Y V} } \]

of categories. Here the functor $j_{spaces}$ is the obvious extension of $j$ to the category $X_{spaces, {\acute{e}tale}}$. Thus the inner square is commutative. In fact $j_{spaces}$ can be identified with the localization functor $j_ X : Y_{spaces, {\acute{e}tale}}/X \to Y_{spaces, {\acute{e}tale}}$ discussed in Sites, Section 7.25. Hence, by Sites, Lemma 7.27.2 the cocontinuous functor $j_{spaces}$ and the functor $v$ of the diagram induce the same morphism of topoi. By Sites, Lemma 7.21.2 the commutativity of the inner square (consisting of cocontinuous functors between sites) gives a commutative diagram of associated morphisms of topoi. Hence, by the construction of $f_{small}$ in Lemma 65.18.8 we win.
$\square$

The lemma above says that the pullback of $\mathcal{G}$ via an étale morphism $f : X \to Y$ of algebraic spaces is simply the restriction of $\mathcal{G}$ to the category $X_{\acute{e}tale}$. We will often use the short hand

65.18.11.1
\begin{equation} \label{spaces-properties-equation-restrict} \mathcal{G}|_{X_{\acute{e}tale}} = f_{small}^{-1}\mathcal{G} \end{equation}

to indicate this. Note that the functor $j : X_{\acute{e}tale}\to Y_{\acute{e}tale}$ of the lemma in this situation is faithful, but not fully faithful in general. We will discuss this in a more technical fashion in Section 65.27.

Lemma 65.18.12. Let $S$ be a scheme. Let

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

be a cartesian square of algebraic spaces over $S$. Let $\mathcal{F}$ be a sheaf on $X_{\acute{e}tale}$. If $g$ is étale, then

$f'_{small, *}(\mathcal{F}|_{X'}) = (f_{small, *}\mathcal{F})|_{Y'}$ in $\mathop{\mathit{Sh}}\nolimits (Y'_{\acute{e}tale})$^{1}, and

if $\mathcal{F}$ is an abelian sheaf, then $R^ if'_{small, *}(\mathcal{F}|_{X'}) = (R^ if_{small, *}\mathcal{F})|_{Y'}$.

**Proof.**
Consider the following diagram of functors

\[ \xymatrix{ X'_{spaces, {\acute{e}tale}} \ar[r]_ j & X_{spaces, {\acute{e}tale}} \\ Y'_{spaces, {\acute{e}tale}} \ar[r]^ j \ar[u]^{V' \mapsto V' \times _{Y'} X'} & Y_{spaces, {\acute{e}tale}} \ar[u]_{V \mapsto V \times _ Y X} } \]

The horizontal arrows are localizations and the vertical arrows induce morphisms of sites. Hence the last statement of Sites, Lemma 7.28.1 gives (1). To see (2) apply (1) to an injective resolution of $\mathcal{F}$ and use that restriction is exact and preserves injectives (see Cohomology on Sites, Lemma 21.7.1).
$\square$

The following lemma says that you can think of a sheaf on the small étale site of an algebraic space as a compatible collection of sheaves on the small étale sites of schemes étale over the space. Please note that all the comparison mappings $c_ f$ in the lemma are isomorphisms, which is compatible with Topologies, Lemma 34.4.20 and the fact that all morphisms between objects of $X_{\acute{e}tale}$ are étale.

Lemma 65.18.13. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. A sheaf $\mathcal{F}$ on $X_{\acute{e}tale}$ is given by the following data:

for every $U \in \mathop{\mathrm{Ob}}\nolimits (X_{\acute{e}tale})$ a sheaf $\mathcal{F}_ U$ on $U_{\acute{e}tale}$,

for every $f : U' \to U$ in $X_{\acute{e}tale}$ an isomorphism $c_ f : f_{small}^{-1}\mathcal{F}_ U \to \mathcal{F}_{U'}$.

These data are subject to the condition that given any $f : U' \to U$ and $g : U'' \to U'$ in $X_{\acute{e}tale}$ the composition $c_ g \circ g_{small}^{-1} c_ f$ is equal to $c_{f \circ g}$.

**Proof.**
We may interpret $g_{small}^{-1}$ as in Lemma 65.18.11. Then the lemma follows from a general fact about sites, see Sites, Lemma 7.26.6.
$\square$

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $X = U/R$ be a presentation of $X$ coming from any surjective étale morphism $\varphi : U \to X$, see Spaces, Definition 64.9.3. In particular, we obtain a groupoid $(U, R, s, t, c, e, i)$ such that $j = (t, s) : R \to U \times _ S U$, see Groupoids, Lemma 39.13.3.

Lemma 65.18.14. 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 sheaf^{2} $\mathcal{G} = \varphi ^{-1}\mathcal{F}$ 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 65.18.14.**
Let $\mathcal{C} = X_{spaces, {\acute{e}tale}}$. By Lemma 65.18.11 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 65.18.14.**
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 65.18.8. 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 $H^0(\{ V' \to V\} , \mathcal{G}) = \mathcal{G}(V)$.
$\square$

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