Theorem 65.28.4. Let $X$, $Y$ be algebraic spaces over $\mathop{\mathrm{Spec}}(\mathbf{Z})$. Let

\[ (g, g^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}), \mathcal{O}_ X) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale}), \mathcal{O}_ Y) \]

be a morphism of locally ringed topoi. Then there exists a unique morphism of algebraic spaces $f : X \to Y$ such that $(g, g^\sharp )$ is isomorphic to $(f_{small}, f^\sharp )$. In other words, the construction

\[ \textit{Spaces}/\mathop{\mathrm{Spec}}(\mathbf{Z}) \longrightarrow \textit{Locally ringed topoi}, \quad X \longrightarrow (X_{\acute{e}tale}, \mathcal{O}_ X) \]

is fully faithful (morphisms up to $2$-isomorphisms on the right hand side).

**Proof.**
The uniqueness we have seen in Lemma 65.28.3. Thus it suffices to prove existence. In this proof we will freely use the identifications of Equation (65.27.0.4) as well as the result of Lemma 65.27.2.

Let $U \in \mathop{\mathrm{Ob}}\nolimits (X_{\acute{e}tale})$, let $V \in \mathop{\mathrm{Ob}}\nolimits (Y_{\acute{e}tale})$ and let $s \in g^{-1}h_ V(U)$ be a section. We may think of $s$ as a map of sheaves $s : h_ U \to g^{-1}h_ V$. By Modules on Sites, Lemma 18.22.3 we obtain a commutative diagram of morphisms of ringed topoi

\[ \xymatrix{ (\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}/U), \mathcal{O}_ U) \ar[rr]_-{(j, j^\sharp )} \ar[d]_{(g_ s, g_ s^\sharp )} & & (\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}), \mathcal{O}_ X) \ar[d]^{(g, g^\sharp )} \\ (\mathop{\mathit{Sh}}\nolimits (V_{\acute{e}tale}), \mathcal{O}_ V) \ar[rr] & & (\mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale}), \mathcal{O}_ Y). } \]

By Étale Cohomology, Theorem 59.40.5 we obtain a unique morphism of schemes $f_ s : U \to V$ such that $(g_ s, g_ s^\sharp )$ is $2$-isomorphic to $(f_{s, small}, f_ s^\sharp )$. The construction $(U, V, s) \leadsto f_ s$ just explained satisfies the following functoriality property: Suppose given morphisms $a : U' \to U$ in $X_{\acute{e}tale}$ and $b : V' \to V$ in $Y_{\acute{e}tale}$ and a map $s' : h_{U'} \to g^{-1}h_{V'}$ such that the diagram

\[ \xymatrix{ h_{U'} \ar[d]_ a \ar[r]_{s'} & g^{-1}h_{V'} \ar[d]^{g^{-1}b} \\ h_ U \ar[r]^ s & g^{-1}h_ V } \]

commutes. Then the diagram

\[ \xymatrix{ U' \ar[r]_-{f_{s'}} \ar[d]_ a & u(V') \ar[d]^{u(b)} \\ U \ar[r]^-{f_ s} & u(V) } \]

of schemes commutes. The reason this is true is that the same condition holds for the morphisms $(g_ s, g_ s^\sharp )$ constructed in Modules on Sites, Lemma 18.22.3 and the uniqueness in Étale Cohomology, Theorem 59.40.5.

The problem is to glue the morphisms $f_ s$ to a morphism of algebraic spaces. To do this first choose a scheme $V$ and a surjective étale morphism $V \to Y$. This means that $h_ V \to *$ is surjective and hence $g^{-1}h_ V \to *$ is surjective too. This means there exists a scheme $U$ and a surjective étale morphism $U \to X$ and a morphism $s : h_ U \to g^{-1}h_ V$. Next, set $R = V \times _ Y V$ and $R' = U \times _ X U$. Then we get $g^{-1}h_ R = g^{-1}h_ V \times g^{-1}h_ V$ as $g^{-1}$ is exact. Thus $s$ induces a morphism $s \times s : h_{R'} \to g^{-1}h_ R$. Applying the constructions above we see that we get a commutative diagram of morphisms of schemes

\[ \xymatrix{ R' \ar@<1ex>[d] \ar@<-1ex>[d] \ar[rr]_{f_{s \times s}} & & R \ar@<1ex>[d] \ar@<-1ex>[d] \\ U \ar[rr]^{f_ s} & & V } \]

Since we have $X = U/R'$ and $Y = V/R$ (see Spaces, Lemma 64.9.1) we conclude that this diagram defines a morphism of algebraic spaces $f : X \to Y$ fitting into an obvious commutative diagram. Now we still have to show that $(f_{small}, f^\sharp )$ is $2$-isomorphic to $(g, g^\sharp )$. Let $t_ V : f_{s, small}^{-1} \to g_ s^{-1}$ and $t_ R : f_{s \times s, small}^{-1} \to g_{s \times s}^{-1}$ be the $2$-isomorphisms which are given to us by the construction above. Let $\mathcal{G}$ be a sheaf on $Y_{\acute{e}tale}$. Then we see that $t_ V$ defines an isomorphism

\[ f_{small}^{-1}\mathcal{G}|_{U_{\acute{e}tale}} = f_{s, small}^{-1}\mathcal{G}|_{V_{\acute{e}tale}} \xrightarrow {t_ V} g_ s^{-1}\mathcal{G}|_{V_{\acute{e}tale}} = g^{-1}\mathcal{G}|_{U_{\acute{e}tale}}. \]

Moreover, this isomorphism pulled back to $R'$ via either projection $R' \to U$ is the isomorphism

\[ f_{small}^{-1}\mathcal{G}|_{R'_{\acute{e}tale}} = f_{s \times s, small}^{-1}\mathcal{G}|_{R_{\acute{e}tale}} \xrightarrow {t_ R} g_{s \times s}^{-1}\mathcal{G}|_{R_{\acute{e}tale}} = g^{-1}\mathcal{G}|_{R'_{\acute{e}tale}}. \]

Since $\{ U \to X\} $ is a covering in the site $X_{spaces, {\acute{e}tale}}$ this means the first displayed isomorphism descends to an isomorphism $t : f_{small}^{-1}\mathcal{G} \to g^{-1}\mathcal{G}$ of sheaves (small detail omitted). The isomorphism is functorial in $\mathcal{G}$ since $t_ V$ and $t_ R$ are transformations of functors. Finally, $t$ is compatible with $f^\sharp $ and $g^\sharp $ as $t_ V$ and $t_ R$ are (some details omitted). This finishes the proof of the theorem.
$\square$

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