The Stacks project

66.28 Recovering morphisms

In this section we prove that the rule which associates to an algebraic space its locally ringed small étale topos is fully faithful in a suitable sense, see Theorem 66.28.4.

Lemma 66.28.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The morphism of ringed topoi $(f_{small}, f^\sharp )$ associated to $f$ is a morphism of locally ringed topoi, see Modules on Sites, Definition 18.40.9.

Proof. Note that the assertion makes sense since we have seen that $(X_{\acute{e}tale}, \mathcal{O}_{X_{\acute{e}tale}})$ and $(Y_{\acute{e}tale}, \mathcal{O}_{Y_{\acute{e}tale}})$ are locally ringed sites, see Lemma 66.22.3. Moreover, we know that $X_{\acute{e}tale}$ has enough points, see Theorem 66.19.12. Hence it suffices to prove that $(f_{small}, f^\sharp )$ satisfies condition (3) of Modules on Sites, Lemma 18.40.8. To see this take a point $p$ of $X_{\acute{e}tale}$. By Lemma 66.19.13 $p$ corresponds to a geometric point $\overline{x}$ of $X$. By Lemma 66.19.9 the point $q = f_{small} \circ p$ corresponds to the geometric point $\overline{y} = f \circ \overline{x}$ of $Y$. Hence the assertion we have to prove is that the induced map of étale local rings

\[ \mathcal{O}_{Y, \overline{y}} \longrightarrow \mathcal{O}_{X, \overline{x}} \]

is a local ring map. You can prove this directly, but instead we deduce it from the corresponding result for schemes. To do this choose a commutative diagram

\[ \xymatrix{ U \ar[d] \ar[r]_\psi & V \ar[d] \\ X \ar[r] & Y } \]

where $U$ and $V$ are schemes, and the vertical arrows are surjective étale (see Spaces, Lemma 65.11.6). Choose a lift $\overline{u} : \overline{x} \to U$ (possible by Lemma 66.19.5). Set $\overline{v} = \psi \circ \overline{u}$. We obtain a commutative diagram of étale local rings

\[ \xymatrix{ \mathcal{O}_{U, \overline{u}} & \mathcal{O}_{V, \overline{v}} \ar[l] \\ \mathcal{O}_{X, \overline{x}} \ar[u] & \mathcal{O}_{Y, \overline{y}}. \ar[l] \ar[u] } \]

By Étale Cohomology, Lemma 59.40.1 the top horizontal arrow is a local ring map. Finally by Lemma 66.22.1 the vertical arrows are isomorphisms. Hence we win. $\square$

Lemma 66.28.2. Let $S$ be a scheme. Let $X$, $Y$ be algebraic spaces over $S$. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $t$ be a $2$-morphism from $(f_{small}, f^\sharp )$ to itself, see Modules on Sites, Definition 18.8.1. Then $t = \text{id}$.

Proof. Let $X'$, resp. $Y'$ be $X$ viewed as an algebraic space over $\mathop{\mathrm{Spec}}(\mathbf{Z})$, see Spaces, Definition 65.16.2. It is clear from the construction that $(X_{small}, \mathcal{O})$ is equal to $(X'_{small}, \mathcal{O})$ and similarly for $Y$. Hence we may work with $X'$ and $Y'$. In other words we may assume that $S = \mathop{\mathrm{Spec}}(\mathbf{Z})$.

Assume $S = \mathop{\mathrm{Spec}}(\mathbf{Z})$, $f : X \to Y$ and $t$ are as in the lemma. This means that $t : f^{-1}_{small} \to f^{-1}_{small}$ is a transformation of functors such that the diagram

\[ \xymatrix{ f_{small}^{-1}\mathcal{O}_ Y \ar[rd]_{f^\sharp } & & f_{small}^{-1}\mathcal{O}_ Y \ar[ll]^ t \ar[ld]^{f^\sharp } \\ & \mathcal{O}_ X } \]

is commutative. Suppose $V \to Y$ is étale with $V$ affine. Write $V = \mathop{\mathrm{Spec}}(B)$. Choose generators $b_ j \in B$, $j \in J$ for $B$ as a $\mathbf{Z}$-algebra. Set $T = \mathop{\mathrm{Spec}}(\mathbf{Z}[\{ x_ j\} _{j \in J}])$. In the following we will use that $\mathop{\mathrm{Mor}}\nolimits _{\mathit{Sch}}(U, T) = \prod _{j \in J} \Gamma (U, \mathcal{O}_ U)$ for any scheme $U$ without further mention. The surjective ring map $\mathbf{Z}[x_ j] \to B$, $x_ j \mapsto b_ j$ corresponds to a closed immersion $V \to T$. We obtain a monomorphism

\[ i : V \longrightarrow T_ Y = T \times Y \]

of algebraic spaces over $Y$. In terms of sheaves on $Y_{\acute{e}tale}$ the morphism $i$ induces an injection $h_ i : h_ V \to \prod _{j \in J} \mathcal{O}_ Y$ of sheaves. The base change $i' : X \times _ Y V \to T_ X$ of $i$ to $X$ is a monomorphism too (Spaces, Lemma 65.5.5). Hence $i' : X \times _ Y V \to T_ X$ is a monomorphism, which in turn means that $h_{i'} : h_{X \times _ Y V} \to \prod _{j \in J} \mathcal{O}_ X$ is an injection of sheaves. Via the identification $f_{small}^{-1}h_ V = h_{X \times _ Y V}$ of Lemma 66.19.9 the map $h_{i'}$ is equal to

\[ \xymatrix{ f_{small}^{-1}h_ V \ar[r]^-{f^{-1}h_ i} & \prod _{j \in J} f_{small}^{-1}\mathcal{O}_ Y \ar[r]^{\prod f^\sharp } & \prod _{j \in J} \mathcal{O}_ X } \]

(verification omitted). This means that the map $t : f_{small}^{-1}h_ V \to f_{small}^{-1}h_ V$ fits into the commutative diagram

\[ \xymatrix{ f_{small}^{-1}h_ V \ar[r]^-{f^{-1}h_ i} \ar[d]^ t & \prod _{j \in J} f_{small}^{-1}\mathcal{O}_ Y \ar[r]^-{\prod f^\sharp } \ar[d]^{\prod t} & \prod _{j \in J} \mathcal{O}_ X \ar[d]^{\text{id}}\\ f_{small}^{-1}h_ V \ar[r]^-{f^{-1}h_ i} & \prod _{j \in J} f_{small}^{-1}\mathcal{O}_ Y \ar[r]^-{\prod f^\sharp } & \prod _{j \in J} \mathcal{O}_ X } \]

The commutativity of the right square holds by our assumption on $t$ explained above. Since the composition of the horizontal arrows is injective by the discussion above we conclude that the left vertical arrow is the identity map as well. Any sheaf of sets on $Y_{\acute{e}tale}$ admits a surjection from a (huge) coproduct of sheaves of the form $h_ V$ with $V$ affine (combine Lemma 66.18.6 with Sites, Lemma 7.12.5). Thus we conclude that $t : f_{small}^{-1} \to f_{small}^{-1}$ is the identity transformation as desired. $\square$

Lemma 66.28.3. Let $S$ be a scheme. Let $X$, $Y$ be algebraic spaces over $S$. Any two morphisms $a, b : X \to Y$ of algebraic spaces over $S$ for which there exists a $2$-isomorphism $(a_{small}, a^\sharp ) \cong (b_{small}, b^\sharp )$ in the $2$-category of ringed topoi are equal.

Proof. Let $t : a_{small}^{-1} \to b_{small}^{-1}$ be the $2$-isomorphism. We may equivalently think of $t$ as a transformation $t : a_{spaces, {\acute{e}tale}}^{-1} \to b_{spaces, {\acute{e}tale}}^{-1}$ since there is not difference between sheaves on $X_{\acute{e}tale}$ and sheaves on $X_{spaces, {\acute{e}tale}}$. Choose a commutative diagram

\[ \xymatrix{ U \ar[d]_ p \ar[r]_\alpha & V \ar[d]^ q \\ X \ar[r]^ a & Y } \]

where $U$ and $V$ are schemes, and $p$ and $q$ are surjective étale. Consider the diagram

\[ \xymatrix{ h_ U \ar[r]_-\alpha \ar@{=}[d] & a_{spaces, {\acute{e}tale}}^{-1}h_ V \ar[d]^ t \\ h_ U \ar@{..>}[r] & b_{spaces, {\acute{e}tale}}^{-1}h_ V } \]

Since the sheaf $b_{spaces, {\acute{e}tale}}^{-1}h_ V$ is isomorphic to $h_{V \times _{Y, b} X}$ we see that the dotted arrow comes from a morphism of schemes $\beta : U \to V$ fitting into a commutative diagram

\[ \xymatrix{ U \ar[d]_ p \ar[r]_\beta & V \ar[d]^ q \\ X \ar[r]^ b & Y } \]

We claim that there exists a sequence of $2$-isomorphisms

\begin{align*} (\alpha _{small}, \alpha ^\sharp ) & \cong (\alpha _{spaces, {\acute{e}tale}}, \alpha ^\sharp ) \\ & \cong (a_{spaces, {\acute{e}tale}, c}, a_ c^\sharp ) \\ & \cong (b_{spaces, {\acute{e}tale}, d}, b_ d^\sharp ) \\ & \cong (\beta _{spaces, {\acute{e}tale}}, \beta ^\sharp ) \\ & \cong (\beta _{small}, \beta ^\sharp ) \end{align*}

The first and the last $2$-isomorphisms come from the identifications between sheaves on $U_{spaces, {\acute{e}tale}}$ and sheaves on $U_{\acute{e}tale}$ and similarly for $V$. The second and fourth $2$-isomorphisms are those of Lemma 66.27.1 with $c : U \to X \times _{a, Y} V$ induced by $\alpha $ and $d : U \to X \times _{b, Y} V$ induced by $\beta $. The middle $2$-isomorphism comes from the transformation $t$. Namely, the functor $a_{spaces, {\acute{e}tale}, c}^{-1}$ corresponds to the functor

\[ (\mathcal{H} \to h_ V) \longmapsto (a_{spaces, {\acute{e}tale}}^{-1}\mathcal{H} \times _{a_{spaces, {\acute{e}tale}}^{-1}h_ V, \alpha } h_ U \to h_ U) \]

and similarly for $b_{spaces, {\acute{e}tale}, d}^{-1}$, see Sites, Lemma 7.28.3. This uses the identification of sheaves on $Y_{spaces, {\acute{e}tale}}/V$ as arrows $(\mathcal{H} \to h_ V)$ in $\mathop{\mathit{Sh}}\nolimits (Y_{spaces, {\acute{e}tale}})$ and similarly for $U/X$, see Sites, Lemma 7.25.4. Via this identification the structure sheaf $\mathcal{O}_ V$ corresponds to the pair $(\mathcal{O}_ Y \times h_ V \to h_ V)$ and similarly for $\mathcal{O}_ U$, see Modules on Sites, Lemma 18.21.3. Since $t$ switches $\alpha $ and $\beta $ we see that $t$ induces an isomorphism

\[ t : a_{spaces, {\acute{e}tale}}^{-1}\mathcal{H} \times _{a_{spaces, {\acute{e}tale}}^{-1}h_ V, \alpha } h_ U \longrightarrow b_{spaces, {\acute{e}tale}}^{-1}\mathcal{H} \times _{b_{spaces, {\acute{e}tale}}^{-1}h_ V, \beta } h_ U \]

over $h_ U$ functorially in $(\mathcal{H} \to h_ V)$. Also, $t$ is compatible with $a_ c^\sharp $ and $b_ d^\sharp $ as $t$ is compatible with $a^\sharp $ and $b^\sharp $ by our description of the structure sheaves $\mathcal{O}_ U$ and $\mathcal{O}_ V$ above. Hence, the morphisms of ringed topoi $(\alpha _{small}, \alpha ^\sharp )$ and $(\beta _{small}, \beta ^\sharp )$ are $2$-isomorphic. By Étale Cohomology, Lemma 59.40.3 we conclude $\alpha = \beta $! Since $p : U \to X$ is a surjection of sheaves it follows that $a = b$. $\square$

Here is the main result of this section.

Theorem 66.28.4. 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 66.28.3. Thus it suffices to prove existence. In this proof we will freely use the identifications of Equation ( as well as the result of Lemma 66.27.2.

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 65.9.1) 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. $\square$

Lemma 66.28.5. Let $X$, $Y$ be algebraic spaces over $\mathbf{Z}$. If

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

is an isomorphism of ringed topoi, then there exists a unique morphism $f : X \to Y$ of algebraic spaces such that $(g, g^\sharp )$ is isomorphic to $(f_{small}, f^\sharp )$ and moreover $f$ is an isomorphism of algebraic spaces.

Proof. By Theorem 66.28.4 it suffices to show that $(g, g^\sharp )$ is a morphism of locally ringed topoi. By Modules on Sites, Lemma 18.40.8 (and since the site $X_{\acute{e}tale}$ has enough points) it suffices to check that the map $\mathcal{O}_{Y, q} \to \mathcal{O}_{X, p}$ induced by $g^\sharp $ is a local ring map where $q = f \circ p$ and $p$ is any point of $X_{\acute{e}tale}$. As it is an isomorphism this is clear. $\square$

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