The Stacks project

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 √Čtale Cohomology, Lemma 59.40.3 we conclude $\alpha = \beta $! Since $p : U \to X$ is a surjection of sheaves it follows that $a = b$. $\square$


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