The Stacks project

Lemma 76.9.2. Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $X \subset X'$ and $Y \subset Y'$ be thickenings of algebraic spaces over $B$. Let $f : X \to Y$ be a morphism of algebraic spaces over $B$. Given any map of $\mathcal{O}_ B$-algebras

\[ \alpha : f_{spaces, {\acute{e}tale}}^{-1}\mathcal{O}_{Y'} \to \mathcal{O}_{X'} \]

such that

\[ \xymatrix{ f_{spaces, {\acute{e}tale}}^{-1}\mathcal{O}_ Y \ar[r]_-{f^\sharp } \ar[r] & \mathcal{O}_ X \\ f_{spaces, {\acute{e}tale}}^{-1}\mathcal{O}_{Y'} \ar[r]^-\alpha \ar[u]^{i_ Y^\sharp } & \mathcal{O}_{X'} \ar[u]_{i_ X^\sharp } } \]

commutes, there exists a unique morphism of $(f, f')$ of thickenings over $B$ such that $\alpha = (f')^\sharp $.

Proof. To find $f'$, by Properties of Spaces, Theorem 66.28.4, all we have to do is show that the morphism of ringed topoi

\[ (f_{spaces, {\acute{e}tale}}, \alpha ) : (\mathop{\mathit{Sh}}\nolimits (X_{spaces, {\acute{e}tale}}), \mathcal{O}_{X'}) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (Y_{spaces, {\acute{e}tale}}), \mathcal{O}_{Y'}) \]

is a morphism of locally ringed topoi. This follows directly from the definition of morphisms of locally ringed topoi (Modules on Sites, Definition 18.40.9), the fact that $(f, f^\sharp )$ is a morphism of locally ringed topoi (Properties of Spaces, Lemma 66.28.1), that $\alpha $ fits into the given commutative diagram, and the fact that the kernels of $i_ X^\sharp $ and $i_ Y^\sharp $ are locally nilpotent. Finally, the fact that $f' \circ i_ X = i_ Y \circ f$ follows from the commutativity of the diagram and another application of Properties of Spaces, Theorem 66.28.4. We omit the verification that $f'$ is a morphism over $B$. $\square$

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