The Stacks project

Lemma 75.17.4. Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $X \subset X'$ and $Y \subset Y'$ be first order thickenings over $B$. Assume given a morphism $a : X \to Y$ and a map $A : a^*\mathcal{C}_{Y/Y'} \to \mathcal{C}_{X/X'}$ of $\mathcal{O}_ X$-modules. For an object $U'$ of $(X')_{spaces, {\acute{e}tale}}$ with $U = X \times _{X'} U'$ consider morphisms $a' : U' \to Y'$ such that

  1. $a'$ is a morphism over $B$,

  2. $a'|_ U = a|_ U$, and

  3. the induced map $a^*\mathcal{C}_{Y/Y'}|_ U \to \mathcal{C}_{X/X'}|_ U$ is the restriction of $A$ to $U$.

Then the rule

75.17.4.1
\begin{equation} \label{spaces-more-morphisms-equation-sheaf} U' \mapsto \{ a' : U' \to Y'\text{ such that (1), (2), (3) hold.}\} \end{equation}

defines a sheaf of sets on $(X')_{spaces, {\acute{e}tale}}$.

Proof. Denote $\mathcal{F}$ the rule of the lemma. The restriction mapping $\mathcal{F}(U') \to \mathcal{F}(V')$ for $V' \subset U' \subset X'$ of $\mathcal{F}$ is really the restriction map $a' \mapsto a'|_{V'}$. With this definition in place it is clear that $\mathcal{F}$ is a sheaf since morphisms of algebraic spaces satisfy ├ętale descent, see Descent on Spaces, Lemma 73.7.2. $\square$


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