Lemma 80.4.3 (046G)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 80: Bootstrap
Section 80.4: Properties of maps of presheaves representable by algebraic spaces
Lemma 80.4.3 (cite)

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Lemma 80.4.3. Let $S$ be a scheme. Let $\mathcal{P}$ be a property as in Definition 80.4.1, and assume $\mathcal{P}$ is stable under composition. Let

$\xymatrix{ F \ar[r]^ a & G \ar[r]^ b & H }$

be maps of presheaves on $(\mathit{Sch}/S)_{fppf}$ . If $a$ , $b$ are representable by algebraic spaces and has $\mathcal{P}$ so does $b \circ a$ .

Proof. Omitted. Hint: See Lemma 80.3.8 and use stability under composition. $\square$

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