Lemma 65.5.5 (02WL)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 65: Algebraic Spaces
Section 65.5: Properties of representable morphisms of presheaves
Lemma 65.5.5 (cite)

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Lemma 65.5.5. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$ . Let $F, G, H : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$ . Let $\mathcal{P}$ be a property as in Definition 65.5.1. Let $a : F \to G$ be a representable transformations of functors. Let $b : H \to G$ be any transformation of functors. Consider the fibre product diagram

$\xymatrix{ H \times _{b, G, a} F \ar[r]_-{b'} \ar[d]_{a'} & F \ar[d]^ a \\ H \ar[r]^ b & G }$

If $a$ has property $\mathcal{P}$ then also the base change $a'$ has property $\mathcal{P}$ .

Proof. Note that the lemma makes sense by Lemma 65.3.3. Proof omitted. $\square$

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