The Stacks project

Lemma 64.5.6. 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 64.5.1. 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

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

Assume that $b$ induces a surjective map of fppf sheaves $H^\# \to G^\# $. In this case, if $a'$ has property $\mathcal{P}$, then also $a$ has property $\mathcal{P}$.

Proof. First we remark that by Lemma 64.3.3 the transformation $a'$ is representable. Let $U \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$, and let $\xi \in G(U)$. By assumption there exists an fppf covering $\{ U_ i \to U\} _{i \in I}$ and elements $\xi _ i \in H(U_ i)$ mapping to $\xi |_ U$ via $b$. From general category theory it follows that for each $i$ we have a fibre product diagram

\[ \xymatrix{ U_ i \times _{\xi _ i, H, a'} (H \times _{b, G, a} F) \ar[r] \ar[d] & U \times _{\xi , G, a} F \ar[d] \\ U_ i \ar[r] & U } \]

By assumption the left vertical arrow is a morphism of schemes which has property $\mathcal{P}$. Since $\mathcal{P}$ is local in the fppf topology this implies that also the right vertical arrow has property $\mathcal{P}$ as desired. $\square$

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