The Stacks project

Lemma 92.10.7. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $\mathcal{X}, \mathcal{Y}, \mathcal{Z}$ be categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Let $\mathcal{P}$ be a property as in Definition 92.10.1. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism representable by algebraic spaces. Let $g : \mathcal{Z} \to \mathcal{Y}$ be any $1$-morphism. Consider the fibre product diagram

\[ \xymatrix{ \mathcal{Z} \times _{g, \mathcal{Y}, f} \mathcal{X} \ar[r]_-{g'} \ar[d]_{f'} & \mathcal{X} \ar[d]^ f \\ \mathcal{Z} \ar[r]^ g & \mathcal{Y} } \]

Assume that for every scheme $U$ and object $x$ of $\mathcal{Y}_ U$, there exists an fppf covering $\{ U_ i \to U\} $ such that $x|_{U_ i}$ is in the essential image of the functor $g : \mathcal{Z}_{U_ i} \to \mathcal{Y}_{U_ i}$. In this case, if $f'$ has $\mathcal{P}$, then $f$ has $\mathcal{P}$.

Proof. Proof omitted. Hint: Compare with the proof of Spaces, Lemma 63.5.6. $\square$

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