The Stacks project

Lemma 35.22.2. Let $\tau \in \{ fpqc, fppf, syntomic, smooth, {\acute{e}tale}, Zariski\} $. Let $\mathcal{P}$ be a property of morphisms which is $\tau $ local on the target. Let $f : X \to Y$ have property $\mathcal{P}$. For any morphism $Y' \to Y$ which is flat, resp. flat and locally of finite presentation, resp. syntomic, resp. ├ętale, resp. an open immersion, the base change $f' : Y' \times _ Y X \to Y'$ of $f$ has property $\mathcal{P}$.

Proof. This is true because we can fit $Y' \to Y$ into a family of morphisms which forms a $\tau $-covering. $\square$

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