Lemma 74.10.2. Let S be a scheme. Let \tau \in \{ fpqc, fppf, syntomic, smooth, {\acute{e}tale}\} . Let \mathcal{P} be a property of morphisms of algebraic spaces over S 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. étale, 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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