The Stacks project

Lemma 35.26.3. Let $\tau \in \{ fppf, syntomic, smooth, {\acute{e}tale}\} $. Let $\mathcal{P}$ be a property of morphisms which is $\tau $ local on the source. For any morphism of schemes $f : X \to Y$ there exists a largest open $W(f) \subset X$ such that the restriction $f|_{W(f)} : W(f) \to Y$ has $\mathcal{P}$. Moreover, if $g : X' \to X$ is flat and locally of finite presentation, syntomic, smooth, or ├ętale and $f' = f \circ g : X' \to Y$, then $g^{-1}(W(f)) = W(f')$.

Proof. Consider the union $W$ of the images $g(X') \subset X$ of morphisms $g : X' \to X$ with the properties:

  1. $g$ is flat and locally of finite presentation, syntomic, smooth, or ├ętale, and

  2. the composition $X' \to X \to Y$ has property $\mathcal{P}$.

Since such a morphism $g$ is open (see Morphisms, Lemma 29.25.10) we see that $W \subset X$ is an open subset of $X$. Since $\mathcal{P}$ is local in the $\tau $ topology the restriction $f|_ W : W \to Y$ has property $\mathcal{P}$ because we are given a $\tau $ covering $\{ X' \to W\} $ of $W$ such that the pullbacks have $\mathcal{P}$. This proves the existence of $W(f)$. The compatibility stated in the last sentence follows immediately from the construction of $W(f)$. $\square$

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