Lemma 35.22.3. Let \tau \in \{ fppf, syntomic, smooth, {\acute{e}tale}\} . Let \mathcal{P} be a property of morphisms which is \tau local on the target. For any morphism of schemes f : X \to Y there exists a largest open W(f) \subset Y such that the restriction X_{W(f)} \to W(f) has \mathcal{P}. Moreover,
if g : Y' \to Y is flat and locally of finite presentation, syntomic, smooth, or étale and the base change f' : X_{Y'} \to Y' has \mathcal{P}, then g(Y') \subset W(f),
if g : Y' \to Y is flat and locally of finite presentation, syntomic, smooth, or étale, then W(f') = g^{-1}(W(f)), and
if \{ g_ i : Y_ i \to Y\} is a \tau -covering, then g_ i^{-1}(W(f)) = W(f_ i), where f_ i is the base change of f by Y_ i \to Y.
Proof.
Consider the union W of the images g(Y') \subset Y of morphisms g : Y' \to Y with the properties:
g is flat and locally of finite presentation, syntomic, smooth, or étale, and
the base change Y' \times _{g, Y} 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 Y is an open subset of Y. Since \mathcal{P} is local in the \tau topology the restriction X_ W \to W has property \mathcal{P} because we are given a covering \{ Y' \to W\} of W such that the pullbacks have \mathcal{P}. This proves the existence and proves that W(f) has property (1). To see property (2) note that W(f') \supset g^{-1}(W(f)) because \mathcal{P} is stable under base change by flat and locally of finite presentation, syntomic, smooth, or étale morphisms, see Lemma 35.22.2. On the other hand, if Y'' \subset Y' is an open such that X_{Y''} \to Y'' has property \mathcal{P}, then Y'' \to Y factors through W by construction, i.e., Y'' \subset g^{-1}(W(f)). This proves (2). Assertion (3) follows from (2) because each morphism Y_ i \to Y is flat and locally of finite presentation, syntomic, smooth, or étale by our definition of a \tau -covering.
\square
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