Lemma 73.10.3. Let $S$ be a scheme. Let $\tau \in \{ fppf, syntomic, smooth, {\acute{e}tale}\}$. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$ which is $\tau$ local on the target. For any morphism of algebraic spaces $f : X \to Y$ over $S$ there exists a largest open subspace $W(f) \subset Y$ such that the restriction $X_{W(f)} \to W(f)$ has $\mathcal{P}$. Moreover,

1. if $g : Y' \to Y$ is a morphism of algebraic spaces which is flat and locally of finite presentation, syntomic, smooth, or étale and the base change $f' : X_{Y'} \to Y'$ has $\mathcal{P}$, then $g$ factors through $W(f)$,

2. if $g : Y' \to Y$ is flat and locally of finite presentation, syntomic, smooth, or étale, then $W(f') = g^{-1}(W(f))$, and

3. 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 $W_{set} \subset |Y|$ of the images $g(|Y'|) \subset |Y|$ of morphisms $g : Y' \to Y$ with the properties:

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

2. the base change $Y' \times _{g, Y} X \to Y'$ has property $\mathcal{P}$.

Since such a morphism $g$ is open (see Morphisms of Spaces, Lemma 66.30.6) we see that $W_{set}$ is an open subset of $|Y|$. Denote $W \subset Y$ the open subspace whose underlying set of points is $W_{set}$, see Properties of Spaces, Lemma 65.4.8. 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 73.10.2. 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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