Lemma 74.10.3. 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,
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 $W(f)$,
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 $W_{set} \subset |Y|$ 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 of Spaces, Lemma 67.30.6) 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 66.4.8. 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 74.10.2. 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.
$\square$
Comments (0)