74.10 Descending properties of morphisms
In this section we introduce the notion of when a property of morphisms of algebraic spaces is local on the target in a topology. Please compare with Descent, Section 35.22.
Definition 74.10.1. Let $S$ be a scheme. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$. Let $\tau \in \{ fpqc, fppf, syntomic, smooth, {\acute{e}tale}\} $. We say $\mathcal{P}$ is $\tau $ local on the base, or $\tau $ local on the target, or local on the base for the $\tau $-topology if for any $\tau $-covering $\{ Y_ i \to Y\} _{i \in I}$ of algebraic spaces and any morphism of algebraic spaces $f : X \to Y$ we have
\[ f \text{ has }\mathcal{P} \Leftrightarrow \text{each }Y_ i \times _ Y X \to Y_ i\text{ has }\mathcal{P}. \]
To be sure, since isomorphisms are always coverings we see (or require) that property $\mathcal{P}$ holds for $X \to Y$ if and only if it holds for any arrow $X' \to Y'$ isomorphic to $X \to Y$. If a property is $\tau $-local on the target then it is preserved by base changes by morphisms which occur in $\tau $-coverings. Here is a formal statement.
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.
$\square$
A simple often used consequence of the above is that if $f : X \to Y$ has property $\mathcal{P}$ which is $\tau $-local on the target and $f(X) \subset V$ for some open subspace $V \subset Y$, then also the induced morphism $X \to V$ has $\mathcal{P}$. Proof: The base change $f$ by $V \to Y$ gives $X \to V$.
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.
$\square$
Lemma 74.10.4. Let $S$ be a scheme. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$. Assume
if $X_ i \to Y_ i$, $i = 1, 2$ have property $\mathcal{P}$ so does $X_1 \amalg X_2 \to Y_1 \amalg Y_2$,
a morphism of algebraic spaces $f : X \to Y$ has property $\mathcal{P}$ if and only if for every affine scheme $Z$ and morphism $Z \to Y$ the base change $Z \times _ Y X \to Z$ of $f$ has property $\mathcal{P}$, and
for any surjective flat morphism of affine schemes $Z' \to Z$ over $S$ and a morphism $f : X \to Z$ from an algebraic space to $Z$ we have
\[ f' : Z' \times _ Z X \to Z'\text{ has }\mathcal{P} \Rightarrow f\text{ has }\mathcal{P}. \]
Then $\mathcal{P}$ is fpqc local on the base.
Proof.
If $\mathcal{P}$ has property (2), then it is automatically stable under any base change. Hence the direct implication in Definition 74.10.1.
Let $\{ Y_ i \to Y\} _{i \in I}$ be an fpqc covering of algebraic spaces over $S$. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume each base change $f_ i : Y_ i \times _ Y X \to Y_ i$ has property $\mathcal{P}$. Our goal is to show that $f$ has $\mathcal{P}$. Let $Z$ be an affine scheme, and let $Z \to Y$ be a morphism. By (2) it suffices to show that the morphism of algebraic spaces $Z \times _ Y X \to Z$ has $\mathcal{P}$. Since $\{ Y_ i \to Y\} _{i \in I}$ is an fpqc covering we know there exists a standard fpqc covering $\{ Z_ j \to Z\} _{j = 1, \ldots , n}$ and morphisms $Z_ j \to Y_{i_ j}$ over $Y$ for suitable indices $i_ j \in I$. Since $f_{i_ j}$ has $\mathcal{P}$ we see that
\[ Z_ j \times _ Y X = Z_ j \times _{Y_{i_ j}} (Y_{i_ j} \times _ Y X) \longrightarrow Z_ j \]
has $\mathcal{P}$ as a base change of $f_{i_ j}$ (see first remark of the proof). Set $Z' = \coprod _{j = 1, \ldots , n} Z_ j$, so that $Z' \to Z$ is a flat and surjective morphism of affine schemes over $S$. By (1) we conclude that $Z' \times _ Y X \to Z'$ has property $\mathcal{P}$. Since this is the base change of the morphism $Z \times _ Y X \to Z$ by the morphism $Z' \to Z$ we conclude that $Z \times _ Y X \to Z$ has property $\mathcal{P}$ as desired.
$\square$
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