35.22 Properties of morphisms local on the target
Suppose that $f : X \to Y$ is a morphism of schemes. Let $g : Y' \to Y$ be a morphism of schemes. Let $f' : X' \to Y'$ be the base change of $f$ by $g$:
\[ \xymatrix{ X' \ar[d]_{f'} \ar[r]_{g'} & X \ar[d]^ f \\ Y' \ar[r]^ g & Y } \]
Let $\mathcal{P}$ be a property of morphisms of schemes. Then we can wonder if (a) $\mathcal{P}(f) \Rightarrow \mathcal{P}(f')$, and also whether the converse (b) $\mathcal{P}(f') \Rightarrow \mathcal{P}(f)$ is true. If (a) holds whenever $g$ is flat, then we say $\mathcal{P}$ is preserved under flat base change. If (b) holds whenever $g$ is surjective and flat, then we say $\mathcal{P}$ descends through flat surjective base changes. If $\mathcal{P}$ is preserved under flat base changes and descends through flat surjective base changes, then we say $\mathcal{P}$ is flat local on the target. Compare with the discussion in Section 35.15. This turns out to be a very important notion which we formalize in the following definition.
Definition 35.22.1. Let $\mathcal{P}$ be a property of morphisms of schemes over a base. Let $\tau \in \{ fpqc, fppf, syntomic, smooth, {\acute{e}tale}, Zariski\} $. 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 $\{ Y_ i \to Y\} _{i \in I}$ (see Topologies, Section 34.2) and any morphism of schemes $f : X \to Y$ over $S$ 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 35.22.2. Let $\tau \in \{ fpqc, fppf, syntomic, smooth, {\acute{e}tale}, Zariski\} $. Let $\mathcal{P}$ be a property of morphisms 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, resp. an open immersion, 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 subscheme $V \subset Y$, 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 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.
$\square$
Lemma 35.22.4. Let $\mathcal{P}$ be a property of morphisms of schemes over a base. Let $\tau \in \{ fpqc, fppf, {\acute{e}tale}, smooth, syntomic\} $. Assume that
the property is preserved under flat, flat and locally of finite presentation, étale, smooth, or syntomic base change depending on whether $\tau $ is fpqc, fppf, étale, smooth, or syntomic (compare with Schemes, Definition 26.18.3),
the property is Zariski local on the base.
for any surjective morphism of affine schemes $S' \to S$ which is flat, flat of finite presentation, étale, smooth or syntomic depending on whether $\tau $ is fpqc, fppf, étale, smooth, or syntomic, and any morphism of schemes $f : X \to S$ property $\mathcal{P}$ holds for $f$ if property $\mathcal{P}$ holds for the base change $f' : X' = S' \times _ S X \to S'$.
Then $\mathcal{P}$ is $\tau $ local on the base.
Proof.
This follows almost immediately from the definition of a $\tau $-covering, see Topologies, Definition 34.9.1 34.7.1 34.4.1 34.5.1, or 34.6.1 and Topologies, Lemma 34.9.9, 34.7.4, 34.4.4, 34.5.4, or 34.6.4. Details omitted.
$\square$
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