35.26 Properties of morphisms local on the source
It often happens one can prove a morphism has a certain property after precomposing with some other morphism. In many cases this implies the morphism has the property too. We formalize this in the following definition.
Definition 35.26.1. Let $\mathcal{P}$ be a property of morphisms of schemes. Let $\tau \in \{ Zariski, \linebreak[0] fpqc, \linebreak[0] fppf, \linebreak[0] {\acute{e}tale}, \linebreak[0] smooth, \linebreak[0] syntomic\} $. We say $\mathcal{P}$ is $\tau $ local on the source, or local on the source for the $\tau $-topology if for any morphism of schemes $f : X \to Y$ over $S$, and any $\tau $-covering $\{ X_ i \to X\} _{i \in I}$ we have
\[ f \text{ has }\mathcal{P} \Leftrightarrow \text{each }X_ i \to Y\text{ has }\mathcal{P}. \]
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 source then it is preserved by precomposing with morphisms which occur in $\tau $-coverings. Here is a formal statement.
Lemma 35.26.2. 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 source. Let $f : X \to Y$ have property $\mathcal{P}$. For any morphism $a : X' \to X$ which is flat, resp. flat and locally of finite presentation, resp. syntomic, resp. étale, resp. an open immersion, the composition $f \circ a : X' \to Y$ has property $\mathcal{P}$.
Proof.
This is true because we can fit $X' \to X$ into a family of morphisms which forms a $\tau $-covering.
$\square$
Lemma 35.26.3. Let $\tau \in \{ fppf, syntomic, smooth, {\acute{e}tale}\} $. Let $\mathcal{P}$ be a property of morphisms which is $\tau $ local on the source. For any morphism of schemes $f : X \to Y$ there exists a largest open $W(f) \subset X$ such that the restriction $f|_{W(f)} : W(f) \to Y$ has $\mathcal{P}$. Moreover, if $g : X' \to X$ is flat and locally of finite presentation, syntomic, smooth, or étale and $f' = f \circ g : X' \to Y$, then $g^{-1}(W(f)) = W(f')$.
Proof.
Consider the union $W$ of the images $g(X') \subset X$ of morphisms $g : X' \to X$ with the properties:
$g$ is flat and locally of finite presentation, syntomic, smooth, or étale, and
the composition $X' \to X \to Y$ has property $\mathcal{P}$.
Since such a morphism $g$ is open (see Morphisms, Lemma 29.25.10) we see that $W \subset X$ is an open subset of $X$. Since $\mathcal{P}$ is local in the $\tau $ topology the restriction $f|_ W : W \to Y$ has property $\mathcal{P}$ because we are given a $\tau $ covering $\{ X' \to W\} $ of $W$ such that the pullbacks have $\mathcal{P}$. This proves the existence of $W(f)$. The compatibility stated in the last sentence follows immediately from the construction of $W(f)$.
$\square$
Lemma 35.26.4. Let $\mathcal{P}$ be a property of morphisms of schemes. Let $\tau \in \{ fpqc, \linebreak[0] fppf, \linebreak[0] {\acute{e}tale}, \linebreak[0] smooth, \linebreak[0] syntomic\} $. Assume that
the property is preserved under precomposing with flat, flat locally of finite presentation, étale, smooth or syntomic morphisms depending on whether $\tau $ is fpqc, fppf, étale, smooth, or syntomic,
the property is Zariski local on the source,
the property is Zariski local on the target,
for any morphism of affine schemes $f : X \to Y$, and any surjective morphism of affine schemes $X' \to X$ which is flat, flat of finite presentation, étale, smooth or syntomic depending on whether $\tau $ is fpqc, fppf, étale, smooth, or syntomic, property $\mathcal{P}$ holds for $f$ if property $\mathcal{P}$ holds for the composition $f' : X' \to Y$.
Then $\mathcal{P}$ is $\tau $ local on the source.
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. (Hint: Use locality on the source and target to reduce the verification of property $\mathcal{P}$ to the case of a morphism between affines. Then apply (1) and (4).)
$\square$
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