Definition 35.23.1. 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

## 35.23 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.

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 35.23.2. 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 35.23.3. 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 35.23.4. 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 34.9.8, 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).)
$\square$

Remark 35.23.5. (This is a repeat of Remarks 35.12.3 and 35.19.5 above.) In Lemma 35.23.4 above if $\tau = smooth$ then in condition (4) we may assume that the morphism is a (surjective) standard smooth morphism. Similarly, when $\tau = syntomic$ or $\tau = {\acute{e}tale}$.

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