Definition 74.14.1. Let $S$ be a scheme. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$. Let $\tau \in \{ fpqc, \linebreak[0] fppf, \linebreak[0] syntomic, \linebreak[0] smooth, \linebreak[0] {\acute{e}tale}\} $. We say $\mathcal{P}$ is *$\tau $ local on the source*, or *local on the source for the $\tau $-topology* if for any morphism $f : X \to Y$ of algebraic spaces over $S$, and any $\tau $-covering $\{ X_ i \to X\} _{i \in I}$ of algebraic spaces we have

## 74.14 Properties of morphisms local on the source

In this section we define what it means for a property of morphisms of algebraic spaces to be local on the source. Please compare with Descent, Section 35.26.

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 74.14.2. Let $S$ be a scheme. Let $\tau \in \{ fpqc, \linebreak[0] fppf, \linebreak[0] syntomic, \linebreak[0] smooth, \linebreak[0] {\acute{e}tale}\} $. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$ 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. smooth, resp. étale, 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 74.14.3. Let $S$ be a scheme. Let $\tau \in \{ fpqc, \linebreak[0] fppf, \linebreak[0] syntomic, \linebreak[0] smooth, \linebreak[0] {\acute{e}tale}\} $. Suppose that $\mathcal{P}$ is a property of morphisms of schemes over $S$ which is étale local on the source-and-target. Denote $\mathcal{P}_{spaces}$ the corresponding property of morphisms of algebraic spaces over $S$, see Morphisms of Spaces, Definition 67.22.2. If $\mathcal{P}$ is local on the source for the $\tau $-topology, then $\mathcal{P}_{spaces}$ is local on the source for the $\tau $-topology.

**Proof.**
Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\{ X_ i \to X\} _{i \in I}$ be a $\tau $-covering of algebraic spaces. Choose a scheme $V$ and a surjective étale morphism $V \to Y$. Choose a scheme $U$ and a surjective étale morphism $U \to X \times _ Y V$. For each $i$ choose a scheme $U_ i$ and a surjective étale morphism $U_ i \to X_ i \times _ X U$.

Note that $\{ X_ i \times _ X U \to U\} _{i \in I}$ is a $\tau $-covering. Note that each $\{ U_ i \to X_ i \times _ X U\} $ is an étale covering, hence a $\tau $-covering. Hence $\{ U_ i \to U\} _{i \in I}$ is a $\tau $-covering of algebraic spaces over $S$. But since $U$ and each $U_ i$ is a scheme we see that $\{ U_ i \to U\} _{i \in I}$ is a $\tau $-covering of schemes over $S$.

Now we have

the first and last equivalence by the definition of $\mathcal{P}_{spaces}$ the middle equivalence because we assumed $\mathcal{P}$ is local on the source in the $\tau $-topology. $\square$

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