The Stacks project

73.14 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.

Definition 73.14.1. 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

\[ f \text{ has }\mathcal{P} \Leftrightarrow \text{each }X_ i \to Y\text{ has }\mathcal{P}. \]

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 73.14.2. 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 73.14.3. 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 66.22.2. 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

\begin{align*} f \text{ has }\mathcal{P}_{spaces} & \Leftrightarrow U \to V \text{ has }\mathcal{P} \\ & \Leftrightarrow \text{each }U_ i \to V \text{ has }\mathcal{P} \\ & \Leftrightarrow \text{each }X_ i \to Y\text{ has }\mathcal{P}_{spaces}. \end{align*}

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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