Lemma 35.27.1. The property $\mathcal{P}(f)=$“$f$ is flat” is fpqc local on the source.

## 35.27 Properties of morphisms local in the fpqc topology on the source

Here are some properties of morphisms that are fpqc local on the source.

**Proof.**
Since flatness is defined in terms of the maps of local rings (Morphisms, Definition 29.25.1) what has to be shown is the following algebraic fact: Suppose $A \to B \to C$ are local homomorphisms of local rings, and assume $B \to C$ is flat. Then $A \to B$ is flat if and only if $A \to C$ is flat. If $A \to B$ is flat, then $A \to C$ is flat by Algebra, Lemma 10.39.4. Conversely, assume $A \to C$ is flat. Note that $B \to C$ is faithfully flat, see Algebra, Lemma 10.39.17. Hence $A \to B$ is flat by Algebra, Lemma 10.39.10. (Also see Morphisms, Lemma 29.25.13 for a direct proof.)
$\square$

Lemma 35.27.2. Then property $\mathcal{P}(f : X \to Y)=$“for every $x \in X$ the map of local rings $\mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X, x}$ is injective” is fpqc local on the source.

**Proof.**
Omitted. This is just a (probably misguided) attempt to be playful.
$\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (2)

Comment #1660 by Lucas Braune on

Comment #1661 by Lucas Braune on