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. $\square$
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