Lemma 35.24.1. The property $\mathcal{P}(f)=$“$f$ is flat” is 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 10.38.4. Conversely, assume $A \to C$ is flat. Note that $B \to C$ is faithfully flat, see Algebra, Lemma 10.38.17. Hence $A \to B$ is flat by Algebra, Lemma 10.38.10. (Also see Morphisms, Lemma 29.25.13 for a direct proof.)
$\square$

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