The Stacks project

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.) $\square$