Lemma 23.8.9. Let $A \to B$ be a flat local homomorphism of Noetherian local rings. Then the following are equivalent

$B$ is a complete intersection,

$A$ and $B/\mathfrak m_ A B$ are complete intersections.

Lemma 23.8.9. Let $A \to B$ be a flat local homomorphism of Noetherian local rings. Then the following are equivalent

$B$ is a complete intersection,

$A$ and $B/\mathfrak m_ A B$ are complete intersections.

**Proof.**
Now that the definition makes sense this is a trivial reformulation of the (nontrivial) Proposition 23.8.4.
$\square$

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