Lemma 10.163.3. Let $R \to S$ be a flat local homomorphism of local Noetherian rings. Then the following are equivalent

$S$ is Cohen-Macaulay, and

$R$ and $S/\mathfrak m_ RS$ are Cohen-Macaulay.

Lemma 10.163.3. Let $R \to S$ be a flat local homomorphism of local Noetherian rings. Then the following are equivalent

$S$ is Cohen-Macaulay, and

$R$ and $S/\mathfrak m_ RS$ are Cohen-Macaulay.

**Proof.**
Follows from the definitions and Lemmas 10.163.2 and 10.112.7.
$\square$

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