The Stacks project

Lemma 10.112.9. Let $R \to S$ be a local homomorphism of Noetherian local rings. Assume $R$ Cohen-Macaulay. If $S$ is finite flat over $R$, or if $S$ is flat over $R$ and $\dim (S) \leq \dim (R)$, then $S$ is Cohen-Macaulay and $\dim (R) = \dim (S)$.

Proof. Let $x_1, \ldots , x_ d \in \mathfrak m_ R$ be a regular sequence of length $d = \dim (R)$. By Lemma 10.68.5 this maps to a regular sequence in $S$. Hence $S$ is Cohen-Macaulay if $\dim (S) \leq d$. This is true if $S$ is finite flat over $R$ by Lemma 10.112.4. And in the second case we assumed it. $\square$

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