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Being Cohen-Macaulay ascends and descends along étale maps.

Proposition 41.19.3. Let $A$, $B$ be Noetherian local rings. Let $f : A \to B$ be an étale homomorphism of local rings. Then $A$ is Cohen-Macaulay if and only if $B$ is so.

Proof. A local ring $A$ is Cohen-Macaulay if and only if $\dim (A) = \text{depth}(A)$. As both of these invariants is preserved under an étale extension, the claim follows. $\square$

Comments (1)

Comment #985 by on

Suggested slogan: Being Cohen-Macaulay for Noetherian local rings ascends and descends along etale maps.

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View Proposition 41.19.3 as pdf