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Let $f \colon R \to S$ be a homomorphism between (local Noetherian) rings which turns $S$ into a finitely generated $R-$module and let $M$ be a finitely generated over $S$.

Is $M$ is Cohen-Macaulay over $S$ if and only if $M$ is Cohen-Macaulay over $R$? Is $\text{depth}_R(M) = \text{depth}_S(M)$?

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    $\begingroup$ Yes to both questions. Unfortunately (?) the only reference I know is in french: Bourbaki, Algèbre commutative X, §2, Proposition 8. $\endgroup$
    – abx
    Commented Dec 19, 2021 at 14:04
  • $\begingroup$ Another way to see this easily is to notice that $H^i_{m_R}(M) = H^i_{m_S}(M)$ since under your hypotheses $\sqrt{m_R S} = m_S$. Then use the description of depth via local cohomology vanishing. $\endgroup$
    – Karl Schwede
    Commented Dec 20, 2021 at 4:58
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    $\begingroup$ Also, see stacks.math.columbia.edu/tag/0AUK for the statement on depth. $\endgroup$
    – Johan
    Commented Dec 20, 2021 at 18:08

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