Lemma 47.11.5. Let $R \to S$ be a flat local ring homomorphism of Noetherian local rings. Denote $\mathfrak m \subset R$ the maximal ideal. Let $I \subset S$ be an ideal. If $S/\mathfrak mS$ is Cohen-Macaulay, then

$\text{depth}_ I(S) \geq \dim (S/\mathfrak mS) - \dim (S/\mathfrak mS + I)$

Proof. By Algebra, Lemma 10.98.3 any sequence in $S$ which maps to a regular sequence in $S/\mathfrak mS$ is a regular sequence in $S$. Thus it suffices to prove the lemma in case $R$ is a field. This is a special case of Lemma 47.11.4. $\square$

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