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 10.99.3 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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