Lemma 10.163.2. Suppose that $R \to S$ is a flat and local ring homomorphism of Noetherian local rings. Then

$\text{depth}(S) = \text{depth}(R) + \text{depth}(S/\mathfrak m_ RS).$

Proof. This is a special case of Lemma 10.163.1. $\square$

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