The Stacks project

Lemma 10.52.14. Let $A \to B \to C$ be flat local homomorphisms of local rings. Then

\[ \text{length}_ B(B/\mathfrak m_ A B) \text{length}_ C(C/\mathfrak m_ B C) = \text{length}_ C(C/\mathfrak m_ A C) \]

Proof. Follows from Lemma 10.52.13 applied to the ring map $B \to C$ and the $B$-module $M = B/\mathfrak m_ A B$ $\square$

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