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The Stacks project

Lemma 10.52.13. Let A \to B be a flat local homomorphism of local rings. Then for any A-module M we have

\text{length}_ A(M) \text{length}_ B(B/\mathfrak m_ AB) = \text{length}_ B(M \otimes _ A B).

In particular, if \text{length}_ B(B/\mathfrak m_ AB) < \infty then M has finite length if and only if M \otimes _ A B has finite length.

Proof. The ring map A \to B is faithfully flat by Lemma 10.39.17. Hence if 0 = M_0 \subset M_1 \subset \ldots \subset M_ n = M is a chain of length n in M, then the corresponding chain 0 = M_0 \otimes _ A B \subset M_1 \otimes _ A B \subset \ldots \subset M_ n \otimes _ A B = M \otimes _ A B has length n also. This proves \text{length}_ A(M) = \infty \Rightarrow \text{length}_ B(M \otimes _ A B) = \infty . Next, assume \text{length}_ A(M) < \infty . In this case we see that M has a filtration of length \ell = \text{length}_ A(M) whose quotients are A/\mathfrak m_ A. Arguing as above we see that M \otimes _ A B has a filtration of length \ell whose quotients are isomorphic to B \otimes _ A A/\mathfrak m_ A = B/\mathfrak m_ AB. Thus the lemma follows. \square


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