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

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

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 $M = B/\mathfrak m_ A B$
$\square$

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)

There are also: