The Stacks project

Lemma 43.15.3. Let $A$ be a Noetherian local ring. Let $I \subset A$ be an ideal of definition. Let $M$ be a finite $A$-module. Let $d \geq \dim (\text{Supp}(M))$. Then

\[ e_ I(M, d) = \sum \text{length}_{A_\mathfrak p}(M_\mathfrak p) e_ I(A/\mathfrak p, d) \]

where the sum is over primes $\mathfrak p \subset A$ with $\dim (A/\mathfrak p) = d$.

Proof. Both the left and side and the right hand side are additive in short exact sequences of modules of dimension $\leq d$, see Lemma 43.15.2 and Algebra, Lemma 10.52.3. Hence by Algebra, Lemma 10.62.1 it suffices to prove this when $M = A/\mathfrak q$ for some prime $\mathfrak q$ of $A$ with $\dim (A/\mathfrak q) \leq d$. This case is obvious. $\square$

Comments (0)

There are also:

  • 1 comment(s) on Section 43.15: Algebraic multiplicities

Post a comment

Your email address will not be published.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.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0AZX. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 43.15.3 as pdf