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. $\square$

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