Lemma 43.15.2. Let $A$ be a Noetherian local ring. Let $I \subset A$ be an ideal of definition. Let $0 \to M' \to M \to M'' \to 0$ be a short exact sequence of finite $A$-modules. Let $d \geq \dim (\text{Supp}(M))$. Then

\[ e_ I(M, d) = e_ I(M', d) + e_ I(M'', d) \]

**Proof.**
Immediate from the definitions and Algebra, Lemma 10.59.10.
$\square$

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