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Definition 43.15.1. In the situation above, assume $d \geq \dim (\text{Supp}(M))$. In this case, if $d > \dim (\text{Supp}(M))$, then we set $e_ I(M, d) = 0$ and if $d = \dim (\text{Supp}(M))$, then we set $e_ I(M, d)$ equal to $d!$ times the leading coefficient of the numerical polynomial $\chi _{I, M}$. Thus in both cases we have

\[ \chi _{I, M}(n) \sim e_ I(M, d) \frac{n^ d}{d!} + \text{lower order terms} \]

The multiplicity of $M$ for the ideal of definition $I$ is $e_ I(M) = e_ I(M, \dim (\text{Supp}(M)))$.

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