Lemma 10.59.7. Let $R$ be a Noetherian local ring. Let $M$ be a finite $R$-module.

1. The degree of the numerical polynomial $\varphi _{I, M}$ is independent of the ideal of definition $I$.

2. The degree of the numerical polynomial $\chi _{I, M}$ is independent of the ideal of definition $I$.

Proof. Part (2) follows immediately from Lemma 10.59.4. Part (1) follows from (2) because $\varphi _{I, M}(n) = \chi _{I, M}(n) - \chi _{I, M}(n - 1)$ for $n \geq 1$. $\square$

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