Lemma 10.59.7 (00K9)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.59: Noetherian local rings
Lemma 10.59.7 (cite)

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

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