Lemma 10.59.9 (00KB)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.59: Noetherian local rings
Lemma 10.59.9 (cite)

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Lemma 10.59.9. Let $R$ be a Noetherian local ring. Let $I \subset R$ be an ideal of definition. Let $M$ be a finite $R$ -module which does not have finite length. If $M' \subset M$ is a submodule with finite colength, then $\chi _{I, M} - \chi _{I, M'}$ is a polynomial of degree $<$ degree of either polynomial.

Proof. Follows from Lemma 10.59.2 by elementary calculus. $\square$

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