Lemma 10.59.4 (00K7)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.59: Noetherian local rings
Lemma 10.59.4 (cite)

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Lemma 10.59.4. Suppose that $I$ , $I'$ are two ideals of definition for the Noetherian local ring $R$ . Let $M$ be a finite $R$ -module. There exists a constant $a$ such that $\chi _{I, M}(n) \leq \chi _{I', M}(an)$ for $n \geq 1$ .

Proof. There exists an integer $c \geq 1$ such that $(I')^ c \subset I$ . Hence we get a surjection $M/(I')^{c(n + 1)}M \to M/I^{n + 1}M$ . Whence the result with $a = 2c - 1$ . $\square$

Comments (2)

Comment #4266 by Manuel Hoff on June 20, 2019 at 12:23

I think the argument doesn't work with $a=c+1$ . If I am not mistaken one needs at least $a=2c-1$ .

Comment #4436 by Johan on September 01, 2019 at 09:23

OK, yes, the choice we had only works for $n$ large enough. Thanks and fixed here.

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