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 $a = 2c - 1$. $\square$

Comment #4266 by Manuel Hoff on

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 on

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

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