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

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