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The Artin-Rees lemma states that if $R$ is a Noetherian ring, $I \subseteq R$ is an ideal and $N \subseteq M$ are finitely generated $R$-modules, then there exists an integer $k$ such that for every $n \ge k$, we have $$I^n M \cap N = I^{n-k} \cdot (I^kM \cap N) \subseteq I^{n-k} N \, .$$

The inclusion $I^n M \cap N \subseteq I^{n-k}N$ is sometimes called the weak Artin-Rees-Lemma.

There exist several versions of this weak Artin-Rees Lemma which are uniform with respect to $I$, i.e., the corresponding integer $k$ depends only on the modules $N \subseteq M$ of $R$ and works for every ideal $I$ of $R$.

Do there exist variants of this result which are also uniform with respect to the modules $N$ and $M$?

I wonder whether it is possible to bound the corresponding $k$'s in terms of invariants of the ring $R$. For example, suppose that $R$ is a regular local ring. Is $k$ bounded by the Krull dimension of $R$?

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There is an answer to your question by Craig Huneke: "Uniform bounds in noetherian rings", Invent. math. 107, 203-223 (1992).

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  • $\begingroup$ Craig Huneke uses a filtration 0 = K_0 \subseteq K_1 \subseteq \ldots \subseteq K_n = M/N such that each quotient K_i/K_{i-1} is isomorphic to R/p_i for some prime p_i of R. Can one bound the length of such a filtration in terms of the Krull dimension of R? $\endgroup$
    – user447241
    Commented Oct 17, 2014 at 14:45
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    $\begingroup$ I found "there is an answer in XXX" a quite frustrating answer (it would rather fit in a comment). $\endgroup$
    – YCor
    Commented Jan 14, 2015 at 22:49

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