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  "body": "\n## Statement\n\nLet $R$ be a commutative [[Noetherian ring|0670T59HGE]] and let $I$ be an [[ideal|096Z9MP2ME]] of $R$.\nLet $M$ be a [[finitely generated|05QY1VEBYD]] $R$-module, and let $N$ be a [[submodule|05WDSN597C]] of $M$.\nThen there is an integer $k \\ge 0$ such that\n$$I^n M \\cap N = I^{n-k} (I^k M \\cap N) \\quad \\text{for all } n \\ge k.$$\n\n## Remarks\n\nThe [[Rees algebra|W8ZAH8TTHS]] $R[It]$ is Noetherian because $R$ is Noetherian and $I$ is finitely generated.\nThe module $\\bigoplus_n I^n M$ is a finitely generated module over $R[It]$, so its graded submodule $\\bigoplus_n (I^n M \\cap N)$ is finitely generated too.\nA finite set of homogeneous generators of degree at most $k$ gives the equality for $n \\ge k$.\nIn words, the [[filtration|WHME2NGANG]] $(I^n M \\cap N)_n$ of $N$ is a stable $I$-filtration.\nConsequently the $I$-adic topology of $M$ induces the $I$-adic topology on $N$, since $I^n M \\cap N \\subseteq I^{n-k} N$.\nExample: for $R = M = \\Z$, $I = 2\\Z$ and $N = 6\\Z$, one has $2^n \\Z \\cap 6\\Z = 3 \\cdot 2^n \\Z = 2^{n-1}(2\\Z \\cap 6\\Z)$ for $n \\ge 1$, so $k = 1$ works.\n\n## Explanations\n\n### L1 human\n\nTake a big module and a smaller module inside it.\nThere are two ways to say that an element of the small module is tiny.\nIt can be a multiple of a high power of $I$ inside the big module, or inside the small module.\nThe lemma says that these two ways agree up to a fixed shift $k$.\n\n### L2 human\n\nThe lemma is the main technical tool for completions of Noetherian rings.\nIt implies the [[Krull intersection theorem|1Q1AH2V41S]], and it shows that $I$-adic completion is exact on finitely generated modules, which gives [[flatness of completion|NPF42DJMBX]].\n",
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