Artin–Rees lemma

Remarks

The Rees algebra $R[It]$ is Noetherian because $R$ is Noetherian and $I$ is finitely generated. The 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. A finite set of homogeneous generators of degree at most $k$ gives the equality for $n \ge k$. In words, the filtration $(I^n M \cap N)_n$ of $N$ is a stable $I$-filtration. Consequently the $I$-adic topology of $M$ induces the $I$-adic topology on $N$, since $I^n M \cap N \subseteq I^{n-k} N$. Example: 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.

Going deeper

For a student

The lemma is the main technical tool for completions of Noetherian rings. It implies the Krull intersection theorem, and it shows that $I$-adic completion is exact on finitely generated modules, which gives flatness of completion.