Artin–Rees lemma

In mathematics, the Artin–Rees lemma is a basic result about modules over a Noetherian ring, along with results such as the Hilbert basis theorem. It was proved in the 1950s in independent works by the mathematicians Emil Artin and David Rees; a special case was known to Oscar Zariski prior to their work.

One consequence of the lemma is the Krull intersection theorem. The result is also used to prove the exactness property of completion (Atiyah & MacDonald 1969, pp. 107–109).

Statement

Let I be an ideal in a Noetherian ring R; let M be a finitely generated R-module and let N a submodule of M. Then there exists an integer k ≥ 1 so that, for n ≥ k,

${\displaystyle I^{n}M\cap N=I^{n-k}((I^{k}M)\cap N).}$

Proof

The lemma immediately follows from the fact that R is Noetherian once necessary notions and notations are set up.[1]

For any ring R and an ideal I in R, we set ${\displaystyle B_{I}R=\textstyle \bigoplus _{n=0}^{\infty }I^{n}}$ (B for blow-up.) We say a decreasing sequence of submodules ${\displaystyle M=M_{0}\supset M_{1}\supset M_{2}\supset \cdots }$ is an I-filtration if ${\displaystyle IM_{n}\subset M_{n+1}}$; moreover, it is stable if ${\displaystyle IM_{n}=M_{n+1}}$ for sufficiently large n. If M is given an I-filtration, we set ${\displaystyle B_{I}M=\textstyle \bigoplus _{n=0}^{\infty }M_{n}}$; it is a graded module over ${\displaystyle B_{I}R}$.

Now, let M be a R-module with the I-filtration ${\displaystyle M_{i}}$ by finitely generated R-modules. We make an observation

${\displaystyle B_{I}M}$ is a finitely generated module over ${\displaystyle B_{I}R}$ if and only if the filtration is I-stable.

Indeed, if the filtration is I-stable, then ${\displaystyle B_{I}M}$ is generated by the first ${\displaystyle k+1}$ terms ${\displaystyle M_{0},\dots ,M_{k}}$ and those terms are finitely generated; thus, ${\displaystyle B_{I}M}$ is finitely generated. Conversely, if it is finitely generated, say, by some homogeneous elements in ${\displaystyle \textstyle \bigoplus _{j=0}^{k}M_{j}}$, then, for ${\displaystyle n\geq k}$, each f in ${\displaystyle M_{n}}$ can be written as

${\displaystyle f=\sum a_{ij}g_{ij},\quad a_{ij}\in I^{n-j}}$

with the generators ${\displaystyle g_{ij}}$ in ${\displaystyle M_{j},j\leq k}$. That is, ${\displaystyle f\in I^{n-k}M_{k}}$.

We can now prove the lemma, assuming R is Noetherian. Let ${\displaystyle M_{n}=I^{n}M}$. Then ${\displaystyle M_{n}}$ are an I-stable filtration. Thus, by the observation, ${\displaystyle B_{I}M}$ is finitely generated over ${\displaystyle B_{I}R}$. But ${\displaystyle B_{I}R\simeq R[It]}$ is a Noetherian ring since R is. (The ring ${\displaystyle R[It]}$ is called the Rees algebra.) Thus, ${\displaystyle B_{I}M}$ is a Noetherian module and any submodule is finitely generated over ${\displaystyle B_{I}R}$; in particular, ${\displaystyle B_{I}N}$ is finitely generated when N is given the induced filtration; i.e., ${\displaystyle N_{n}=M_{n}\cap N}$. Then the induced filtration is I-stable again by the observation.

Proof of Krull's intersection theorem

Besides the use in completion of a ring, a typical application of the lemma is the proof of the Krull's intersection theorem, which says: ${\displaystyle \textstyle \bigcap _{n=1}^{\infty }I^{n}=0}$ for a proper ideal I in a commutative Noetherian local ring. By the lemma applied to the intersection N, we find k such that for ${\displaystyle n\geq k}$,

${\displaystyle I^{n}\cap N=I^{n-k}(I^{k}\cap N).}$

But then ${\displaystyle N=IN}$ and thus ${\displaystyle N=0}$ by Nakayama.

References

1. ^ Eisenbud, Lemma 5.1