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,

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

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 $B_I R = \textstyle\bigoplus_{n=0}^\infty I^n$ (B for blow-up.) We say a decreasing sequence of submodules $M = M_0 \supset M_1 \supset M_2 \supset \cdots$ is an I-filtration if $I M_n \subset M_{n+1}$; moreover, it is stable if $I M_n = M_{n+1}$ for sufficiently large n. If M is given an I-filtration, we set $B_I M = \textstyle\bigoplus_{n=0}^\infty M_n$; it is a graded module over $B_I R$.

Now, let M be a R-module with the I-filtration $M_i$ by finitely generated R-modules. We make an observation

$B_I M$ is a finitely generated module over $B_I R$ if and only if the filtration is I-stable.

Indeed, if the filtration is I-stable, then $B_I M$ is generated by the first $k+1$ terms $M_0, \dots, M_k$ and those terms are finitely generated; thus, $B_I M$ is finitely generated. Conversely, if it is finitely generated, say, by $\textstyle\bigoplus_{j=0}^k M_j$, then, for $n \ge k$, each f in $M_n$ can be written as

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

with the generators $g_{ij}$ in $M_j, j \le k$. That is, $f \in I^{n-k} M_k$.

We can now prove the lemma, assuming R is Noetherian. Let $M_n = I^n M$. Then $M_n$ are an I-stable filtration. Thus, by the observation, $B_I M$ is finitely generated over $B_I R$. But $B_I R \simeq R[It]$ is a Noetherian ring since R is. (The ring $R[It]$ is called the Rees algebra.) Thus, $B_I M$ is a Noetherian module and any submodule is finitely generated over $B_I R$; in particular, $B_I N$ is finitely generated when N is given the induced filtration; i.e., $N_n = M_n \cap N$. 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: $\textstyle\bigcap_{n=1}^\infty I^n = 0$ for a proper ideal I in a Noetherian local ring. By the lemma applied to the intersection N, we find k such that for $n \ge k$,

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

But then $N = IN$ and thus $N = 0$ by Nakayama.

References

1. Eisenbud, Lemma 5.1

• Atiyah, Michael Francis; Macdonald, I.G. (1969), Introduction to Commutative Algebra, Westview Press, ISBN 978-0-201-40751-8
• Eisenbud, David, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, 1995, ISBN 0-387-94268-8.

External links

• Artin-Rees Theorem at PlanetMath.org.