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. The lemma also plays a key role in the study of ℓ-adic sheaves.

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.

For any ring R and an ideal I in R, we set ${\textstyle B_{I}R=\bigoplus _{n=0}^{\infty }I^{n}}$ (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 $IM_{n}\subset M_{n+1}$; moreover, it is stable if $IM_{n}=M_{n+1}$ for sufficiently large n. If M is given an I-filtration, we set ${\textstyle B_{I}M=\bigoplus _{n=0}^{\infty }M_{n}}$; 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 some homogeneous elements in ${\textstyle \bigoplus _{j=0}^{k}M_{j}}$, then, for $n\geq k$, each f in $M_{n}$ can be written as

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

with the generators $g_{j}$ in $M_{j},j\leq k$. 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.

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 commutative Noetherian ring that is either a local ring or an integral domain. By the lemma applied to the intersection $N$, we find k such that for $n\geq k$,

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

Taking $n=k+1$, this means $I^{k+1}\cap N=I(I^{k}\cap N)$ or $N=IN$. Thus, if A is local, $N=0$ by Nakayama's lemma. If A is an integral domain, then one uses the determinant trick (that is a variant of the Cayley–Hamilton theorem and yields Nakayama's lemma):

Theorem — Let u be an endomorphism of an A-module N generated by n elements and I an ideal of A such that $u(N)\subset IN$. Then there is a relation:

$$u^{n}+a_{1}u^{n-1}+\cdots +a_{n-1}u+a_{n}=0,\,a_{i}\in I^{i}.$$

In the setup here, take u to be the identity operator on N; that will yield a nonzero element x in A such that $xN=0$, which implies $N=0$, as $x$ is a nonzerodivisor.

For both a local ring and an integral domain, the "Noetherian" cannot be dropped from the assumption: for the local ring case, see local ring#Commutative case. For the integral domain case, take $A$ to be the ring of algebraic integers (i.e., the integral closure of $\mathbb {Z}$ in $\mathbb {C}$). If ${\mathfrak {p}}$ is a prime ideal of A, then we have: ${\mathfrak {p}}^{n}={\mathfrak {p}}$ for every integer $n>0$. Indeed, if $y\in {\mathfrak {p}}$, then $y=\alpha ^{n}$ for some complex number $\alpha$. Now, $\alpha$ is integral over $\mathbb {Z}$; thus in $A$ and then in ${\mathfrak {p}}$, proving the claim.

References

Atiyah, Michael Francis; MacDonald, I.G. (1969). Introduction to Commutative Algebra. Westview Press. pp. 107–109. ISBN 978-0-201-40751-8.

Further reading

• Conrad, Brian; de Jong, Aise Johan (2002). "Approximation of versal deformations" (PDF). Journal of Algebra. 255 (2): 489–515. doi:10.1016/S0021-8693(02)00144-8. MR 1935511. gives a somehow more precise version of the Artin–Rees lemma.

External links

• "Artin-Rees Theorem". PlanetMath.