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;^[1][2] a special case was known to Oscar Zariski prior to their work.

An intuitive characterization of the lemma involves the notion that a submodule N of a module M over some ring A with specified ideal I holds a priori two topologies: one induced by the topology on M, and the other when considered with the I-adic topology over A. Then Artin-Rees dictates that these topologies actually coincide, at least when A is Noetherian and M finitely-generated.

One consequence of the lemma is the Krull intersection theorem. The result is also used to prove the exactness property of completion.^[3] 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,

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

Proof

The lemma immediately follows from the fact that R is Noetherian once necessary notions and notations are set up.^[4]

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 ${\displaystyle M=M_{0}\supset M_{1}\supset M_{2}\supset \cdots }$ 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 ${\textstyle B_{I}M=\bigoplus _{n=0}^{\infty }M_{n}}$; 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 ${\textstyle \bigoplus _{j=0}^{k}M_{j}}$, then, for ${\displaystyle n\geq k}$, each f in ${\displaystyle M_{n}}$ can be written as $${\displaystyle f=\sum a_{j}g_{j},\quad a_{j}\in I^{n-j}}$$ with the generators ${\displaystyle g_{j}}$ 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.

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 ${\displaystyle N}$, we find k such that for ${\displaystyle n\geq k}$, $${\displaystyle I^{n}\cap N=I^{n-k}(I^{k}\cap N).}$$ Taking ${\displaystyle n=k+1}$, this means ${\displaystyle I^{k+1}\cap N=I(I^{k}\cap N)}$ or ${\displaystyle N=IN}$. Thus, if A is local, ${\displaystyle N=0}$ by Nakayama's lemma. If A is an integral domain, then one uses the determinant trick ^[5] (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 ${\displaystyle u(N)\subset IN}$. Then there is a relation: $${\displaystyle u^{n}+a_{1}u^{n-1}+\cdots +a_{n-1}u+a_{n}=0,\,a_{i}\in I^{i}.}$$

In the setup here, take u to be the identity operator on N; that will yield a nonzero element x in A such that ${\displaystyle xN=0}$, which implies ${\displaystyle N=0}$, as ${\displaystyle x}$ 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 ${\displaystyle A}$ to be the ring of algebraic integers (i.e., the integral closure of ${\displaystyle \mathbb {Z} }$ in ${\displaystyle \mathbb {C} }$). If ${\displaystyle {\mathfrak {p}}}$ is a prime ideal of A, then we have: ${\displaystyle {\mathfrak {p}}^{n}={\mathfrak {p}}}$ for every integer ${\displaystyle n>0}$. Indeed, if ${\displaystyle y\in {\mathfrak {p}}}$, then ${\displaystyle y=\alpha ^{n}}$ for some complex number ${\displaystyle \alpha }$. Now, ${\displaystyle \alpha }$ is integral over ${\displaystyle \mathbb {Z} }$; thus in ${\displaystyle A}$ and then in ${\displaystyle {\mathfrak {p}}}$, proving the claim.

References

1. David Rees (1956). "Two classical theorems of ideal theory". Proc. Camb. Phil. Soc. 52 (1): 155–157. Bibcode:1956PCPS...52..155R. doi:10.1017/s0305004100031091. S2CID 121827047. Here: Lemma 1
2. Sharp, R. Y. (2015). "David Rees. 29 May 1918 — 16 August 2013". Biographical Memoirs of Fellows of the Royal Society. 61: 379–401. doi:10.1098/rsbm.2015.0010. S2CID 123809696. Here: Sect.7, Lemma 7.2, p.10
3. Atiyah & MacDonald 1969, pp. 107–109
4. Eisenbud, David (1995). Commutative Algebra with a View Toward Algebraic Geometry. Graduate Texts in Mathematics. Vol. 150. Springer-Verlag. Lemma 5.1. doi:10.1007/978-1-4612-5350-1. ISBN 0-387-94268-8.
5. Atiyah & MacDonald 1969, Proposition 2.4.

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.