Artin–Rees lemma explained

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.

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. 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} (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 $B\_I\; R\; =\; \backslash bigoplus\_^\backslash infty\; I^n$ (B for blow-up.) We say a decreasing sequence of submodules

M = M _{0} \supset M _{1} \supset M _{2} \supset … 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 $B\_I\; M\; =\; \backslash bigoplus\_^\backslash infty\; M\_n$; 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 observationB _{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,} ... , 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 $\backslash bigoplus\_^k\; M\_j$, then, for n \ge k , each f in M _{n} can be written as$$f\; =\; \backslash sum\; a\_\; g\_,\; \backslash quad\; a\_\; \backslash in\; I^$$with the generators g _{j} 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.

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: $\backslash bigcap\_^\backslash infty\; I^n\; =\; 0$ 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 \ge k ,$$I^\; \backslash cap\; N\; =\; I^\; (I^\; \backslash cap\; N).$$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):

In the setup here, take u to be the identity operator on N; that will yield a nonzero element x in A such that

x N = 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 Z in C ). If ak p is a prime ideal of A, then we have: ak{p} ^{n} = ak{p} for every integer n > 0 . Indeed, if y \in ak p , then y = \alpha ^{n} for some complex number \alpha . Now, \alpha is integral over Z ; thus in A and then in ak{p} , proving the claim.

References

• Rees. David. David Rees (mathematician). 1956. Two classical theorems of ideal theory. Mathematical Proceedings of the Cambridge Philosophical Society. 52. 1 . 155–157. 10.1017/s0305004100031091. 1956PCPS...52..155R . 121827047.
• 10.1098/rsbm.2015.0010. David Rees. 29 May 1918 — 16 August 2013. Biographical Memoirs of Fellows of the Royal Society. 61. 379–401. 2015. Sharp. R. Y.. 123809696 . free.
• Book: Atiyah . Michael Francis . Michael Atiyah . MacDonald . I.G. . Ian G. Macdonald . Introduction to Commutative Algebra . Westview Press . 978-0-201-40751-8 . 1969 . 107–109.
• Book: David Eisenbud. Eisenbud. David. Commutative Algebra with a View Toward Algebraic Geometry. Graduate Texts in Mathematics. 150. Springer-Verlag. 1995. 0-387-94268-8. 10.1007/978-1-4612-5350-1.
• Conrad . Brian . Brian Conrad . de Jong . Aise Johan . Aise Johan de Jong . 10.1016/S0021-8693(02)00144-8 . 2 . . 1935511 . 489–515 . Approximation of versal deformations . 255 . 2002. gives a somehow more precise version of the Artin–Rees lemma.