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 I-adic topology on M, and the other when considered as a module in its own right 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,

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 for blow-up.) We say a decreasing sequence of submodules  is an I-filtration if ; moreover, it is stable if  for sufficiently large n. If M is given an I-filtration, we set ; it is a graded module over .

Now, let M be a R-module with the I-filtration  by finitely generated R-modules. We make an observation
 is a finitely generated module over  if and only if the filtration is I-stable.
Indeed, if the filtration is I-stable, then  is generated by the first  terms  and those terms are finitely generated; thus,  is finitely generated. Conversely, if it is finitely generated, say, by some homogeneous elements in , then, for , each f in  can be written as

with the generators  in . That is, .

We can now prove the lemma, assuming R is Noetherian. Let . Then  are an I-stable filtration. Thus, by the observation,  is finitely generated over . But  is a Noetherian ring since R is. (The ring  is called the Rees algebra.) Thus,  is a Noetherian module and any submodule is finitely generated over ; in particular,  is finitely generated when N is given the induced filtration; i.e., . 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:  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 , we find k such that for ,

Taking , this means  or . Thus, if A is local,  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 , which implies , as  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  to be the ring of algebraic integers (i.e., the integral closure of  in ). If  is a prime ideal of A, then we have:  for every integer . Indeed, if , then  for some complex number . Now,  is integral over ; thus in  and then in , proving the claim.

References

Further reading 
 gives a somehow more precise version of the Artin–Rees lemma.

External links
 

Commutative algebra
Lemmas in algebra
Module theory
Theorems in ring theory