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.
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, 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
We say a decreasing sequence of submodules
for sufficiently large n. If M is given an I-filtration, we set
by finitely generated R-modules.
We make an observation Indeed, if the filtration is I-stable, then
and those terms are finitely generated; thus,
Conversely, if it is finitely generated, say, by some homogeneous elements in
We can now prove the lemma, assuming R is Noetherian.
are an I-stable filtration.
is called the Rees algebra.)
is a Noetherian module and any submodule is finitely generated over
is finitely generated when N is given the induced filtration; i.e.,
Then the induced filtration is I-stable again by the observation.
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
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
In the setup here, take u to be the identity operator on N; that will yield a nonzero element x in A such that
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
, proving the claim.