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,

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 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
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. 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,
But then. Thus, if A is local, by Nakayama's lemma. If A is an integral domain, then one uses the determinant trick :
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.
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. 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.