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.
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,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, then it is generated by for some. Then, for, each f in can be written as
with 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,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 :
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. 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.
Both the cases of the Noetherian ring being local and the Noetherian ring being an integral domain are consequences of a more general version of Krull's intersection theorem, which is also a consequence of the Artin–Rees and Nakayama lemmata: