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.
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.
Proof of 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 local ring. By the lemma applied to the intersection N, we find k such that for ,
But then and thus by Nakayama.
- Eisenbud, Lemma 5.1
- Atiyah, Michael Francis; Macdonald, I.G. (1969), Introduction to Commutative Algebra, Westview Press, ISBN 978-0-201-40751-8
- Eisenbud, David, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, 1995, ISBN 0-387-94268-8.