Artin–Rees lemma

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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 (Atiyah & MacDonald 1969, pp. 107–109).

Statement[edit]

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[edit]

The lemma immediately follows from the fact that R is Noetherian once necessary notions and notations are set up.[1]

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[edit]

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.

References[edit]

  1. ^ Eisenbud, Lemma 5.1

External links[edit]