Artin–Rees lemma - Alchetron, The Free Social Encyclopedia

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.

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,

I n M ∩ N = I n − k ( ( I k M ) ∩ N ) .

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 B I R = ⨁ n = 0 ∞ I n (B for blow-up.) We say a decreasing sequence of submodules M = M 0 ⊃ M 1 ⊃ M 2 ⊃ ⋯ is an I-filtration if I M n ⊂ M n + 1 ; moreover, it is stable if I M n = M n + 1 for sufficiently large n. If M is given an I-filtration, we set B I M = ⨁ n = 0 ∞ M n ; it is a graded module over B I R .

Now, let M be a R-module with the I-filtration M i by finitely generated R-modules. We make an observation

B I M is a finitely generated module over B I R if and only if the filtration is I-stable.

Indeed, if the filtration is I-stable, then B I M is generated by the first k + 1 terms M 0 , … , M k and those terms are finitely generated; thus, B I M is finitely generated. Conversely, if it is finitely generated, say, by some homogeneous elements in ⨁ j = 0 k M j , then, for n ≥ k , each f in M n can be written as

f = ∑ a i j g i j , a i j ∈ I n − j

with the generators g i j in M j , j ≤ k . That is, f ∈ I n − k M k .

We can now prove the lemma, assuming R is Noetherian. Let M n = I n M . Then M n are an I-stable filtration. Thus, by the observation, B I M is finitely generated over B I R . But B I R ≃ R [ I t ] is a Noetherian ring since R is. (The ring R [ I t ] is called the Rees algebra.) Thus, B I M is a Noetherian module and any submodule is finitely generated over B I R ; in particular, B I N is finitely generated when N is given the induced filtration; i.e., N n = M n ∩ N . 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: ⋂ n = 1 ∞ I n = 0 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 n ≥ k ,

But then N = I N and thus N = 0 by Nakayama.

References

Artin–Rees lemma Wikipedia

(Text) CC BY-SA

Contents

Statement

Proof

Proof of Krull's intersection theorem

References