Artin–Rees lemma

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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,

I^{n} M \cap N = I^{n - k} ((I^{k} M) \cap N).

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_I R = \oplus_0^\infty I^n (B for blow-up.) We say a decreasing sequence of submodules M = M_0 \supset M_1 \supset M_2 \supset \cdots is an I-filtration if I M_n \subset 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 = \oplus_0^\infty 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, \dots, M_k and those terms are finitely generated; thus, B_I M is finitely generated. Conversely, if it is finitely generated, say, by \oplus_0^k M_j, then, for n \ge k, each f in M_n can be written as

f = \sum a_{ij} g_{ij}, \quad a_{ij} \in I^{n-j}

with the generators g_{ij} in M_j, j \le k. That is, f \in 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 \simeq R[It] is a Noetherian ring since R is. (The ring R[It] 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 \cap N. 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: \cap_1^\infty I^n = 0 for a proper ideal I in a Noetherian local ring. By the lemma applied to the intersection N, we find k such that for n \ge k,

I^{n} \cap N = I^{n - k} (I^{k} \cap N).

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

References[edit]

  1. ^ Eisenbud, Lemma 5.1

External links[edit]