Artin–Rees lemma

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 \cap N = I^{n - k} (I^{k} M \cap 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 = \bigoplus_{n=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 = \bigoplus_{n=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, then it is generated by \bigoplus_{j=0}^k M_j for some k \ge 0. Then, for n k, each f in M_n can be written as f = \sum a_{j} g_{j}, \quad a_{j} \in I^{n-j} with g_{j} 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.

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: \bigcap_{n=1}^\infty I^n = 0 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 N, we find k such that for n \ge k, I^{n} \cap N = I^{n - k} (I^{k} \cap N). Taking n = k+1, this means I^{k+1}\cap N = I(I^{k}\cap N) or N = IN. Thus, if A is local, N = 0 by Nakayama's lemma. If A is an integral domain, then one uses the determinant trick (that is, a variant of the Cayley–Hamilton theorem that yields Nakayama's lemma):

In the setup here, take u to be the identity operator on N; that will yield a nonzero element x in A such that x N = 0, which implies N = 0, as x 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 A to be the ring of algebraic integers (i.e., the integral closure of \mathbb{Z} in \mathbb{C}). If \mathfrak p is a prime ideal of A, then we have: \mathfrak{p}^n = \mathfrak{p} for every integer n 0. Indeed, if y \in \mathfrak p, then y = \alpha^n for some complex number \alpha. Now, \alpha is integral over \mathbb{Z}; thus in A and then in \mathfrak{p}, 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:

Footnotes

References

• gives a somehow more precise version of the Artin–Rees lemma.

References

1. {{harvnb. Rees. 1956
2. {{harvnb. Sharp. 2015
3. {{harvnb. Atiyah. MacDonald. 1969
4. {{harvnb. Eisenbud. 1995
5. {{harvnb. Atiyah. MacDonald. 1969
6. {{harvnb. Atiyah. MacDonald. 1969