Artin–Rees lemma

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;^[1]^[2] a special case was known to Oscar Zariski prior to their work.

An intuitive characterization of the lemma involves the notion that a submodule N of a module M over some ring A with specified ideal I holds a priori two topologies: one induced by the topology on M, and the other when considered with the I-adic topology over A. Then Artin-Rees dictates that these topologies actually coincide, at least when A is Noetherian and M finitely-generated.

One consequence of the lemma is the Krull intersection theorem. The result is also used to prove the exactness property of completion.^[3] The lemma also plays a key role in the study of ℓ-adic sheaves.

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.^[4]

For any ring R and an ideal I in R, we set ${\textstyle B_{I}R=\bigoplus _{n=0}^{\infty }I^{n}}$ (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 $IM_{n}\subset M_{n+1}$ ; moreover, it is stable if $IM_{n}=M_{n+1}$ for sufficiently large n. If M is given an I-filtration, we set ${\textstyle B_{I}M=\bigoplus _{n=0}^{\infty }M_{n}}$ ; 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, say, by some homogeneous elements in ${\textstyle \bigoplus _{j=0}^{k}M_{j}}$ , then, for $n\geq k$ , each f in $M_{n}$ can be written as

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

with the generators

g_{j}

M_{j},j\leq 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: ${\textstyle \bigcap _{n=1}^{\infty }I^{n}=0}$ 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 $n\geq k$ ,

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)

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 ^[5] (that is a variant of the Cayley–Hamilton theorem and yields Nakayama's lemma):

Theorem — Let u be an endomorphism of an A-module N generated by n elements and I an ideal of A such that $u(N)\subset IN$ . Then there is a relation:

u^{n}+a_{1}u^{n-1}+\cdots +a_{n-1}u+a_{n}=0,\,a_{i}\in I^{i}.

In the setup here, take u to be the identity operator on N; that will yield a nonzero element x in A such that $xN=0$ , 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.

References

^ David Rees (1956). "Two classical theorems of ideal theory". Proc. Camb. Phil. Soc. 52 (1): 155–157. Bibcode:1956PCPS...52..155R. doi:10.1017/s0305004100031091. S2CID 121827047. Here: Lemma 1
^ Sharp, R. Y. (2015). "David Rees. 29 May 1918 — 16 August 2013". Biographical Memoirs of Fellows of the Royal Society. 61: 379–401. doi:10.1098/rsbm.2015.0010. S2CID 123809696. Here: Sect.7, Lemma 7.2, p.10
^ Atiyah & MacDonald 1969, pp. 107–109
^ Eisenbud, David (1995). Commutative Algebra with a View Toward Algebraic Geometry. Graduate Texts in Mathematics. Vol. 150. Springer-Verlag. Lemma 5.1. doi:10.1007/978-1-4612-5350-1. ISBN 0-387-94268-8.
^ Atiyah & MacDonald 1969, Proposition 2.4.