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.

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. 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 ∩ N = I n − k ( I k M ∩ N ) . {\displaystyle 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 = ⨁ n = 0 ∞ I n {\textstyle B_{I}R=\bigoplus _{n=0}^{\infty }I^{n}} (B for blow-up.) We say a decreasing sequence of submodules M = M 0 ⊃ M 1 ⊃ M 2 ⊃ ⋯ {\displaystyle M=M_{0}\supset M_{1}\supset M_{2}\supset \cdots } is an I-filtration if I M n ⊂ M n + 1 {\displaystyle IM_{n}\subset M_{n+1}}; moreover, it is stable if I M n = M n + 1 {\displaystyle IM_{n}=M_{n+1}} for sufficiently large n. If M is given an I-filtration, we set B I M = ⨁ n = 0 ∞ M n {\textstyle B_{I}M=\bigoplus _{n=0}^{\infty }M_{n}}; it is a graded module over B I R {\displaystyle B_{I}R}.

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

B I M {\displaystyle B_{I}M} is a finitely generated module over B I R {\displaystyle B_{I}R} if and only if the filtration is I-stable.

Indeed, if the filtration is I-stable, then B I M {\displaystyle B_{I}M} is generated by the first k + 1 {\displaystyle k+1} terms M 0 , … , M k {\displaystyle M_{0},\dots ,M_{k}} and those terms are finitely generated; thus, B I M {\displaystyle B_{I}M} is finitely generated. Conversely, if it is finitely generated, then it is generated by ⨁ j = 0 k M j {\textstyle \bigoplus _{j=0}^{k}M_{j}} for some k ≥ 0 {\displaystyle k\geq 0}. Then, for n > k {\displaystyle n>k}, each f in M n {\displaystyle M_{n}} can be written as f = ∑ a j g j , a j ∈ I n − j {\displaystyle f=\sum a_{j}g_{j},\quad a_{j}\in I^{n-j}} with g j {\displaystyle g_{j}} in M j , j ≤ k {\displaystyle M_{j},j\leq k}. That is, f ∈ I n − k M k {\displaystyle f\in I^{n-k}M_{k}}.

We can now prove the lemma, assuming R is Noetherian. Let M n = I n M {\displaystyle M_{n}=I^{n}M}. Then M n {\displaystyle M_{n}} are an I-stable filtration. Thus, by the observation, B I M {\displaystyle B_{I}M} is finitely generated over B I R {\displaystyle B_{I}R}. But B I R ≃ R [ I t ] {\displaystyle B_{I}R\simeq R[It]} is a Noetherian ring since R is. (The ring R [ I t ] {\displaystyle R[It]} is called the Rees algebra.) Thus, B I M {\displaystyle B_{I}M} is a Noetherian module and any submodule is finitely generated over B I R {\displaystyle B_{I}R}; in particular, B I N {\displaystyle B_{I}N} is finitely generated when N is given the induced filtration; i.e., N n = M n ∩ N {\displaystyle 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: ⋂ n = 1 ∞ I n = 0 {\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 {\displaystyle N}, we find k such that for n ≥ k {\displaystyle n\geq k}, I n ∩ N = I n − k ( I k ∩ N ) . {\displaystyle I^{n}\cap N=I^{n-k}(I^{k}\cap N).} Taking n = k + 1 {\displaystyle n=k+1}, this means I k + 1 ∩ N = I ( I k ∩ N ) {\displaystyle I^{k+1}\cap N=I(I^{k}\cap N)} or N = I N {\displaystyle N=IN}. Thus, if A is local, N = 0 {\displaystyle 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):

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 ) ⊂ I N {\displaystyle u(N)\subset IN}. Then there is a relation: u n + a 1 u n − 1 + ⋯ + a n − 1 u + a n = 0 , a i ∈ I i . {\displaystyle 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 x N = 0 {\displaystyle xN=0}, which implies N = 0 {\displaystyle N=0}, as x {\displaystyle 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 {\displaystyle A} to be the ring of algebraic integers (i.e., the integral closure of Z {\displaystyle \mathbb {Z} } in C {\displaystyle \mathbb {C} }). If p {\displaystyle {\mathfrak {p}}} is a prime ideal of A, then we have: p n = p {\displaystyle {\mathfrak {p}}^{n}={\mathfrak {p}}} for every integer n > 0 {\displaystyle n>0}. Indeed, if y ∈ p {\displaystyle y\in {\mathfrak {p}}}, then y = α n {\displaystyle y=\alpha ^{n}} for some complex number α {\displaystyle \alpha }. Now, α {\displaystyle \alpha } is integral over Z {\displaystyle \mathbb {Z} }; thus in A {\displaystyle A} and then in p {\displaystyle {\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:

Theorem (Krull Intersection)—Let I be a ideal of commutative Noetherian ring A and M be a finitely generated A-module. Set N := ⋂ n = 1 ∞ I n M . {\displaystyle N:=\textstyle \bigcap _{n=1}^{\infty }I^{n}M.} Then there exists x ∈ 1 + I {\displaystyle x\in 1+I} such that x annihilates N.

Footnotes

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