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; 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,

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

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 ${\textstyle B_{I}R=\bigoplus _{n=0}^{\infty }I^{n}}$ (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, then it is generated by ${\textstyle \bigoplus _{j=0}^{k}M_{j}}$ for some $k\geq 0$. 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 $M_{j},j\leq k$. 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 $I^{k+1}\cap N=I(I^{k}\cap N)$ 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):

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.

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 $N:=\textstyle \bigcap _{n=1}^{\infty }I^{n}M.$ Then there exists $x\in 1+I$ such that x annihilates N.

References

• Rees, David (1956). "Two classical theorems of ideal theory". Mathematical Proceedings of the Cambridge Philosophical Society. 52 (1): 155–157. Bibcode:1956PCPS...52..155R. doi:10.1017/s0305004100031091. S2CID 121827047.
• 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.
• Atiyah, Michael Francis; MacDonald, I.G. (1969). Introduction to Commutative Algebra. Westview Press. pp. 107–109. ISBN 978-0-201-40751-8.
• Eisenbud, David (1995). Commutative Algebra with a View Toward Algebraic Geometry. Graduate Texts in Mathematics. Vol. 150. Springer-Verlag. doi:10.1007/978-1-4612-5350-1. ISBN 0-387-94268-8.
• Conrad, Brian; de Jong, Aise Johan (2002). "Approximation of versal deformations" (PDF). Journal of Algebra. 255 (2): 489–515. doi:10.1016/S0021-8693(02)00144-8. MR 1935511. gives a somehow more precise version of the Artin–Rees lemma.

External links

• "Artin-Rees Theorem". PlanetMath.