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.

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) harv error: no target: CITEREFAtiyahMacDonald1969 (help). 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,

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

Proof

The lemma immediately follows from the fact that R is Noetherian once necessary notions and notations are set up.[3]

For any ring R and an ideal I in R, we set ${\displaystyle B_{I}R=\textstyle \bigoplus _{n=0}^{\infty }I^{n}}$ (B for blow-up.) We say a decreasing sequence of submodules ${\displaystyle M=M_{0}\supset M_{1}\supset M_{2}\supset \cdots }$ is an I-filtration if ${\displaystyle IM_{n}\subset M_{n+1}}$; moreover, it is stable if ${\displaystyle IM_{n}=M_{n+1}}$ for sufficiently large n. If M is given an I-filtration, we set ${\displaystyle B_{I}M=\textstyle \bigoplus _{n=0}^{\infty }M_{n}}$; it is a graded module over ${\displaystyle B_{I}R}$.

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

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

Indeed, if the filtration is I-stable, then ${\displaystyle B_{I}M}$ is generated by the first ${\displaystyle k+1}$ terms ${\displaystyle M_{0},\dots ,M_{k}}$ and those terms are finitely generated; thus, ${\displaystyle B_{I}M}$ is finitely generated. Conversely, if it is finitely generated, say, by some homogeneous elements in ${\displaystyle \textstyle \bigoplus _{j=0}^{k}M_{j}}$, then, for ${\displaystyle n\geq k}$, each f in ${\displaystyle M_{n}}$ can be written as

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

with the generators ${\displaystyle g_{j}}$ in ${\displaystyle M_{j},j\leq k}$. That is, ${\displaystyle f\in I^{n-k}M_{k}}$.

We can now prove the lemma, assuming R is Noetherian. Let ${\displaystyle M_{n}=I^{n}M}$. Then ${\displaystyle M_{n}}$ are an I-stable filtration. Thus, by the observation, ${\displaystyle B_{I}M}$ is finitely generated over ${\displaystyle B_{I}R}$. But ${\displaystyle B_{I}R\simeq R[It]}$ is a Noetherian ring since R is. (The ring ${\displaystyle R[It]}$ is called the Rees algebra.) Thus, ${\displaystyle B_{I}M}$ is a Noetherian module and any submodule is finitely generated over ${\displaystyle B_{I}R}$; in particular, ${\displaystyle B_{I}N}$ is finitely generated when N is given the induced filtration; i.e., ${\displaystyle N_{n}=M_{n}\cap N}$. 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: ${\displaystyle \textstyle \bigcap _{n=1}^{\infty }I^{n}=0}$ 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 ${\displaystyle N}$, we find k such that for ${\displaystyle n\geq k}$,

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

But then ${\displaystyle N=IN}$. Thus, if A is local, ${\displaystyle N=0}$ 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 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 ${\displaystyle u(N)\subset IN}$. Then there is a relation:

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

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

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 ${\displaystyle A}$ to be the ring of algebraic integers (i.e., the integral closure of ${\displaystyle \mathbb {Z} }$ in ${\displaystyle \mathbb {C} }$). If ${\displaystyle {\mathfrak {p}}}$ is a prime ideal of A, then we have: ${\displaystyle {\mathfrak {p}}^{n}={\mathfrak {p}}}$ for every integer ${\displaystyle n>0}$. Indeed, if ${\displaystyle y\in {\mathfrak {p}}}$, then ${\displaystyle y=\alpha ^{n}}$ for some complex number ${\displaystyle \alpha }$. Now, ${\displaystyle \alpha }$ is integral over ${\displaystyle \mathbb {Z} }$; thus in ${\displaystyle A}$ and then in ${\displaystyle {\mathfrak {p}}}$, proving the claim.

References

1. 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. Here: Lemma 1
2. 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. Here: Sect.7, Lemma 7.2, p.10
3. Eisenbud, Lemma 5.1 harvnb error: no target: CITEREFEisenbud (help)

• Atiyah, Michael Francis; Macdonald, I.G. (1969). Introduction to Commutative Algebra. Westview Press. ISBN 978-0-201-40751-8.
• Eisenbud, David (1995). Commutative Algebra with a View Toward Algebraic Geometry. Graduate Texts in Mathematics. 150. Springer-Verlag. doi:10.1007/978-1-4612-5350-1. ISBN 0-387-94268-8.

Further reading

• 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.