Artin–Rees lemma

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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,

InMN=Ink(IkMN).

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 BIR=n=0In (B for blow-up.) We say a decreasing sequence of submodules M=M0M1M2 is an I-filtration if IMnMn+1; moreover, it is stable if IMn=Mn+1 for sufficiently large n. If M is given an I-filtration, we set BIM=n=0Mn; it is a graded module over BIR.

Now, let M be a R-module with the I-filtration Mi by finitely generated R-modules. We make an observation

BIM is a finitely generated module over BIR if and only if the filtration is I-stable.

Indeed, if the filtration is I-stable, then BIM is generated by the first k+1 terms M0,,Mk and those terms are finitely generated; thus, BIM is finitely generated. Conversely, if it is finitely generated, say, by some homogeneous elements in j=0kMj, then, for nk, each f in Mn can be written as f=ajgj,ajInj with the generators gj in Mj,jk. That is, fInkMk.

We can now prove the lemma, assuming R is Noetherian. Let Mn=InM. Then Mn are an I-stable filtration. Thus, by the observation, BIM is finitely generated over BIR. But BIRR[It] is a Noetherian ring since R is. (The ring R[It] is called the Rees algebra.) Thus, BIM is a Noetherian module and any submodule is finitely generated over BIR; in particular, BIN is finitely generated when N is given the induced filtration; i.e., Nn=MnN. 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=1In=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 nk, InN=Ink(IkN). Taking n=k+1, this means Ik+1N=I(IkN) 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 [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)IN. Then there is a relation: un+a1un1++an1u+an=0,aiIi.

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 in ). If 𝔭 is a prime ideal of A, then we have: 𝔭n=𝔭 for every integer n>0. Indeed, if y𝔭, then y=αn for some complex number α. Now, α is integral over ; thus in A and then in 𝔭, proving the claim.

References

  1. David Rees (1956). "Two classical theorems of ideal theory". Proc. Camb. Phil. Soc. 52 (1): 155–157. doi:10.1017/s0305004100031091. Bibcode1956PCPS...52..155R.  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. Atiyah & MacDonald 1969, pp. 107–109
  4. Eisenbud, David (1995). Commutative Algebra with a View Toward Algebraic Geometry. Graduate Texts in Mathematics. 150. Springer-Verlag. Lemma 5.1. doi:10.1007/978-1-4612-5350-1. ISBN 0-387-94268-8. 
  5. Atiyah & MacDonald 1969, Proposition 2.4.

Atiyah, Michael Francis; MacDonald, I.G. (1969). Introduction to Commutative Algebra. Westview Press. pp. 107–109. ISBN 978-0-201-40751-8. 

Further reading