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

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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,

 

Proof

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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   (B for blow-up.) We say a decreasing sequence of submodules   is an I-filtration if  ; moreover, it is stable if   for sufficiently large n. If M is given an I-filtration, we set  ; it is a graded module over  .

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

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

Indeed, if the filtration is I-stable, then   is generated by the first   terms   and those terms are finitely generated; thus,   is finitely generated. Conversely, if it is finitely generated, say, by some homogeneous elements in  , then, for  , each f in   can be written as   with the generators   in  . That is,  .

We can now prove the lemma, assuming R is Noetherian. Let  . Then   are an I-stable filtration. Thus, by the observation,   is finitely generated over  . But   is a Noetherian ring since R is. (The ring   is called the Rees algebra.) Thus,   is a Noetherian module and any submodule is finitely generated over  ; in particular,   is finitely generated when N is given the induced filtration; i.e.,  . Then the induced filtration is I-stable again by the observation.

Krull's intersection theorem

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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:   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  , we find k such that for  ,   Taking  , this means   or  . Thus, if A is local,   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  . Then there is a relation:  

In the setup here, take u to be the identity operator on N; that will yield a nonzero element x in A such that  , which implies  , as   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   to be the ring of algebraic integers (i.e., the integral closure of   in  ). If   is a prime ideal of A, then we have:   for every integer  . Indeed, if  , then   for some complex number  . Now,   is integral over  ; thus in   and then in  , proving the claim.

References

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  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. S2CID 121827047. 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. S2CID 123809696. 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. Vol. 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

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