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. 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,
- [math]\displaystyle{ I^{n} M \cap N = I^{n - k} (I^{k} M \cap N). }[/math]
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 [math]\displaystyle{ B_I R = \bigoplus_{n=0}^\infty I^n }[/math] (B for blow-up.) We say a decreasing sequence of submodules [math]\displaystyle{ M = M_0 \supset M_1 \supset M_2 \supset \cdots }[/math] is an I-filtration if [math]\displaystyle{ I M_n \subset M_{n+1} }[/math]; moreover, it is stable if [math]\displaystyle{ I M_n = M_{n+1} }[/math] for sufficiently large n. If M is given an I-filtration, we set [math]\displaystyle{ B_I M = \bigoplus_{n=0}^\infty M_n }[/math]; it is a graded module over [math]\displaystyle{ B_I R }[/math].
Now, let M be a R-module with the I-filtration [math]\displaystyle{ M_i }[/math] by finitely generated R-modules. We make an observation
- [math]\displaystyle{ B_I M }[/math] is a finitely generated module over [math]\displaystyle{ B_I R }[/math] if and only if the filtration is I-stable.
Indeed, if the filtration is I-stable, then [math]\displaystyle{ B_I M }[/math] is generated by the first [math]\displaystyle{ k+1 }[/math] terms [math]\displaystyle{ M_0, \dots, M_k }[/math] and those terms are finitely generated; thus, [math]\displaystyle{ B_I M }[/math] is finitely generated. Conversely, if it is finitely generated, say, by some homogeneous elements in [math]\displaystyle{ \bigoplus_{j=0}^k M_j }[/math], then, for [math]\displaystyle{ n \ge k }[/math], each f in [math]\displaystyle{ M_n }[/math] can be written as [math]\displaystyle{ f = \sum a_{j} g_{j}, \quad a_{j} \in I^{n-j} }[/math] with the generators [math]\displaystyle{ g_{j} }[/math] in [math]\displaystyle{ M_j, j \le k }[/math]. That is, [math]\displaystyle{ f \in I^{n-k} M_k }[/math].
We can now prove the lemma, assuming R is Noetherian. Let [math]\displaystyle{ M_n = I^n M }[/math]. Then [math]\displaystyle{ M_n }[/math] are an I-stable filtration. Thus, by the observation, [math]\displaystyle{ B_I M }[/math] is finitely generated over [math]\displaystyle{ B_I R }[/math]. But [math]\displaystyle{ B_I R \simeq R[It] }[/math] is a Noetherian ring since R is. (The ring [math]\displaystyle{ R[It] }[/math] is called the Rees algebra.) Thus, [math]\displaystyle{ B_I M }[/math] is a Noetherian module and any submodule is finitely generated over [math]\displaystyle{ B_I R }[/math]; in particular, [math]\displaystyle{ B_I N }[/math] is finitely generated when N is given the induced filtration; i.e., [math]\displaystyle{ N_n = M_n \cap N }[/math]. 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: [math]\displaystyle{ \bigcap_{n=1}^\infty I^n = 0 }[/math] 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 [math]\displaystyle{ N }[/math], we find k such that for [math]\displaystyle{ n \ge k }[/math], [math]\displaystyle{ I^{n} \cap N = I^{n - k} (I^{k} \cap N). }[/math] Taking [math]\displaystyle{ n = k+1 }[/math], this means [math]\displaystyle{ I^{k+1}\cap N = I(I^{k}\cap N) }[/math] or [math]\displaystyle{ N = IN }[/math]. Thus, if A is local, [math]\displaystyle{ N = 0 }[/math] 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 [math]\displaystyle{ u(N) \subset IN }[/math]. Then there is a relation: [math]\displaystyle{ u^n + a_1 u^{n-1} + \cdots + a_{n-1} u + a_n = 0, \, a_i \in I^i. }[/math]
In the setup here, take u to be the identity operator on N; that will yield a nonzero element x in A such that [math]\displaystyle{ x N = 0 }[/math], which implies [math]\displaystyle{ N = 0 }[/math], as [math]\displaystyle{ x }[/math] 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 [math]\displaystyle{ A }[/math] to be the ring of algebraic integers (i.e., the integral closure of [math]\displaystyle{ \mathbb{Z} }[/math] in [math]\displaystyle{ \mathbb{C} }[/math]). If [math]\displaystyle{ \mathfrak p }[/math] is a prime ideal of A, then we have: [math]\displaystyle{ \mathfrak{p}^n = \mathfrak{p} }[/math] for every integer [math]\displaystyle{ n \gt 0 }[/math]. Indeed, if [math]\displaystyle{ y \in \mathfrak p }[/math], then [math]\displaystyle{ y = \alpha^n }[/math] for some complex number [math]\displaystyle{ \alpha }[/math]. Now, [math]\displaystyle{ \alpha }[/math] is integral over [math]\displaystyle{ \mathbb{Z} }[/math]; thus in [math]\displaystyle{ A }[/math] and then in [math]\displaystyle{ \mathfrak{p} }[/math], proving the claim.
References
- ↑ David Rees (1956). "Two classical theorems of ideal theory". Proc. Camb. Phil. Soc. 52 (1): 155–157. doi:10.1017/s0305004100031091. Bibcode: 1956PCPS...52..155R. Here: Lemma 1
- ↑ 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
- ↑ Atiyah & MacDonald 1969, pp. 107–109
- ↑ 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.
- ↑ 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
- Conrad, Brian; de Jong, Aise Johan (2002). "Approximation of versal deformations". Journal of Algebra 255 (2): 489–515. doi:10.1016/S0021-8693(02)00144-8. https://math.stanford.edu/~conrad/papers/approx.pdf. gives a somehow more precise version of the Artin–Rees lemma.
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