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. 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. 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). 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, 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 B I R = ⨁ n = 0 ∞ I n {displaystyle B_{I}R= extstyle igoplus _{n=0}^{infty }I^{n}} (B for blow-up.) We say a decreasing sequence of submodules M = M 0 ⊃ M 1 ⊃ M 2 ⊃ ⋯ {displaystyle M=M_{0}supset M_{1}supset M_{2}supset cdots } is an I-filtration if I M n ⊂ M n + 1 {displaystyle IM_{n}subset M_{n+1}} ; moreover, it is stable if I M n = M n + 1 {displaystyle IM_{n}=M_{n+1}} for sufficiently large n. If M is given an I-filtration, we set B I M = ⨁ n = 0 ∞ M n {displaystyle B_{I}M= extstyle igoplus _{n=0}^{infty }M_{n}} ; it is a graded module over B I R {displaystyle B_{I}R} . Now, let M be a R-module with the I-filtration M i {displaystyle M_{i}} by finitely generated R-modules. We make an observation Indeed, if the filtration is I-stable, then B I M {displaystyle B_{I}M} is generated by the first k + 1 {displaystyle k+1} terms M 0 , … , M k {displaystyle M_{0},dots ,M_{k}} and those terms are finitely generated; thus, B I M {displaystyle B_{I}M} is finitely generated. Conversely, if it is finitely generated, say, by some homogeneous elements in ⨁ j = 0 k M j {displaystyle extstyle igoplus _{j=0}^{k}M_{j}} , then, for n ≥ k {displaystyle ngeq k} , each f in M n {displaystyle M_{n}} can be written as with the generators g j {displaystyle g_{j}} in M j , j ≤ k {displaystyle M_{j},jleq k} . That is, f ∈ I n − k M k {displaystyle fin I^{n-k}M_{k}} . We can now prove the lemma, assuming R is Noetherian. Let M n = I n M {displaystyle M_{n}=I^{n}M} . Then M n {displaystyle M_{n}} are an I-stable filtration. Thus, by the observation, B I M {displaystyle B_{I}M} is finitely generated over B I R {displaystyle B_{I}R} . But B I R ≃ R [ I t ] {displaystyle B_{I}Rsimeq R} is a Noetherian ring since R is. (The ring R [ I t ] {displaystyle R} is called the Rees algebra.) Thus, B I M {displaystyle B_{I}M} is a Noetherian module and any submodule is finitely generated over B I R {displaystyle B_{I}R} ; in particular, B I N {displaystyle B_{I}N} is finitely generated when N is given the induced filtration; i.e., N n = M n ∩ N {displaystyle N_{n}=M_{n}cap N} . Then the induced filtration is I-stable again by the observation.