Artin-Rees' theorem

The Artin-Rees theorem , named after Emil Artin and David Rees , is a theorem from commutative algebra . It makes a statement about the products of powers of ideals of a Noetherian ring and finitely generated modules . The sentence can be used to demonstrate a certain topology of a sub-module as a relative topology .

Formulation of the sentence

It is an ideal in a commutative, Noetherian ring . Next there is a finitely generated module and a sub-module. Then there is a number such that it applies to all : ${\displaystyle {\mathfrak {a}}\subset R}$ ${\displaystyle R}$ ${\displaystyle M}$ ${\displaystyle R}$ ${\displaystyle N\subset M}$ ${\displaystyle k\in \mathbb {N} }$ ${\displaystyle i\geq k}$

${\displaystyle {\mathfrak {a}}^{i}M\cap N={\mathfrak {a}}^{i-k}({\mathfrak {a}}^{k}M\cap N)}$.

Applications

If there is any module, then define the powers ${\displaystyle M}$ ${\displaystyle R}$

${\displaystyle {\mathfrak {a}}^{1}M\supset {\mathfrak {a}}^{2}M\supset {\mathfrak {a}}^{3}M\supset \ldots }$

a null environment base in and thus a topology, the so-called -adic topology. In this a lot is if and open , if for every one is having . In the situation of the sentence above, and the sub-module carry the -adic topology, but as a subset also carry the relative topology of the -adic topology of . With the help of Artin-Rees' theorem, it is no longer difficult to show the equality of these two topologies . ${\displaystyle M}$ ${\displaystyle {\mathfrak {a}}}$ ${\displaystyle U\subset M}$ ${\displaystyle x\in U}$ ${\displaystyle i\in N}$ ${\displaystyle x+{\mathfrak {a}}^{i}M\subset U}$ ${\displaystyle M}$ ${\displaystyle N}$ ${\displaystyle {\mathfrak {a}}}$ ${\displaystyle N}$ ${\displaystyle {\mathfrak {a}}}$ ${\displaystyle M}$ ${\displaystyle N}$

Artin-Rees' Theorem can also be used to prove Krull's Average Theorem .

Individual evidence

1. ^ Siegfried Bosch : Algebraic Geometry and Commutative Algebra. Springer-Verlag, 2012, ISBN 1-4471-4828-2 , 2.3. Lemma 1.