Artin-Rees' theorem

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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 :

.

Applications

If there is any module, then define the powers

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 .

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.