Average theorem from Krull

The average rate of Krull named after Wolfgang Krull , is a set of the commutative algebra that deals with powers of ideals of a noetherian ring busy. The consequence of this is that a certain topology on finitely generated modules over a Noetherian ring is Hausdorffian .

Formulation of the sentence

Let it be an ideal in a commutative, Noetherian ring and a finitely generated module. ${\displaystyle {\mathfrak {a}}\subset R}$ ${\displaystyle R}$ ${\displaystyle M}$ ${\displaystyle R}$

• For true . ${\displaystyle \textstyle N:=\bigcap _{i\in N}{\mathfrak {a}}^{i}M}$ ${\displaystyle {\mathfrak {a}}N=N}$

• Is also included in the Jacobson radical , so is . ${\displaystyle {\mathfrak {a}}}$ ${\displaystyle \textstyle N:=\bigcap _{i\in N}{\mathfrak {a}}^{i}M=\{0\}}$

The proof is a simple application of Artin-Rees' theorem . After the latter there is a , so that applies to all : ${\displaystyle k\in \mathbb {N} }$ ${\displaystyle i>k}$

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

It follows for ${\displaystyle i=k+1}$

${\displaystyle N\subset {\mathfrak {a}}^{k+1}M\cap N={\mathfrak {a}}^{1}({\mathfrak {a}}^{k}M\cap N)\subset {\mathfrak {a}}N\subset N}$

and with it the first claim. The second follows from the first and the lemma of Nakayama .

application

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 . ${\displaystyle M}$ ${\displaystyle {\mathfrak {a}}}$ ${\displaystyle U\subset M}$ ${\displaystyle x\in U}$ ${\displaystyle i\in N}$ ${\displaystyle x+{\mathfrak {a}}^{i}M\subset U}$

If there is a finitely generated module and an ideal contained in the Jacobson radical, then with the -adic topology is a Hausdorff space. If two different elements are out , then and is therefore for sufficiently large . Then and are disjoint neighborhoods of and . ${\displaystyle M}$ ${\displaystyle R}$ ${\displaystyle {\mathfrak {a}}}$ ${\displaystyle M}$ ${\displaystyle {\mathfrak {a}}}$ ${\displaystyle x,y}$ ${\displaystyle M}$ ${\displaystyle x-y\not =0}$ ${\displaystyle x-y\notin {\mathfrak {a}}^{i}M}$ ${\displaystyle i}$ ${\displaystyle x+{\mathfrak {a}}^{i}M}$ ${\displaystyle y+{\mathfrak {a}}^{i}M}$ ${\displaystyle x}$ ${\displaystyle y}$

Individual evidence

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