Artin-Rees lemma: Difference between revisions

Revision as of 15:50, 27 February 2008

This article is about the statement of a simple but indispensable lemma in commutative algebra
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Suppose $A$ is a Noetherian commutative unital ring and $M$ is a finitely generated $A$ -module and $N$ a submodule of $M$ .

Suppose:

$M = M_{0} \supset M_{1} \supset M_{2} \supset \dots$

is an essentially $I$ -adic filtration (in other words, there exists $n_{0}$ such that for all $n \geq n_{0}$ , $I M_{n} = M_{n + 1}$ ).

Then the filtration of $N$ given by:

$N = N \cap M_{0} \supset N \cap M_{1} \supset N \cap M_{2} \supset \dots$

is also essentially $I$ -adic.

@@ Line 9: / Line 9: @@
 <math>M = M_0 \supset M_1 \supset M_2 \supset \ldots</math>
-is an essentially <math>I</math>-adic filtration (in other words, there exists <math>n_0</math> such that for all <math>n \ge n_0</math>, <math>IM_n = M_{n+1</math>).
+is an essentially <math>I</math>-adic filtration (in other words, there exists <math>n_0</math> such that for all <math>n \ge n_0</math>, <math>IM_n = M_{n+1}</math>).
 Then the filtration of <math>N</math> given by: