Krull intersection theorem for modules: Difference between revisions

Latest revision as of 16:26, 12 May 2008

This article defines a result where the base ring (or one or more of the rings involved) is Noetherian
View more results involving Noetherianness or Read a survey article on applying Noetherianness

This fact is an application of the following pivotal fact/result/idea: Artin-Rees lemma
View other applications of Artin-Rees lemma OR Read a survey article on applying Artin-Rees lemma

This fact is an application of the following pivotal fact/result/idea: Cayley-Hamilton theorem
View other applications of Cayley-Hamilton theorem OR Read a survey article on applying Cayley-Hamilton theorem

Statement

Let $R$ be a Noetherian ring and $I$ be an ideal inside $R$ . Suppose $M$ is a finitely generated module over $R$ . Then, we have the following:

Let $N=\bigcap _{j=1}^{\infty }I^{j}M$ . Then, $IN=N$
There exists $r\in I$ such that $(1-r)N=0$

Results used

Applications

Krull intersection theorem for Jacobson radical, also covers the case of a local ring
Krull intersection theorem for domains

Proof

The intersection equals its product with $I$

We first show that the intersection equals its product with $I$ . This is the step where we se the Artin-Rees lemma.

Let:

$N:=\bigcap _{1}^{\infty }I^{j}M$

Now consider the filtration:

$M\supset IM\supset I^{2}M\supset \ldots$

this is an $I$ -adic filtration and the underlying ring is Noetherian, hence by the Artin-Rees lemma, the following filtration is also $I$ -adic:

$N\supset IM\cap N\supset I^{2}M\cap N\supset \ldots$

Since each $I^{j}M$ contains $N$ , the filtration below is the same as the filtration:

$N\supset N\supset N\supset \ldots$

This being $I$ -adic forces that $IN=N$ .

Finding the element $r$

Since $IN=N$ , we can find an element $r\in I$ such that $(1-r)N=0$ . This is an application of the Cayley-Hamilton theorem: we first find the Cayley-Hamilton polynomial, then observe that $1$ is a root of the polynomial, and then take the negative of the sum of all coefficients of higher degree terms.

References

''Dimensionstheorie in Stellenringen by Wolfgang Krull, 1938

Textbook references

Book:Eisenbud, Page 152

@@ Line 1: / Line 1: @@
+{{Noetherian ring result}}
 {{applicationof|Artin-Rees lemma}}
-{{applicationof|Nakayama's lemma}}
+{{applicationof|Cayley-Hamilton theorem}}
 ==Statement==
-Let <math>R</math> be a [[Noetherian ring]] and <math>I</math> be an [[ideal]] inside <math>R</math>.
+Let <math>R</math> be a [[Noetherian ring]] and <math>I</math> be an [[ideal]] inside <math>R</math>. Suppose <math>M</math> is a [[finitely generated module]] over <math>R</math>. Then, we have the following:
-* If <math>M</math> is a finitely generated <math>R</math>-module, then there exists <math>r \in I</math> such that:
+# Let <math>N = \bigcap_{j=1}^\infty I^j M</math>. Then, <math>IN = N</math>
+# There exists <math>r \in I</math> such that <math>(1 - r)N = 0</math>
-<math>(1-r)(\bigcap_1^\infty I^jM) = 0</math>
+==Results used==
-* If <math>R</math> is an [[integral domain]] or a [[local ring]] and <math>I</math> is a [[proper ideal]] then:
+* [[Artin-Rees lemma]]
+* [[Cayley-Hamilton theorem]]
-<math>\bigcap_1^\infty I^j = 0</math>
+==Applications==
-==Results used==
+* [[Krull intersection theorem for Jacobson radical]], also covers the case of a [[local ring]]
+* [[Krull intersection theorem for domains]]
+==Proof==
+===The intersection equals its product with <math>I</math>===
+We first show that the intersection equals its product with <math>I</math>. This is the step where we se the Artin-Rees lemma.
+Let:
+<math>N := \bigcap_1^\infty I^jM</math>
+Now consider the filtration:
+<math>M \supset IM \supset I^2M \supset \ldots</math>
+this is an <math>I</math>-adic filtration and the underlying ring is [[Noetherian ring|Noetherian]], hence by the [[Artin-Rees lemma]], the following filtration is also <math>I</math>-adic:
+<math>N \supset IM \cap N \supset I^2M \cap N\supset \ldots</math>
+Since each <math>I^jM</math> contains <math>N</math>, the filtration below is the same as the filtration:
+<math>N \supset N \supset N \supset \ldots</math>
+This being <math>I</math>-adic forces that <math>IN = N</math>.
+===Finding the element <math>r</math>===
-* [[Artin-Rees lemma]]
+Since <math>IN = N</math>, we can find an element <math>r \in I</math> such that <math>(1 - r)N = 0</math>. This is an application of the [[Cayley-Hamilton theorem]]: we first find the Cayley-Hamilton polynomial, then observe that <math>1</math> is a root of the polynomial, and then take the negative of the sum of all coefficients of higher degree terms.
-* [[Nakayama's lemma]]
 ==References==
-* ''''Dimensionstheorie in Stellenringen'' by [[Wolfgang Krull], 1938
+* ''''Dimensionstheorie in Stellenringen'' by [[Wolfgang Krull]], 1938
 ===Textbook references===
 * {{booklink|Eisenbud}}, Page 152