Localization respects associated primes for Noetherian rings - Revision history

Vipul: 3 revisions

2008-05-12T16:26:52Z

3 revisions

← Older revision	Revision as of 16:26, 12 May 2008
(No difference)

Vipul at 17:17, 27 February 2008

2008-02-27T17:17:42Z

← Older revision		Revision as of 17:17, 27 February 2008
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	{{Noetherian ~~base~~ result}}		{{Noetherian ring result}}

	==Statement==		==Statement==

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	<math>Ass_{S^{-1}A}S^{-1}M = Ass_AM \cap Spec(S^{-1}A)</math>		<math>Ass_{S^{-1}A}S^{-1}M = Ass_AM \cap Spec(S^{-1}A)</math>

			==Proof==

			The key ingredient in the proof is the fact that if <math>m \in M</math>, the union of annihilators of all elements of <math>Sm</math>, can be realized as the annihilator of a ''single'' element <math>sm</math>.

Vipul at 16:52, 27 February 2008

2008-02-27T16:52:04Z

← Older revision		Revision as of 16:52, 27 February 2008
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			{{Noetherian base result}}
	==Statement==		==Statement==

Vipul: New page: ==Statement== Suppose $A$ is a Noetherian commutative unital ring and $M$ is any $A$ -module (not necessarily finitely generate...

2008-02-27T16:51:03Z

New page: ==Statement== Suppose <math>A</math> is a Noetherian commutative unital ring and <math>M</math> is any <math>A</math>-module (not necessarily finitely generate...

New page

==Statement==

Suppose <math>A</math> is a [[Noetherian ring|Noetherian]] [[commutative unital ring]] and <math>M</math> is any <math>A</math>-[[module]] (not necessarily finitely generated. Let <math>S</math> be a [[multiplicatively closed subset]] of <math>A</math>.

There is a natural inclusion on [[spectrum|spectra]]:

<math>Spec(S^{-1}A) \to Spec(A)</math>

The set of [[associated prime]]s for <math>S^{-1}M</math> as an <math>S^{-1}A</math>-module is the inverse image in <math>Spec(S^{-1}A)</math> of the set of [[associated prime]]s for <math>M</math> as an <math>A</math>-module.

If we identify <math>Spec(S^{-1}A)</math> with its image, a subset of <math>Spec(A)</math>, then we can write:

<math>Ass_{S^{-1}A}S^{-1}M = Ass_AM \cap Spec(S^{-1}A)</math>

Localization respects associated primes for Noetherian rings - Revision history

Vipul: 3 revisions

Vipul at 17:17, 27 February 2008

Vipul at 16:52, 27 February 2008

Vipul: New page: ==Statement== Suppose A is a Noetherian commutative unital ring and M is any A-module (not necessarily finitely generate...

Vipul: New page: ==Statement== Suppose $A$ is a Noetherian commutative unital ring and $M$ is any $A$ -module (not necessarily finitely generate...