Noether normalization theorem - Revision history

Vipul: 4 revisions

2008-05-12T16:27:42Z

4 revisions

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Vipul at 21:13, 2 February 2008

2008-02-02T21:13:30Z

← Older revision		Revision as of 21:13, 2 February 2008
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	==Statement==		==Statement==

	Let <math>k</math> be a [[field]] and <math>R</math> a finitely generated <math>k</math>-algebra~~. Let <math>P_r \ge P_{r-1} \ge \ldots \ge P_0</math> be a chain of descending [[prime ideal]]s, maximal in the sense that no further prime ideals can be inserted in the chain~~. Then ~~there~~ exists a subring <math>S</math> of <math>R</math> ~~with~~ <math>S \cong k[x_1, x_2, \ldots, x_r]</math> ~~sich that~~ <math>R</math> is a finitely generated <math>S</math>-module and <math>P_i \cap S = (x_1, x_2, \ldots, x_i)</math>.		Let <math>k</math> be a [[field]] and <math>R</math> a finitely generated <math>k</math>-algebra. Then:

			* ''Weak version'': There exists a subring <math>S</math> of <math>R</math> such that <math>S \cong k[x_1, x_2, \ldots, x_r]</math> and <math>R</math> is a finitely generated <math>S</math>-module
			* ''Strong version'': Suppose <math>P_0 \le P_1 \le \ldots P_r</math> is an ascending chain of [[prime ideal]]s, such that no further prime ideals can be and <math>P_i \cap S = (x_1, x_2, \ldots, x_i)</math> Then we can choose <math>S</math> as above, with the further constraint that <math>P_i \cap S = (x_1,\ldots,x_i)</math>.

	==Applications==		==Applications==

	* [[~~Hilbert's~~ nullstellensatz]]		* [[weak nullstellensatz for arbitrary fields]] follows very directly from the weak version of the Noether normalization theorem.

Vipul at 19:22, 2 February 2008

2008-02-02T19:22:53Z

← Older revision		Revision as of 19:22, 2 February 2008
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	==Statement==		==Statement==

	Let <math>k</math> be a [[field]] and <math>R</math> a finitely generated <math>k</math>-algebra. Let <math>P_r \ge P_{r-1} \ge \ldots \ge P_0</math> be a chain of descending ]]prime ideal]]s, maximal in the sense that no further prime ideals can be inserted in the chain. Then there exists a subring <math>S</math> of <math>R</math> with <math>S \cong k[x_1, x_2, \ldots, x_r]</math> sich that <math>R</math> is a finitely generated <math>S</math>-module and <math>P_i \cap S = (x_1, x_2, \ldots, x_i)</math>.		Let <math>k</math> be a [[field]] and <math>R</math> a finitely generated <math>k</math>-algebra. Let <math>P_r \ge P_{r-1} \ge \ldots \ge P_0</math> be a chain of descending [[prime ideal]]s, maximal in the sense that no further prime ideals can be inserted in the chain. Then there exists a subring <math>S</math> of <math>R</math> with <math>S \cong k[x_1, x_2, \ldots, x_r]</math> sich that <math>R</math> is a finitely generated <math>S</math>-module and <math>P_i \cap S = (x_1, x_2, \ldots, x_i)</math>.

	==Applications==		==Applications==

	* [[Hilbert's nullstellensatz]]		* [[Hilbert's nullstellensatz]]

Vipul at 00:05, 8 January 2008

2008-01-08T00:05:06Z

← Older revision		Revision as of 00:05, 8 January 2008
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	==Statement==		==Statement==

	Let <math>k</math> be a [[field]] and <math>R</math> a finitely ~~generaetd~~ <math>k</math>-algebra. Let <math>P_r \ge P_{r-1} \ge \ldots \ge P_0</math> be a chain of descending ]]prime ideal]]s, maximal in the sense that no further prime ideals can be inserted in the chain. Then there exists a subring <math>S</math> of <math>R</math> with <math>S \cong k[x_1, x_2, \ldots, x_r]</math> sich that <math>R</math> is a finitely generated <math>S</math>-module and <math>P_i \cap S = (x_1, x_2, \ldots, x_i)</math>.		Let <math>k</math> be a [[field]] and <math>R</math> a finitely generated <math>k</math>-algebra. Let <math>P_r \ge P_{r-1} \ge \ldots \ge P_0</math> be a chain of descending ]]prime ideal]]s, maximal in the sense that no further prime ideals can be inserted in the chain. Then there exists a subring <math>S</math> of <math>R</math> with <math>S \cong k[x_1, x_2, \ldots, x_r]</math> sich that <math>R</math> is a finitely generated <math>S</math>-module and <math>P_i \cap S = (x_1, x_2, \ldots, x_i)</math>.

	==Applications==		==Applications==

	* [[Hilbert's nullstellensatz]]		* [[Hilbert's nullstellensatz]]

Vipul at 12:34, 7 August 2007

2007-08-07T12:34:10Z

New page

==Statement==

Let <math>k</math> be a [[field]] and <math>R</math> a finitely generaetd <math>k</math>-algebra. Let <math>P_r \ge P_{r-1} \ge \ldots \ge P_0</math> be a chain of descending ]]prime ideal]]s, maximal in the sense that no further prime ideals can be inserted in the chain. Then there exists a subring <math>S</math> of <math>R</math> with <math>S \cong k[x_1, x_2, \ldots, x_r]</math> sich that <math>R</math> is a finitely generated <math>S</math>-module and <math>P_i \cap S = (x_1, x_2, \ldots, x_i)</math>.

==Applications==

* [[Hilbert's nullstellensatz]]