Equivalence of dimension notions for affine domain - Revision history

Vipul: 5 revisions

2008-05-12T16:19:58Z

5 revisions

← Older revision	Revision as of 16:19, 12 May 2008
(No difference)

Vipul: /* Statement */

2008-03-17T12:46:43Z

Statement

← Older revision		Revision as of 12:46, 17 March 2008
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	==Statement==		==Statement==

	Let <math>A</math> be an [[affine domain]] over a [[field]] <math>k</math>, i.e. a finitely generated algebra over <math>k</math>, that also happens to be an integral domain. Then, the following are ~~equivalent~~:		Let <math>A</math> be an [[affine domain]] over a [[field]] <math>k</math>, i.e. a finitely generated algebra over <math>k</math>, that also happens to be an integral domain. Then, the following are equal:

	* The [[Krull dimension]] of <math>A</math>		* The [[Krull dimension]] of <math>A</math>

Vipul: /* Facts used */

2008-03-17T01:58:45Z

Facts used

← Older revision		Revision as of 01:58, 17 March 2008
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	* Proving that the Krull dimension of the polynomial ring in <math>n</math> variables, is equal to exactly <math>n</math>		* Proving that the Krull dimension of the polynomial ring in <math>n</math> variables, is equal to exactly <math>n</math>
			* [[Noether normalization theorem]]
	* [[Going up theorem]]		* [[Going up theorem]]
	* [[Going down for integral extensions of normal domains]]		* [[Going down for integral extensions of normal domains]]

Vipul at 01:58, 17 March 2008

2008-03-17T01:58:24Z

← Older revision		Revision as of 01:58, 17 March 2008
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			{{applicationof\|Noether normalization theorem}}
			{{applicationof\|going down}}
	==Statement==		==Statement==

Vipul: /* Facts used */

2008-03-17T01:57:51Z

Facts used

← Older revision		Revision as of 01:57, 17 March 2008
Line 11:		Line 11:
	* Proving that the Krull dimension of the polynomial ring in <math>n</math> variables, is equal to exactly <math>n</math>		* Proving that the Krull dimension of the polynomial ring in <math>n</math> variables, is equal to exactly <math>n</math>
	* [[Going up theorem]]		* [[Going up theorem]]
	* [[Going down ~~theorem~~]]		* [[Going down for integral extensions of normal domains]]

	==Proof==		==Proof==

	{{fillin}}		{{fillin}}

Vipul: New page: ==Statement== Let $A$ be an affine domain over a field $k$ , i.e. a finitely generated algebra over $k$ , that also happens to be an integral domain...

2008-03-17T01:57:22Z

New page: ==Statement== Let <math>A</math> be an affine domain over a field <math>k</math>, i.e. a finitely generated algebra over <math>k</math>, that also happens to be an integral domain...

New page

==Statement==

Let <math>A</math> be an [[affine domain]] over a [[field]] <math>k</math>, i.e. a finitely generated algebra over <math>k</math>, that also happens to be an integral domain. Then, the following are equivalent:

* The [[Krull dimension]] of <math>A</math>
* The [[Krull dimension]] of the localization of <math>A</math> at ''any'' [[maximal ideal]] (which is the same as that obtained using the Hilbert-Samuel polynomial)
* The transcendence degree of the field of fractions of <math>A</math>, over <math>k</math>

==Facts used==

* Proving that the Krull dimension of the polynomial ring in <math>n</math> variables, is equal to exactly <math>n</math>
* [[Going up theorem]]
* [[Going down theorem]]

==Proof==

{{fillin}}

Equivalence of dimension notions for affine domain - Revision history

Vipul: 5 revisions

Vipul: /* Statement */

Vipul: /* Facts used */

Vipul at 01:58, 17 March 2008

Vipul: /* Facts used */

Vipul: New page: ==Statement== Let A be an affine domain over a field k, i.e. a finitely generated algebra over k, that also happens to be an integral domain...

Vipul: New page: ==Statement== Let $A$ be an affine domain over a field $k$ , i.e. a finitely generated algebra over $k$ , that also happens to be an integral domain...