Artin-Tate lemma - Revision history

Vipul: 5 revisions

2008-05-12T16:18:39Z

5 revisions

← Older revision	Revision as of 16:18, 12 May 2008
(No difference)

Vipul at 17:02, 27 February 2008

2008-02-27T17:02:31Z

← Older revision		Revision as of 17:02, 27 February 2008
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	{{indispensable lemma}}		{{indispensable lemma}}
			{{Noetherian ring result}}
	==Statement==		==Statement==

Vipul at 22:15, 20 January 2008

2008-01-20T22:15:29Z

← Older revision		Revision as of 22:15, 20 January 2008
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			{{indispensable lemma}}

	==Statement==		==Statement==

Vipul at 20:42, 19 January 2008

2008-01-19T20:42:15Z

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 * <math>B</math> and <math>C</math> are <math>A</math>-algebras, with <math>B</math> a subalgebra of <math>C</math>
 * <math>C</math> is finitely generated as an <math>A</math>-algebra (i.e. <math>C</math> is an [[algebra of finite type]] over <math>A</math>)
-* <math>C</math> is finitely generated as a <math>B</math>-module (i.e. <math>C</math> is [[[finiteness as an algebra|finite as an algebra]] over <math>B</math>)
+* <math>C</math> is finitely generated as a <math>B</math>-module (i.e. <math>C</math> is [[finiteness as an algebra|finite as an algebra]] over <math>B</math>)
 Then: <math>B</math> is finitely generated as an <math>A</math>-algebra
@@ Line 15: / Line 15: @@
 An analogous result in group theory is the statement that [[gp:finite index in finitely generated implies finitely generated|any subgroup of finite index in a finitely generated group, is also finitely generated]].

Vipul at 20:13, 19 January 2008

2008-01-19T20:13:20Z

← Older revision		Revision as of 20:13, 19 January 2008
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	* <math>A</math> is [[Noetherian ring\|Noetherian]]		* <math>A</math> is [[Noetherian ring\|Noetherian]]
	* <math>B</math> and <math>C</math> are <math>A</math>-algebras, with <math>B</math> a subalgebra of <math>C</math>		* <math>B</math> and <math>C</math> are <math>A</math>-algebras, with <math>B</math> a subalgebra of <math>C</math>
	* <math>C</math> is finitely generated as an <math>A</~~matH~~>-algebra		* <math>C</math> is finitely generated as an <math>A</math>-algebra (i.e. <math>C</math> is an [[algebra of finite type]] over <math>A</math>)
	* <math>C</math> is finitely generated as a <math>B</math>-module		* <math>C</math> is finitely generated as a <math>B</math>-module (i.e. <math>C</math> is [[[finiteness as an algebra\|finite as an algebra]] over <math>B</math>)

	Then: <math>B</math> is finitely generated as an <math>A</math>-algebra		Then: <math>B</math> is finitely generated as an <math>A</math>-algebra

			==Explanation==

			The Artin-Tate lemma is related to the general philosophy that if an object is ''finitely generated'', then any subobject of it is also finitely generated provided the subobject is ''almost'' the whole object. In our case, <math>C</math> is finitely generated as an <math>A</math>-algebra, but it is not necessary that ''all'' subalgebras of <math>C</math> are finitely generated as <math>A</math>-algebras. However, if the subalgebra is ''large'' enough inside <math>C</math>, i.e. it is ''not too deep'', then it behaves like <math>C</math>, and we can try to show it is finitely generated.

			An analogous result in group theory is the statement that [[gp:finite index in finitely generated implies finitely generated\|any subgroup of finite index in a finitely generated group, is also finitely generated]].

Vipul at 15:42, 14 January 2008

2008-01-14T15:42:29Z

New page

==Statement==

Suppose <math>A, B, C</math> are [[commutative unital ring]]s, such that:

* <math>A</math> is [[Noetherian ring|Noetherian]]
* <math>B</math> and <math>C</math> are <math>A</math>-algebras, with <math>B</math> a subalgebra of <math>C</math>
* <math>C</math> is finitely generated as an <math>A</matH>-algebra
* <math>C</math> is finitely generated as a <math>B</math>-module

Then: <math>B</math> is finitely generated as an <math>A</math>-algebra