Regular sequence on a module - Revision history

Vipul: 8 revisions

2008-05-12T16:34:20Z

8 revisions

← Older revision	Revision as of 16:34, 12 May 2008
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Vipul at 14:04, 5 May 2008

2008-05-05T14:04:20Z

← Older revision		Revision as of 14:04, 5 May 2008
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	* For <math>1 \le i \le n</math>, <math>x_i</math> is a [[nonzerodivisor on a module\|nonzerodivisor]] on <math>M/(x_1,x_2,\ldots,x_{i-1})M</math>		* For <math>1 \le i \le n</math>, <math>x_i</math> is a [[nonzerodivisor on a module\|nonzerodivisor]] on <math>M/(x_1,x_2,\ldots,x_{i-1})M</math>

	When no module is specified, we assume the module to be <math>R</math> itself.		When no module is specified, we assume the module to be <math>R</math> itself. {{further\|[[regular sequence in a ring]]}}

	==Facts==		==Facts==

Vipul: Regular sequence moved to Regular sequence on a module

2008-05-05T14:02:27Z

Regular sequence moved to Regular sequence on a module

← Older revision	Revision as of 14:02, 5 May 2008
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Vipul at 14:01, 5 May 2008

2008-05-05T14:01:15Z

← Older revision		Revision as of 14:01, 5 May 2008
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	==Definition==		==Definition==

	Let <math>R</math> be a [[commutative unital ring]], <math>M</math> a <math>R</math>-[[module]], and <math>x_1, x_2, \ldots, x_n</math> be a sequence of elements in <math>R</math>. We say that the <math>x_i</math>s form a regular sequence on <math>M</math> if the following two conditions hold:		Let <math>R</math> be a [[commutative unital ring]], <math>M</math> a <math>R</math>-[[module]], and <math>x_1, x_2, \ldots, x_n</math> be a sequence of elements in <math>R</math>. We say that the <math>x_i</math>s form a '''regular sequence''' on <math>M</math> if the following two conditions hold:

	* <math>(x_1,x_2,\ldots,x_n)M \ne M</math>		* <math>(x_1,x_2,\ldots,x_n)M \ne M</math>
	* For <math>1 \le i \le n</math>, <math>x_i</math> is a [[nonzerodivisor on a module\|nonzerodivisor]] on <math>M/(x_1,x_2,\ldots,x_{i-1})</math>		* For <math>1 \le i \le n</math>, <math>x_i</math> is a [[nonzerodivisor on a module\|nonzerodivisor]] on <math>M/(x_1,x_2,\ldots,x_{i-1})M</math>

			When no module is specified, we assume the module to be <math>R</math> itself.

	==Facts==		==Facts==

Vipul: /* Facts */

2008-03-17T00:38:34Z

Facts

← Older revision		Revision as of 00:38, 17 March 2008
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	==Facts==		==Facts==

	If <math>R</math> is a [[Noetherian ~~ring\|Noetherian]] [[~~local ring]] and <math>x_1, x_2, \ldots, x_n</math> form a regular sequence in its unique [[maximal ideal]], then any permutation of the <math>x_i</math>s also forms a regular sequence in the maximal ideal. In general, a permutation of a regular sequence need not be regular. {{proofat\|[[Permutation of regular sequence is not necessarily regular]]}}		* If <math>R</math> is a [[Noetherian local ring]] and <math>x_1, x_2, \ldots, x_n</math> form a regular sequence in its unique [[maximal ideal]], then any permutation of the <math>x_i</math>s also forms a regular sequence in the maximal ideal. In general, a permutation of a regular sequence need not be regular. {{proofat\|[[Permutation of regular sequence is not necessarily regular]]}}
			* If <math>R</math> is a graded ring, and <math>x_1, x_2, \ldots, x_n</math> form a regular sequence and all the <math>x_i</math>s are homogeneous elements, then any permutation of the <math>x_i</math>s is also a regular sequence.
			* If <math>x_1, x_2, \ldots, x_d</math> are a regular sequence on a module <math>M</math> over a [[Noetherian local ring]], then the difference of degrees of the Hilbert-Samuel polynomial for <math>M</math> and for <math>M/(x_1,x_2,\ldots,x_d)</math> is at least <math>d</math>.

Vipul at 21:44, 16 March 2008

2008-03-16T21:44:18Z

← Older revision		Revision as of 21:44, 16 March 2008
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	==Facts==		==Facts==

	If <math>R</math> is a [[Noetherian ring\|Noetherian]] [[local ring]] and <math>x_1, x_2, \ldots, x_n</math> form a regular sequence in its unique [[maximal ideal]], then any permutation of the <math>x_i</math>s also forms a regular sequence in the maximal ideal.		If <math>R</math> is a [[Noetherian ring\|Noetherian]] [[local ring]] and <math>x_1, x_2, \ldots, x_n</math> form a regular sequence in its unique [[maximal ideal]], then any permutation of the <math>x_i</math>s also forms a regular sequence in the maximal ideal. In general, a permutation of a regular sequence need not be regular. {{proofat\|[[Permutation of regular sequence is not necessarily regular]]}}

Vipul: /* Definition */

2008-03-16T21:37:46Z

Definition

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	* <math>(x_1,x_2,\ldots,x_n)M \ne M</math>		* <math>(x_1,x_2,\ldots,x_n)M \ne M</math>
	* For <math>1 \le i \le n</math>, <math>x_i</math> is a nonzerodivisor on <math>M/(x_1,x_2,\ldots,x_{i-1})</math>		* For <math>1 \le i \le n</math>, <math>x_i</math> is a [[nonzerodivisor on a module\|nonzerodivisor]] on <math>M/(x_1,x_2,\ldots,x_{i-1})</math>

	==Facts==		==Facts==

	If <math>R</math> is a [[Noetherian ring\|Noetherian]] [[local ring]] and <math>x_1, x_2, \ldots, x_n</math> form a regular sequence in its unique [[maximal ideal]], then any permutation of the <math>x_i</math>s also forms a regular sequence in the maximal ideal.		If <math>R</math> is a [[Noetherian ring\|Noetherian]] [[local ring]] and <math>x_1, x_2, \ldots, x_n</math> form a regular sequence in its unique [[maximal ideal]], then any permutation of the <math>x_i</math>s also forms a regular sequence in the maximal ideal.

Vipul at 02:51, 9 August 2007

2007-08-09T02:51:26Z

← Older revision		Revision as of 02:51, 9 August 2007
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	* <math>(x_1,x_2,\ldots,x_n)M \ne M</math>		* <math>(x_1,x_2,\ldots,x_n)M \ne M</math>
	* For <math>1 \le i \le n</math>, <math>x_i</math> is a nonzerodivisor on <math>M/(x_1,x_2,\ldots,x_{i-1})</math>		* For <math>1 \le i \le n</math>, <math>x_i</math> is a nonzerodivisor on <math>M/(x_1,x_2,\ldots,x_{i-1})</math>

			==Facts==

			If <math>R</math> is a [[Noetherian ring\|Noetherian]] [[local ring]] and <math>x_1, x_2, \ldots, x_n</math> form a regular sequence in its unique [[maximal ideal]], then any permutation of the <math>x_i</math>s also forms a regular sequence in the maximal ideal.

Vipul at 02:49, 9 August 2007

2007-08-09T02:49:18Z

New page

==Definition==

Let <math>R</math> be a [[commutative unital ring]], <math>M</math> a <math>R</math>-[[module]], and <math>x_1, x_2, \ldots, x_n</math> be a sequence of elements in <math>R</math>. We say that the <math>x_i</math>s form a regular sequence on <math>M</math> if the following two conditions hold:

* <math>(x_1,x_2,\ldots,x_n)M \ne M</math>
* For <math>1 \le i \le n</math>, <math>x_i</math> is a nonzerodivisor on <math>M/(x_1,x_2,\ldots,x_{i-1})</math>