Regular sequence on a module: Difference between revisions
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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== | ||
Revision as of 14:01, 5 May 2008
Definition
Let be a commutative unital ring, a -module, and be a sequence of elements in . We say that the s form a regular sequence on if the following two conditions hold:
- For , is a nonzerodivisor on
When no module is specified, we assume the module to be itself.
Facts
- If is a Noetherian local ring and form a regular sequence in its unique maximal ideal, then any permutation of the s also forms a regular sequence in the maximal ideal. In general, a permutation of a regular sequence need not be regular. For full proof, refer: Permutation of regular sequence is not necessarily regular
- If is a graded ring, and form a regular sequence and all the s are homogeneous elements, then any permutation of the s is also a regular sequence.
- If are a regular sequence on a module over a Noetherian local ring, then the difference of degrees of the Hilbert-Samuel polynomial for and for is at least .
