Regular sequence on a module: Difference between revisions

Latest revision as of 16:34, 12 May 2008

Definition

Let $R$ be a commutative unital ring, $M$ a $R$ -module, and $x_{1},x_{2},\ldots ,x_{n}$ be a sequence of elements in $R$ . We say that the $x_{i}$ s form a regular sequence on $M$ if the following two conditions hold:

$(x_{1},x_{2},\ldots ,x_{n})M\neq M$
For $1\leq i\leq n$ , $x_{i}$ is a nonzerodivisor on $M/(x_{1},x_{2},\ldots ,x_{i-1})M$

When no module is specified, we assume the module to be $R$ itself. Further information: regular sequence in a ring

Facts

If $R$ is a Noetherian local ring and $x_{1},x_{2},\ldots ,x_{n}$ form a regular sequence in its unique maximal ideal, then any permutation of the $x_{i}$ 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 $R$ is a graded ring, and $x_{1},x_{2},\ldots ,x_{n}$ form a regular sequence and all the $x_{i}$ s are homogeneous elements, then any permutation of the $x_{i}$ s is also a regular sequence.
If $x_{1},x_{2},\ldots ,x_{d}$ are a regular sequence on a module $M$ over a Noetherian local ring, then the difference of degrees of the Hilbert-Samuel polynomial for $M$ and for $M/(x_{1},x_{2},\ldots ,x_{d})$ is at least $d$ .

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 ==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>
-* 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})M</math>
+When no module is specified, we assume the module to be <math>R</math> itself. {{further|[[regular sequence in a ring]]}}
+==Facts==
+* 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>.