PlanetMath: homogeneous system of parameters

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homogeneous system of parameters

(Definition)

Let be a field, let be an $\mathbb{N}^m$ -graded -algebra, and let be a $\mathbb{Z}^m$ -graded -module.

Let $\mathcal{H}(R_+)$ be the homogeneous union of the irrelevant ideal of .

A partial homogeneous system of parameters for is a finite sequence of elements $\theta_1, \theta_2, \ldots, \theta_r\in\mathcal{H}(R_+)$ such that

$\displaystyle \dim\left(M/\left(\sum_{i=1}^r \theta_iM\right)\right)=\dim(M)-r,$

where $\dim$ gives the Krull dimension.

A (complete) homogeneous system of parameters is a partial homogeneous system of parameters such that $r=\dim(M)$ .

A sequence $\theta_1,\ldots,\theta_r\in\mathcal{H}(R_+)$ is a homogeneous -sequence if for all with $0\leq i<r$ , we have that $\theta_{i+1}$ is not a zero-divisor in

$\displaystyle M/\left(\sum_{j=1}^i \theta_iM\right).$

Finally, view as being $\mathbb{Z}$ -graded by using any specialization of the above $\mathbb{Z}^m$ -grading. Then we define the depth of to be the length of the longest homogeneous -sequence.

Bibliography

1: Richard P. Stanley, Combinatorics and Commutative Algebra, Second edition, Birkhauser Press. Boston, MA. 1986.

"homogeneous system of parameters" is owned by mathcam.

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Also defines:

partial homogeneous system of parameters, complete homogeneous system of parameters, homogeneous $M$

-sequence, depth, depth of a module

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Cross-references: homogeneous, length, specialization, sequence, Krull dimension, finite sequence, irrelevant ideal, homogeneous union, field
There are 12 references to this entry.

This is version 2 of homogeneous system of parameters, born on 2004-03-12, modified 2004-03-12.
Object id is 5695, canonical name is HomogeneousSystemOfParameters.
Accessed 7112 times total.

Classification:

AMS MSC:

13A02 (Commutative rings and algebras :: General commutative ring theory :: Graded rings)

Pending Errata and Addenda

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