# homogeneous system of parameters

Let $k$ be a field, let $R$ be an ${\mathbb{N}}^{m}$-graded (http://planetmath.org/GradedAlgebra) $k$-algebra, and let $M$ be a ${\mathbb{Z}}^{m}$-graded $R$-module.

Let $\mathscr{H}({R}_{+})$ be the homogeneous union of the irrelevant ideal of $R$.

A *partial homogeneous system of parameters* for $M$ is a finite sequence^{} of elements ${\theta}_{1},{\theta}_{2},\mathrm{\dots},{\theta}_{r}\in \mathscr{H}({R}_{+})$ such that

$dim\left(M/\left({\displaystyle \sum _{i=1}^{r}}{\theta}_{i}M\right)\right)=dim(M)-r,$ |

where $dim$ gives the Krull dimension^{}.

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

A sequence ${\theta}_{1},\mathrm{\dots},{\theta}_{r}\in \mathscr{H}({R}_{+})$ is a * $M$-sequence* if for all $i$ with $$, we have that ${\theta}_{i+1}$ is not a zero-divisor in

$M/\left({\displaystyle \sum _{j=1}^{i}}{\theta}_{i}M\right).$ |

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

## References

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

Title | homogeneous system of parameters |
---|---|

Canonical name | HomogeneousSystemOfParameters |

Date of creation | 2013-03-22 14:14:55 |

Last modified on | 2013-03-22 14:14:55 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 5 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 13A02 |

Defines | partial homogeneous system of parameters |

Defines | complete homogeneous system of parameters |

Defines | homogeneous $M$-sequence |

Defines | depth |

Defines | depth of a module |