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

A partial homogeneous system of parameters for $M$ is a finite sequence of elements $\theta_{1},\theta_{2},\ldots,\theta_{r}\in\mathcal{H}(R_{+})$ such that

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

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

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

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 HomogeneousSystemOfParameters 2013-03-22 14:14:55 2013-03-22 14:14:55 mathcam (2727) mathcam (2727) 5 mathcam (2727) Definition msc 13A02 partial homogeneous system of parameters complete homogeneous system of parameters homogeneous $M$-sequence depth depth of a module