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.