13A02-HomogeneousSystemOfParameters.tex

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\pmtitle{homogeneous system of parameters}
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\pmtype{Definition}
\pmcomment{trigger rebuild}
\pmclassification{msc}{13A02}
\pmdefines{partial homogeneous system of parameters}
\pmdefines{complete homogeneous system of parameters}
\pmdefines{homogeneous $M$-sequence}
\pmdefines{depth}
\pmdefines{depth of a module}

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\begin{document}
Let $k$ be a field, let $R$ be an $\mb{N}^m$-\PMlinkname{graded}{GradedAlgebra} $k$-algebra, and let $M$ be a $\Z^m$-graded $R$-module.

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

A \emph{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 

\begin{align*}
\dim\left(M/\left(\sum_{i=1}^r \theta_iM\right)\right)=\dim(M)-r,
\end{align*}

where $\dim$ gives the Krull dimension.

A (\PMlinkescapetext{complete}) \emph{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 \emph{\PMlinkescapetext{homogeneous} $M$-sequence} if for all $i$ with $0\leq i<r$, we have that $\theta_{i+1}$ is not a zero-divisor in 

\begin{align*}
M/\left(\sum_{j=1}^i \theta_iM\right).
\end{align*}

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

\begin{thebibliography}{9}
\bibitem{Stan} Richard P. Stanley, {\em Combinatorics and Commutative Algebra}, Second edition, Birkhauser Press.  Boston, MA.  1986.
\end{thebibliography}
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