# Filtrations of graded modules

Let $S$ be a graded noetherian ring and $M$ a finitely generated graded $S$- module. Then I know that there exists a filtration

$$0=M_0 \subseteq M_1\subseteq \cdots \subseteq M_r=M$$ by graded submodules such that for each $i$, $M_i/M_{i-1}=(S/P_i)(l_i)$ for some homogeneous prime ideal $P_i$ of $S$ and $l_i$ integer, where $(S/P_i)(l_i)$ shift module.

I wonder if $M$ can be expressed in the form $(S/P)(l)$ for some homogeneous prime ideal of $S$. Is it possible?

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## 1 Answer

I think it is generally not true.

Let $S=k$ be a field with trivial grading $S_0=k,S_i=\{0\}, i\neq 0.$ Let $M=k[x]/\langle x^2\rangle$ be the quotient of the polynomial ring in one variable over $S,$ with the natural grading. The only homogeneous prime of $S$ is $P=\langle 0\rangle,$ and it is certainly not possible that $M=(S/P)(l)\cong k$ for any shift $l.$

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thanks Andrew. If $S$ is a polynomial ring, is it possible? –  Sang Cheol Lee Sep 7 '12 at 7:15
Take $M=S\oplus S$ for any $S$. –  user18119 Sep 7 '12 at 11:29