Suppose $M= \bigoplus\_{n\in \mathbb Z} M\_n$ is a finitely generated graded module over a Noetherian graded commutative ring $A=\bigoplus\_{n\in \mathbb Z}A\_n$. 


If $A$ is positively graded ($A\_n=0$ if $n<0$), then each $M\_n$ is finitely generated as $A\_0$-module (Atiyah, McDonald: Introduction to commutative algebra, beginning of chap. 11). 


But what happens, if $A$ is not assumed to be positively graded, like $A= \mathbb{Q}[t,t^{-1}]$ ? Is $M\_n\;(n \in \mathbb Z)$ also finitely generated over $A\_0$ in this case ? 



Are homogeneous components of f.g graded modules f.g ? 
The answer is yes. Instead of showing that $M\_n$ is finitely generated we may show it has the
property that any ascending sequence of $A\_0$-submodules stabilizes. If $N$ is an $A\_0$-submodule of $M\_n$, consider $M\_n\cap AN$. It is a sum of the $M\_n\cap NA\_i$. One sees it is 
$N$ itself. So $N$ can be recovered from $AN$. Now if $N\_1\subset N\_2\dots$ is an ascending sequence of $A\_0$-submodules of $M\_n$, the ascending sequence $AN\_1\subset AN\_2\dots$ stabilizes, hence so do the $N\_i=M\_n\cap AN\_i$. 


