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 ?