Is each component of a graded module over a $k$-algebra a finite-dimensional vector space?

Question

I have some problems with an argument in a proof of a lemma:

Let $M = \oplus_{-\infty}^{\infty} M_n$ be a finitely generated graded $A$-module and $A=\oplus_{n\geq 0} A_n$ a graded commutative ring where $A_0$ is a field.

Then each component $M_n$ is a $A_0$-vector space. Are the assumptions made above sufficient to ensure that each component is finite-dimensional, i.e. $\dim_{A_0} M_n < \infty$ for all $n$?

Don't we still need $A$ to be Noetherian for $A$ to be finitely generated as $A_0$-algebra or can we somehow deduce that because $A_0$ is a field?

source: lemma 5 in https://www.math.uni-tuebingen.de/user/keilen/download/Lehre/CASS05/nakayama.pdf

Answer 1

$A=K[x_1,x_2,\ldots]$ is a graded $A=K[x_1,x_2,\ldots]$-module, where $A_0=K$ is a field, but none of the $A_n$ ($n>0$) is a finite-dimensional $K$-vector space. Therefore one needs some assumption like $A$ Noetherian.