More accurately, let $\displaystyle A=\sum\_{i=0}^{\infty}A\_i$ be a finitely generated graded algebra over say $\mathbb{Q}$ but $\dim A\_i=\infty$ for each $i.$ Is it possible?



Need an example of  finitely generated graded algebra such that  each its graded subspace has  infinite dimension.
$\mathbb Q[x,y]$ with $x$ in degree 0, and $y$ in degree 1.  



If you want your generators to be in positive degrees, then what you're asking for is impossible.


