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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?

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Think about how you can generate the elements of $A_1$ if it is infinite-dimensional. – Mariano Suárez-Alvarez Oct 5 '10 at 19:00
This question has already been answered, so maybe it's not that important, but the title is wrong. It says "example of a not finitely generated graded algebra", when the question asks for a finitely generated graded algebra. – arsmath Oct 16 '10 at 16:56
yes, I already correct it – Melania Oct 17 '10 at 10:23
up vote 7 down vote accepted

$\mathbb Q[x,y]$ 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.

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