MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

Need an example of not finitely generated graded algebra such that its Poincaré series is a rational function.

Is it possible ?

-

Let $A$ be the algebra over the field $K$ generated by elements $a_1,a_2,\ldots,$ with $a_i$ in dimension $i$ and with $a_ia_j=0$ for all $i$ and $j$. This is an incredibly uninteresting example, but since each graded piece is one-dimensional, its Poincare series is $\sum_{n=0}^\infty t^n=1/(1-t)$.
@Melania: how about start as Robin did, and then kill all $a_ia_ja_k$? – Hailong Dao Sep 29 2010 at 21:23
Yes..The Poincare series will be $\dfrac{1}{(1-t)(1-t^2)}.$ Thanks! – Melania Sep 29 2010 at 21:54