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

Is it possible ?

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## 1 Answer

Rather obviously yes.

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 $\sum_{n=0}^\infty t^n=1/(1-t)$.

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I agree but I am interested with an algebra with non-trivial multiplication. –  Melania Sep 29 '10 at 20:47
@Melania: how about start as Robin did, and then kill all $a_ia_ja_k$? –  Hailong Dao Sep 29 '10 at 21:23
Yes..The Poincare series will be $\dfrac{1}{(1-t)(1-t^2)}.$ Thanks! –  Melania Sep 29 '10 at 21:54
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