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I am trying to find the Poincare series and Hilbert polynomial for graded $S$-modules $I=S \cdot T^m$ and $M=S/I$ where $S=k[T]$ is the graded polynomial algebra and $m \geq 1$.

I am not particularly comfortable with this material, so I was hoping for some help. Thanks very much!

user 3462
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Since $M$ is free of rank $m$ (an $S$-basis is $\{1,t,\dots,t^{m-1}\}$) you get the Hilbert series $1+X+\cdots+X^{m-1}$ and the the Hilbert polynomial is $0$.

  • Apologies, I just saw this, and it is indeed helpful. – user 3462 Dec 03 '13 at 04:45
  • What of $I=S \cdot T^{m}$? Your answer is helpful for me, but is quite succinct and may not be as helpful as possible to future readers. No need to worry :) – user 3462 Dec 03 '13 at 08:34
  • @user3462 My answer is succinct because there is not much to say. If you want to know $H_I$ just look at the exact sequence $0\to I\to S\to M=S/I\to 0$ and deduce that $H_I+H_M=H_S$, hence once you know $H_M$ you know $H_I$, too. (But I think all I said is something obvious, right?) –  Dec 03 '13 at 08:39
  • True enough. That doesn't require explanation. – user 3462 Dec 03 '13 at 09:03
  • It seems that there will be integers $n \geq 0$ such that $H_{M}(n) \neq \mathrm{dim}{k}M{n}$. When would this occur? – user 3462 Dec 03 '13 at 18:36