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Let $R=\oplus_i R_i,$ $R_0=\mathbb{C}$ be graded finitelly generaded algebra with the Krull dimension $r$ and the Hilbert series $H(A,t).$ Then the Laurent series expansion of $H(A,t)$ at $t=1$ has the form $$ H(A,t)=\frac{a_0}{(t-1)^r}+\frac{a_1}{(t-1)^{r-1}}+\cdots $$

Questions:

  1. What are the correct name of the numbers $a_0,a_1,... $ ?
  2. Why the numbers are so important and useful?
  3. The same questions in the case if $A$ is Cohen-Macaulay and Gorestein algebra.
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@1 What about "Coefficients of the Laurent series expansion" ? –  Johannes Hahn Feb 3 '13 at 15:07

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