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C.2 Hilbert function

Let M $=\bigoplus_{i\in Z} M_i$ be a graded module over $K[x_1,..,x_n]$ with respect to weights $(w_1,..w_n)$. The Hilbert function of $M$, $H_M$, is defined (on the integers) by

\begin{displaymath}H_M(k) :=dim_K M_k.\end{displaymath}

The Hilbert-Poincare series of $M$ is the power series

\begin{displaymath}\hbox{HP}_M(t) :=\sum_{i=-\infty}^\infty
H_M(i)t^i=\sum_{i=-\infty}^\infty dim_K M_i \cdot t^i.\end{displaymath}

It turns out that $\hbox{HP}_M(t)$ can be written in two useful ways for weights $(1,..,1)$:

\begin{displaymath}\hbox{HP}_M(t)={Q(t)\over (1-t)^n}={P(t)\over (1-t)^{dim(M)}}\end{displaymath}

where $Q(t)$ and $P(t)$ are polynomials in ${\bf Z}[t]$. $Q(t)$ is called the first Hilbert series, and $P(t)$ the second Hilbert series. If $P(t)=\sum_{k=0}^N a_k t^k$, and $d = dim(M)$, then $H_M(s)=\sum_{k=0}^N a_k$ ${d+s-k-1}\choose{d-1}$ (the Hilbert polynomial) for $s \ge N$.

Generalizing this to quasihomogeneous modules we get

\begin{displaymath}\hbox{HP}_M(t)={Q(t)\over {\Pi_{i=1}^n(1-t^{w_i})}}\end{displaymath}

where $Q(t)$ is a polynomial in ${\bf Z}[t]$. $Q(t)$ is called the first (weighted) Hilbert series of M.