= Hilbert-Serre theorem
{c}
{wiki}
Let $S=k[x_1,\ldots,x_r]$ be graded with positive degrees $d_i=\deg x_i$, and let $M$ be a finitely generated graded $S$-module with finite-dimensional graded pieces. Then
$$
H_M(t)=\frac{P(t)}{\prod_{i=1}^r(1-t^{d_i})}
$$
for a Laurent polynomial $P(t)\in\mathbb Z[t,t^{-1}]$. For a standard grading, $\dim_kM_n$ consequently agrees with a polynomial for all sufficiently large $n$.
The proof is by induction on $r$. For $x=x_r$ of degree $d$, the exact sequence of graded modules
$$
0\longrightarrow(0:_Mx)(-d)\longrightarrow M(-d)
\xrightarrow{x}M\longrightarrow M/xM\longrightarrow0
$$
gives
$$
(1-t^d)H_M(t)=H_{M/xM}(t)-t^dH_{(0:_Mx)}(t).
$$
Both modules on the right are finitely generated over $k[x_1,\ldots,x_{r-1}]$, so induction supplies the asserted denominator.
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