 Hilbert-Serre theorem

Let $S=k[x_1,\dots ,x_r]$ be graded with positive degrees $d_i=\mathrm{deg}{x}_i$, and let $M$ be a finitely generated graded $S$-module with finite-dimensional graded pieces. Thenfor a Laurent polynomial $P(t)\in \mathbb{Z}[t,t^{-1}]$. For a standard grading, ${\dim }_k{M}_n$ 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 modulesgivesBoth modules on the right are finitely generated over $k[x_1,\dots ,x_{r-1}]$, so induction supplies the asserted denominator.