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For a finitely generated graded module over a standard graded algebra over a field, with finite-dimensional graded pieces, the Hilbert function $h(n)=\dim M_n$ is eventually a polynomial in $n$. Its generating series is the Hilbert series. Thus the polynomial describes eventual graded growth, while the series records all graded pieces.

The Poincare series of a graded module isIt is the generating function of the Hilbert function $n\mapsto {\dim }_k{A}_n$.

Let $S=k[x_1,\dots ,x_r]$ with positive degrees $d_i=\mathrm{deg}{x}_i$, and let $M$ be a finitely generated graded $S$-module whose graded pieces are finite-dimensional over $k$. The Hilbert-Serre theorem states thatfor some Laurent polynomial $P(t)\in \mathbb{Z}[t,t^{-1}]$. For the standard grading, the Hilbert function $n\mapsto {\dim }_k{M}_n$ consequently agrees for all sufficiently large $n$ with a polynomial in $n$.

For the proof, induct on $r$. When $r=0$, $M$ is finite-dimensional and its Hilbert series is a Laurent polynomial. For $r>0$, multiplication by $x_r$ gives an exact sequence of graded modulesBoth $K$ and $C$ are annihilated by $x_r$, so they are finitely generated graded modules over $k[x_1,\dots ,x_{r-1}]$. Additivity of the Hilbert series yieldsThe induction hypothesis supplies the required denominator for the right side and proves the rational formula. When all $d_i=1$, expanding $(1-t{)}^{-r}$ shows that its coefficients are binomial polynomials in $n$, which proves eventual polynomiality.