A finite graded module over a standard graded algebra has rational Hilbert series and eventual polynomial growth

Induct on $r$. If $r=0$, the standard grading makes $S=A$ concentrated in degree zero. Choose a finite homogeneous generating list for $M$ by splitting any finite generating list into its finitely many homogeneous components. Only the finitely many degrees of those generators occur in $M$, and each such degree piece is finite over $A$, hence has finite length by [L3]. Therefore ${\mathrm{HS}}_M(t)$ is a Laurent polynomial and its coefficients are zero for all sufficiently large $n$. Both conclusions hold in this base case.

Assume $r>0$ and the assertion for standard graded algebras presented using $r-1$ variables. By [L3] and [L4], $S$ is Noetherian; hence the kernel of multiplication by $x_r$ on the finite module $M(-1)$ is finite, as is its cokernel. Write $S^{\mathrm{\prime }}=A[x_1,\dots ,x_{r-1}]\mathrm{/}(J\cap A[x_1,\dots ,x_{r-1}])$. Multiplication by the degree-one class of $x_r$ gives an exact sequence of graded $S$-modules $0\to K\to M(-1)\stackrel{x_r}{\to }M\to C\to 0.$ Both $K$ and $C$ are annihilated by $x_r$ and are therefore finite graded modules over the $r-1$-variable algebra $S^{\mathrm{\prime }}$: split their finite $S$-generating lists into homogeneous components, and the same lists generate after the action factors through $S\mathrm{/}(x_r)$, a quotient of $S^{\mathrm{\prime }}$.

Every degree piece of $M$, $K$, and $C$ is a finite $A$-module: finite homogeneous generators and finitely many monomials of any fixed degree suffice. By [L3], it has finite length. Taking degree-$n$ pieces in the exact sequence of step 1.2 and using [L2] yields ${\mathrm{\ell }}_A(C_n)-{\mathrm{\ell }}_A(K_n)={\mathrm{\ell }}_A(M_n)-{\mathrm{\ell }}_A(M_{n-1})$ for every $n$. In Hilbert-series form this is $(1-t){\mathrm{HS}}_M(t)={\mathrm{HS}}_C(t)-{\mathrm{HS}}_K(t)$ by [L1].

The induction hypothesis applies to the finite graded $S^{\mathrm{\prime }}$-modules $K$ and $C$ from step 1.2. Their series have Laurent-polynomial numerators over $(1-t{)}^{r-1}$. Equation 2.1 therefore writes ${\mathrm{HS}}_M(t)$ with a Laurent-polynomial numerator over $(1-t{)}^r$.

Write the numerator from step 3.1 as $p(t)={\sum }_{j=a}^b{c}_j{t}^j$ with finite integers $c_j$. For $r>0$, the coefficient of $t^n$ in $t^j(1-t{)}^{-r}$ is $\left(\genfrac{}{}{0px}{}{n-j+r-1}{r-1}\right)$ for $n\ge j$. For all $n\ge b$, the Hilbert function is therefore ${\sum }_{j=a}^b{c}_j\left(\genfrac{}{}{0px}{}{n-j+r-1}{r-1}\right)$, a polynomial in $n$ with rational coefficients.

The base case is step 1.1, and steps 1.2 through 4.1 prove the induction step. Thus both claims hold for every finite number of variables.