Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 101 / 5 / a

The positive-degree part $A_{+}={⨁}_{n>0}{A}_n$ is an ideal and $A/A_{+}\simeq A_0$, so $A_0$ is Noetherian. Since $A$ is Noetherian, $A_{+}$ has finitely many homogeneous generators $x_1,\dots ,x_s$. Induction on degree shows that every positive-degree homogeneous element is a polynomial in the $x_i$ over $A_0$. Thus

For an additive length function $\lambda$ finite on the graded pieces, define the Poincare series of a graded moduleThe Hilbert-Serre theorem states thatfor a Laurent polynomial $f(t)\in \mathbb{Z}[t,t^{-1}]$.

Induct on $s$. For the last generator $x_s$ of degree $k_s$, multiplication gives an exact sequence whose kernel is the $x_s$-torsion and whose cokernel is $M/x_{s}M$. Additivity of $\lambda$ yieldsBoth modules on the right are finite graded modules over the algebra generated by $x_1,\dots ,x_{s-1}$. The induction hypothesis gives the asserted denominator. The case $s=0$ is a finite Laurent polynomial because $M$ is finitely generated over $A_0$.

When every $k_i=1$, cancel common factors to writewhere $d$ is the pole order at $t=1$. Sincethe coefficient of $t^n$ in $P(M,t)$ is, for all sufficiently large $n$, a fixed linear combination of shifted binomial polynomials. It is therefore a polynomial in $n$ of degree exactly $d-1$ unless $P=0$, in which case its degree is $-1$.