Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 101 / 5 / c

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

A Hilbert polynomial of $A$ is a polynomial $h_A(X)\in \mathbb{Q}[X]$ such thatfor every sufficiently large integer $n$. Eventual equality makes this polynomial unique.

Take $A=k[x]$ with $\mathrm{deg}x=2$. This is a Noetherian graded algebra with $A_0=k$, butNo polynomial can agree eventually with these alternating values, so $A$ has no Hilbert polynomial.

A sufficient condition is that $A$ be a standard graded algebra: it is generated as a $k$-algebra by finitely many elements of degree one. The Hilbert-Serre theorem then makes $P_A(t)$ a rational function whose denominator, after cancellation, is a power of $1-t$. When the eventual Hilbert polynomial is nonzero,where the right side uses the order of the pole at $t=1$.

The degree-$n$ component isDimensions therefore satisfy the Cauchy product rule, giving