Choose homogeneous algebra generators $x_1,\ldots,x_r$ of positive degrees $d_1,\ldots,d_r$ for $S$; they exist because $S$ is Noetherian. For a finitely generated graded module $V=\bigoplus_jV_j$, its <Poincare series of a graded module> is
$$
P_V(t)=\sum_j(\dim_kV_j)t^j.
$$
The <Hilbert-Serre theorem> states that
$$
\boxed{P_V(t)=\frac{Q(t)}{\prod_{i=1}^r(1-t^{d_i})}}
$$
for a Laurent polynomial $Q(t)\in\mathbb Z[t,t^{-1}]$.

We prove this by induction on $r$. For $r=0$, $S=k$ and $V$ is finite-dimensional, so its series is a Laurent polynomial. For $x=x_r$ of degree $d=d_r$, multiplication gives the exact sequence of graded modules
$$
0\longrightarrow(0:_Vx)(-d)\longrightarrow V(-d)
\xrightarrow{x}V\longrightarrow V/xV\longrightarrow0.
$$
Additivity of the <Hilbert series> yields
$$
(1-t^d)P_V(t)=P_{V/xV}(t)-t^dP_{(0:_Vx)}(t).
$$
Both modules on the right are finitely generated over $k[x_1,\ldots,x_{r-1}]$, so the induction hypothesis proves the formula.

For the finitely generated commutative algebra $R$, let $R_j$ be the stated degree filtration. Its <associated graded ring>
$$
\operatorname{gr}R=\bigoplus_{j\geq0}R_j/R_{j-1}
$$
is a <standard graded algebra> generated by the initial forms of $x_1,\ldots,x_n$. Since
$$
\dim_kR_j=\sum_{i=0}^j\dim_k(R_i/R_{i-1}),
$$
Hilbert–Serre shows that \b[the quantity $\dim_kR_j$ agrees with a polynomial for all sufficiently large $j$]. The <growth of a finitely generated commutative algebra> has polynomial degree $\dim R$, independently of the chosen finite generating set.

For
$$
R=k[Y_1,Y_1^{-1},Y_2,Y_2^{-1}]
$$
with the four displayed generators, a basis of $R_j$ consists of the Laurent monomials $Y_1^aY_2^b$ satisfying $|a|+|b|\leq j$. There are $4r$ points with $|a|+|b|=r>0$, and one point at the origin. Hence the <growth of the two-variable Laurent polynomial algebra> is
$$
\boxed{\dim_kR_j=1+4\sum_{r=1}^jr=2j^2+2j+1.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 148 / 6 / Solution

Let a commutative $k$-algebra $R$ be generated by $x_1,\ldots,x_n$, and let $R_j$ be the span of words of total degree at most $j$ in those generators. The <associated graded ring> of this filtration is a <standard graded algebra>, and
$$
\dim_kR_j=\sum_{i=0}^j\dim_k(R_i/R_{i-1}).
$$
The <Hilbert-Serre theorem> therefore makes $\dim_kR_j$ eventually a polynomial in $j$. Its degree is $\dim R$; this follows by applying the Rees-ring deformation, whose special fibers are $R$ and the associated graded ring, so they have the same <Krull dimension>.

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