Exercise 3, § 4

(a) Give an example of a graded algebra $H$ of type $Z$, over a field $K$, generated by a finite number of homogeneous elements of degrees $> 0$, and such that the mapping $n \mapsto \dim_K H_n$ is not polynomial at infinity. (b) With the notation and hypotheses of Theorem 1 of § 4, No. 2, let $D$ be the $\mathrm{ppcm}$ of the $d_i$. Show that $H$ is a finitely generated module over the ring $H_0[(X_i^{D/d_i})_{i \in I}]$.

Deduce that for every $n_0 \in N$, the series $\sum_{n \in Z} \operatorname{long}_{H_0}(M_{n_0 + nD}) \; T^n$ is of the form $\frac{P_{n_0}(T)}{(1 - T)^c}$ where $c = \operatorname{Card}(I)$ and $P_{n_0}(T) \in Q[T, T^{-1}]$. Deduce that the mapping $n \mapsto \operatorname{long}_{H_0}(M_{n_0 + nD})$ is polynomial at infinity.