Exercise 12, § 6

Let $H$ be a graded algebra of finite type, such that $H_0$ is a field, and let $(\xi_a)_{a \in A}$ be homogeneous elements of $H$ which generate $H$. Suppose that there exists a family of strictly positive integers $(d_i)_{i \in I}$ such that $P_H = \prod_{i \in I} (1 - T^{d_i})^{-1}$. We denote by $\delta_a > 0$ the degree of $\xi_a$. a) Now prove that there exists an injection $\varphi : I \to A$ such that for every $i$, $d_i$ divides $\delta_{\varphi(i)}$. (One may use § 4, No. 2, Theorem 1.) b) Now prove that one has $\inf_{a \in A} \delta_a \leq \inf_{i \in I} d_i$. c) Deduce that if the $\delta_a$ are equal to one another, the graded algebra $H$ is regular. d) Suppose that $A$ has two elements $a, a'$ such that $\delta_a < \delta_{a'}$ and that $\delta_{a'}$ is not a multiple of $\delta_a$. Now prove that the graded algebra $H$ is regular.