Exercise 2, § 1

Let $A$ be a graded commutative ring of type $\mathbf{Z}$ such that $A_n = 0$ for $n < 0$. For all $n \geq 0$, let $A_{[n]}$ denote the graded ideal $\bigoplus_{m \geq n} A_m$ of $A$; the $A$-algebra $A^h = \bigoplus_{n \geq 0} A_{[n]}$ is a ring graded by the $A_{[n]} = A_n^h$; for all $f \in A, (d > 0)$, let $f^h$ denote the element of $A^h$ whose component are zero except for that of degree $d$, which is equal to $f$.

(a) Show that, iff $\in A_d$, where $d > 0$, the ring $A_{(f^h)}^h$ is isomorphic to $(A_f)^{\geq}$ (notation of Exercise 1).

(b) For $A^h$ to be a finitely generated $A$-algebra, it is necessary and sufficient that $A$ be a finitely generated $A_{(f)}$-algebra.

(c) In order that $A_{n+1}^h = A_1^h A_n^h$ (resp. $A_n^h = (A_1^h)^n$) for $n \geq n_0$, it is necessary and sufficient that $A_{n+1} = A_1 A_n$ (resp. $A_n = (A_1)^n$) for $n \geq n_0$.