Exercise 1, § 1

Let $A$ be a graded commutative ring of type $\mathbf{Z}$ and $(A_n)$ its graduation; we set $A^{\geq} = \bigoplus_{n \geq 0} A_n, \quad A^{\leq} = \bigoplus_{n \geq 0} A_n$; these are graded subrings of $A$.

(a) For every homogeneous element $f$ of $A$ of degree $d$, the ring of fractions $A_f$ (corresponding to the multiplicative subset consisting of the $f^n$, where $n \geq 0$) has a canonical graded ring structure (Algebra, Chapter 11, § 11); we denote by $A_{(f)}$ the subring of $A$, consisting of elements of degree 0. Show that, if $d > 0$, then $(A''_f) = A_{(f)}$, $A_{(f)}$ is isomorphic to $A^{(d)}/(f - 1)A^{(d)}$ and $((A_f)^{\geq})_{/f/1}$ is a graded ring isomorphic to $A_{(f)}$.

(b) Let the polynomial ring $B = A[X] = A \otimes_{\mathbf{Z}} \mathbf{Z}[X]$ be given the graduation the tensor product of those on $A$ and $\mathbf{Z}[X]$, a graduation compatible with the ring structure on $B$. Show that, if $d > 0$, $B_{(f)}$ is isomorphic to $(A_n)^{\leq}$ and $(A^{(d)})_f$ is a graded ring isomorphic to $A_{(f)} \otimes_{\mathbf{Z}} \mathbf{Z}[X, X^{-1}]$.

(c) Let $g$ be another homogeneous element of $A$ of degree $e$. Show that, if $d > 0$ and $e > 0$, $A_{(f,g)}$ is isomorphic to $(A_{(f)})_{g^d/f^e}$.

(d) Suppose $d > 0$ and $A_n = \{0\}$ for $n < 0$. Show that, if $A$ is Noetherian, so is $A_{(f)}$ (use § 2, no. 10, Corollary 4 to Theorem 2). \*