Let $A$ be a $\mathbb{N}$-graded ring: $A = \bigoplus_{i \geq 0} A_i$. Fix $d > 0$ and consider $A^{(d)} = B = \bigoplus_{i \geq 0} A_{id} \subseteq A$ as sets, but with different graduations: $A^{(d)}_i = A_{di}$ and
- $B_i = 0$ if $d$ does not divide $i$;
- $B_i = A_i = A^{(d)}_{i/d}$ if $d$ divides $i$.
1) Prove that $Proj A^{(d)} \simeq Proj B$.
2) Observe that $B$ is a graded subring of $A$ and consider the morphism $\phi$ induced by the inclusion $B \hookrightarrow A$. Prove that $\phi$ is an isomorphism between $Proj \ B$ and $Proj \ A$. (Hint: what is the domain of $\phi$? How does $\phi$ act on principal affine open subsets?)