The Stacks project

Exercise 111.37.6. Maps of $\text{Proj}$. Let $R$ and $S$ be graded rings. Suppose we have a ring map

and an integer $e \geq 1$ such that $\psi (R_ d) \subset S_{de}$ for all $d \geq 0$. (By our conventions this is not a homomorphism of graded rings, unless $e = 1$.)

1. For which elements $\mathfrak p \in \text{Proj}(S)$ is there a well-defined corresponding point in $\text{Proj}(R)$? In other words, find a suitable open $U \subset \text{Proj}(S)$ such that $\psi$ defines a continuous map $r_\psi : U \to \text{Proj}(R)$.

2. Give an example where $U \not= \text{Proj}(S)$.

3. Give an example where $U = \text{Proj}(S)$.

4. (Do not write this down.) Convince yourself that the continuous map $U \to \text{Proj}(R)$ comes canonically with a map on sheaves so that $r_\psi$ is a morphism of schemes:

   $$\text{Proj}(S) \supset U \longrightarrow \text{Proj}(R).$$

5. What can you say about this map if $R = \bigoplus _{d \geq 0} S_{de}$ (as a graded ring with $S_ e$, $S_{2e}$, etc in degree $1$, $2$, etc) and $\psi$ is the inclusion mapping?