The Stacks project

Lemma 27.11.1. Let $A$, $B$ be two graded rings. Set $X = \text{Proj}(A)$ and $Y = \text{Proj}(B)$. Let $\psi : A \to B$ be a graded ring map. Set

\[ U(\psi ) = \bigcup \nolimits _{f \in A_{+}\ \text{homogeneous}} D_{+}(\psi (f)) \subset Y. \]

Then there is a canonical morphism of schemes

\[ r_\psi : U(\psi ) \longrightarrow X \]

and a map of $\mathbf{Z}$-graded $\mathcal{O}_{U(\psi )}$-algebras

\[ \theta = \theta _\psi : r_\psi ^*\left( \bigoplus \nolimits _{d \in \mathbf{Z}} \mathcal{O}_ X(d) \right) \longrightarrow \bigoplus \nolimits _{d \in \mathbf{Z}} \mathcal{O}_{U(\psi )}(d). \]

The triple $(U(\psi ), r_\psi , \theta )$ is characterized by the following properties:

  1. For every $d \geq 0$ the diagram

    \[ \xymatrix{ A_ d \ar[d] \ar[rr]_{\psi } & & B_ d \ar[d] \\ \Gamma (X, \mathcal{O}_ X(d)) \ar[r]^-\theta & \Gamma (U(\psi ), \mathcal{O}_ Y(d)) & \Gamma (Y, \mathcal{O}_ Y(d)) \ar[l] } \]

    is commutative.

  2. For any $f \in A_{+}$ homogeneous we have $r_\psi ^{-1}(D_{+}(f)) = D_{+}(\psi (f))$ and the restriction of $r_\psi $ to $D_{+}(\psi (f))$ corresponds to the ring map $A_{(f)} \to B_{(\psi (f))}$ induced by $\psi $.

Proof. Clearly condition (2) uniquely determines the morphism of schemes and the open subset $U(\psi )$. Pick $f \in A_ d$ with $d \geq 1$. Note that $\mathcal{O}_ X(n)|_{D_{+}(f)}$ corresponds to the $A_{(f)}$-module $(A_ f)_ n$ and that $\mathcal{O}_ Y(n)|_{D_{+}(\psi (f))}$ corresponds to the $B_{(\psi (f))}$-module $(B_{\psi (f)})_ n$. In other words $\theta $ when restricted to $D_{+}(\psi (f))$ corresponds to a map of $\mathbf{Z}$-graded $B_{(\psi (f))}$-algebras

\[ A_ f \otimes _{A_{(f)}} B_{(\psi (f))} \longrightarrow B_{\psi (f)} \]

Condition (1) determines the images of all elements of $A$. Since $f$ is an invertible element which is mapped to $\psi (f)$ we see that $1/f^ m$ is mapped to $1/\psi (f)^ m$. It easily follows from this that $\theta $ is uniquely determined, namely it is given by the rule

\[ a/f^ m \otimes b/\psi (f)^ e \longmapsto \psi (a)b/\psi (f)^{m + e}. \]

To show existence we remark that the proof of uniqueness above gave a well defined prescription for the morphism $r$ and the map $\theta $ when restricted to every standard open of the form $D_{+}(\psi (f)) \subset U(\psi )$ into $D_{+}(f)$. Call these $r_ f$ and $\theta _ f$. Hence we only need to verify that if $D_{+}(f) \subset D_{+}(g)$ for some $f, g \in A_{+}$ homogeneous, then the restriction of $r_ g$ to $D_{+}(\psi (f))$ matches $r_ f$. This is clear from the formulas given for $r$ and $\theta $ above. $\square$

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