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:
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.
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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