The Stacks project

Lemma 27.15.2. In Situation 27.15.1. Suppose $U \subset U' \subset S$ are affine opens. Let $A = \mathcal{A}(U)$ and $A' = \mathcal{A}(U')$. The map of graded rings $A' \to A$ induces a morphism $r : \text{Proj}(A) \to \text{Proj}(A')$, and the diagram

\[ \xymatrix{ \text{Proj}(A) \ar[r] \ar[d] & \text{Proj}(A') \ar[d] \\ U \ar[r] & U' } \]

is cartesian. Moreover there are canonical isomorphisms $\theta : r^*\mathcal{O}_{\text{Proj}(A')}(n) \to \mathcal{O}_{\text{Proj}(A)}(n)$ compatible with multiplication maps.

Proof. Let $R = \mathcal{O}_ S(U)$ and $R' = \mathcal{O}_ S(U')$. Note that the map $R \otimes _{R'} A' \to A$ is an isomorphism as $\mathcal{A}$ is quasi-coherent (see Schemes, Lemma 26.7.3 for example). Hence the lemma follows from Lemma 27.11.6. $\square$

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