Lemma 27.16.10. Let $S$ be a scheme and $\mathcal{A}$ be a quasi-coherent sheaf of graded $\mathcal{O}_ S$-algebras. Let $g : S' \to S$ be any morphism of schemes. Then there is a canonical isomorphism

$r : \underline{\text{Proj}}_{S'}(g^*\mathcal{A}) \longrightarrow S' \times _ S \underline{\text{Proj}}_ S(\mathcal{A})$

as well as a corresponding isomorphism

$\theta : r^*\text{pr}_2^*\left(\bigoplus \nolimits _{d \in \mathbf{Z}} \mathcal{O}_{\underline{\text{Proj}}_ S(\mathcal{A})}(d)\right) \longrightarrow \bigoplus \nolimits _{d \in \mathbf{Z}} \mathcal{O}_{\underline{\text{Proj}}_{S'}(g^*\mathcal{A})}(d)$

of $\mathbf{Z}$-graded $\mathcal{O}_{\underline{\text{Proj}}_{S'}(g^*\mathcal{A})}$-algebras.

Proof. This follows from Lemma 27.16.1 and the construction of $\underline{\text{Proj}}_ S(\mathcal{A})$ in Lemma 27.16.5 as the union of the schemes $U_ d$ representing the functors $F_ d$. In terms of the construction of relative Proj via glueing this isomorphism is given by the isomorphisms constructed in Lemma 27.11.6 which provides us with the isomorphism $\theta$. Some details omitted. $\square$

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