Lemma 71.11.5. Let S be a scheme. Let g : X' \to X be a morphism of algebraic spaces over S and let \mathcal{A} be a quasi-coherent sheaf of graded \mathcal{O}_ X-algebras. Then there is a canonical isomorphism
r : \underline{\text{Proj}}_{X'}(g^*\mathcal{A}) \longrightarrow X' \times _ X \underline{\text{Proj}}_ X(\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}}_ X(\mathcal{A})}(d)\right) \longrightarrow \bigoplus \nolimits _{d \in \mathbf{Z}} \mathcal{O}_{\underline{\text{Proj}}_{X'}(g^*\mathcal{A})}(d)
of \mathbf{Z}-graded \mathcal{O}_{\underline{\text{Proj}}_{X'}(g^*\mathcal{A})}-algebras.
Proof.
Let F be the functor (71.11.1.1) and let F' be the corresponding functor defined using g^*\mathcal{A} on X'. We claim there is a canonical isomorphism r : F' \to X' \times _ X F of functors (and of course r is the isomorphism of the lemma). It suffices to construct the bijection r : F'(T) \to X'(T) \times _{X(T)} F(T) for quasi-compact schemes T over S. First, if \xi = (d', f', \mathcal{L}', \psi ') is a quadruple over T for F', then we can set r(\xi ) = (f', (d', g \circ f', \mathcal{L}', \psi ')). This makes sense as (g \circ f')^*\mathcal{A}^{(d)} = (f')^*(g^*\mathcal{A})^{(d)}. The inverse map sends the pair (f', (d, f, \mathcal{L}, \psi )) to the quadruple (d, f', \mathcal{L}, \psi ). We omit the proof of the final assertion (hint: reduce to the case of schemes by étale localization and apply Constructions, Lemma 27.16.10).
\square
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