Lemma 71.11.2. In Situation 71.11.1. The functor F above is an algebraic space. For any morphism g : Z \to X where Z is a scheme there is a canonical isomorphism \underline{\text{Proj}}_ Z(g^*\mathcal{A}) = Z \times _ X F compatible with further base change.
Proof. It suffices to prove the second assertion, see Spaces, Lemma 65.11.3. Let g : Z \to X be a morphism where Z is a scheme. Let F' be the functor of quadruples associated to the graded quasi-coherent \mathcal{O}_ Z-algebra g^*\mathcal{A}. Then there is a canonical isomorphism F' = Z \times _ X F, sending a quadruple (d, f : T \to Z, \mathcal{L}, \psi ) for F' to (d, g \circ f, \mathcal{L}, \psi ) (details omitted, see proof of Constructions, Lemma 27.16.1). By Constructions, Lemmas 27.16.4, 27.16.5, and 27.16.6 and Definition 27.16.7 we see that F' is representable by \underline{\text{Proj}}_ Z(g^*\mathcal{A}). \square
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