The Stacks project

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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