Lemma 71.11.11. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{A}$ be a quasi-coherent sheaf of graded $\mathcal{O}_ X$-modules generated as an $\mathcal{A}_0$-algebra by $\mathcal{A}_1$. With $P = \underline{\text{Proj}}_ X(\mathcal{A})$ we have

$P$ represents the functor $F_1$ which associates to $T$ over $S$ the set of isomorphism classes of triples $(f, \mathcal{L}, \psi )$, where $f : T \to X$ is a morphism over $S$, $\mathcal{L}$ is an invertible $\mathcal{O}_ T$-module, and $\psi : f^*\mathcal{A} \to \bigoplus _{n \geq 0} \mathcal{L}^{\otimes n}$ is a map of graded $\mathcal{O}_ T$-algebras inducing a surjection $f^*\mathcal{A}_1 \to \mathcal{L}$,

the canonical map $\pi ^*\mathcal{A}_1 \to \mathcal{O}_ P(1)$ is surjective, and

each $\mathcal{O}_ P(n)$ is invertible and the multiplication maps induce isomorphisms $\mathcal{O}_ P(n) \otimes _{\mathcal{O}_ P} \mathcal{O}_ P(m) = \mathcal{O}_ P(n + m)$.

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