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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