The Stacks project

Lemma 27.16.11. Let $S$ be a scheme. Let $\mathcal{A}$ be a quasi-coherent sheaf of graded $\mathcal{O}_ S$-modules generated as an $\mathcal{A}_0$-algebra by $\mathcal{A}_1$. In this case the scheme $X = \underline{\text{Proj}}_ S(\mathcal{A})$ represents the functor $F_1$ which associates to a scheme $f : T \to S$ over $S$ the set of pairs $(\mathcal{L}, \psi )$, where

  1. $\mathcal{L}$ is an invertible $\mathcal{O}_ T$-module, and

  2. $\psi : f^*\mathcal{A} \to \bigoplus _{n \geq 0} \mathcal{L}^{\otimes n}$ is a graded $\mathcal{O}_ T$-algebra homomorphism such that $f^*\mathcal{A}_1 \to \mathcal{L}$ is surjective

up to strict equivalence as above. Moreover, in this case all the quasi-coherent sheaves $\mathcal{O}_{\underline{\text{Proj}}(\mathcal{A})}(n)$ are invertible $\mathcal{O}_{\underline{\text{Proj}}(\mathcal{A})}$-modules and the multiplication maps induce isomorphisms $ \mathcal{O}_{\underline{\text{Proj}}(\mathcal{A})}(n) \otimes _{\mathcal{O}_{\underline{\text{Proj}}(\mathcal{A})}} \mathcal{O}_{\underline{\text{Proj}}(\mathcal{A})}(m) = \mathcal{O}_{\underline{\text{Proj}}(\mathcal{A})}(n + m)$.

Proof. Under the assumptions of the lemma the sheaves $\mathcal{O}_{\underline{\text{Proj}}(\mathcal{A})}(n)$ are invertible and the multiplication maps isomorphisms by Lemma 27.16.5 and Lemma 27.12.3 over affine opens of $S$. Thus $X$ actually represents the functor $F_1$, see proof of Lemma 27.16.5. $\square$

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