Lemma 27.15.5. In Situation 27.15.1. The morphism $\pi : \underline{\text{Proj}}_ S(\mathcal{A}) \to S$ of Lemma 27.15.4 comes with the following additional structure. There exists a quasi-coherent $\mathbf{Z}$-graded sheaf of $\mathcal{O}_{\underline{\text{Proj}}_ S(\mathcal{A})}$-algebras $\bigoplus \nolimits _{n \in \mathbf{Z}} \mathcal{O}_{\underline{\text{Proj}}_ S(\mathcal{A})}(n)$, and a morphism of graded $\mathcal{O}_ S$-algebras

$\psi : \mathcal{A} \longrightarrow \bigoplus \nolimits _{n \geq 0} \pi _*\left(\mathcal{O}_{\underline{\text{Proj}}_ S(\mathcal{A})}(n)\right)$

uniquely determined by the following property: For every affine open $U \subset S$ with $A = \mathcal{A}(U)$ there is an isomorphism

$\theta _ U : i_ U^*\left( \bigoplus \nolimits _{n \in \mathbf{Z}} \mathcal{O}_{\text{Proj}(A)}(n) \right) \longrightarrow \left( \bigoplus \nolimits _{n \in \mathbf{Z}} \mathcal{O}_{\underline{\text{Proj}}_ S(\mathcal{A})}(n) \right)|_{\pi ^{-1}(U)}$

of $\mathbf{Z}$-graded $\mathcal{O}_{\pi ^{-1}(U)}$-algebras such that

$\xymatrix{ A_ n \ar[rr]_\psi \ar[dr]_-{(01MP)} & & \Gamma (\pi ^{-1}(U), \mathcal{O}_{\underline{\text{Proj}}_ S(\mathcal{A})}(n)) \\ & \Gamma (\text{Proj}(A), \mathcal{O}_{\text{Proj}(A)}(n)) \ar[ru]_-{\theta _ U} & }$

is commutative.

Proof. We are going to use Lemma 27.2.2 to glue the sheaves of $\mathbf{Z}$-graded algebras $\bigoplus _{n \in \mathbf{Z}} \mathcal{O}_{\text{Proj}(A)}(n)$ for $A = \mathcal{A}(U)$, $U \subset S$ affine open over the scheme $\underline{\text{Proj}}_ S(\mathcal{A})$. We have constructed the data necessary for this in Lemma 27.15.2 and we have checked condition (d) of Lemma 27.2.2 in Lemma 27.15.3. Hence we get the sheaf of $\mathbf{Z}$-graded $\mathcal{O}_{\underline{\text{Proj}}_ S(\mathcal{A})}$-algebras $\bigoplus _{n \in \mathbf{Z}} \mathcal{O}_{\underline{\text{Proj}}_ S(\mathcal{A})}(n)$ together with the isomorphisms $\theta _ U$ for all $U \subset S$ affine open and all $n \in \mathbf{Z}$. For every affine open $U \subset S$ with $A = \mathcal{A}(U)$ we have a map $A \to \Gamma (\text{Proj}(A), \bigoplus _{n \geq 0} \mathcal{O}_{\text{Proj}(A)}(n))$. Hence the map $\psi$ exists by functoriality of relative glueing, see Remark 27.2.3. The diagram of the lemma commutes by construction. This characterizes the sheaf of $\mathbf{Z}$-graded $\mathcal{O}_{\underline{\text{Proj}}_ S(\mathcal{A})}$-algebras $\bigoplus \mathcal{O}_{\underline{\text{Proj}}_ S(\mathcal{A})}(n)$ because the proof of Lemma 27.11.1 shows that having these diagrams commute uniquely determines the maps $\theta _ U$. Some details omitted. $\square$

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