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
Comments (2)
Comment #7952 by Raffaele Lamagna on
Comment #8189 by Stacks Project on