Situation 27.15.1. Here $S$ is a scheme, and $\mathcal{A}$ is a quasi-coherent graded $\mathcal{O}_ S$-algebra.
27.15 Relative Proj via glueing
In this section we outline how to construct a morphism of schemes
\[ \underline{\text{Proj}}_ S(\mathcal{A}) \longrightarrow S \]
by glueing the homogeneous spectra $\text{Proj}(\Gamma (U, \mathcal{A}))$ where $U$ ranges over the affine opens of $S$. We first show that the homogeneous spectra of the values of $\mathcal{A}$ over affines form a suitable collection of schemes, as in Lemma 27.2.1.
Lemma 27.15.2. In Situation 27.15.1. Suppose $U \subset U' \subset S$ are affine opens. Let $A = \mathcal{A}(U)$ and $A' = \mathcal{A}(U')$. The map of graded rings $A' \to A$ induces a morphism $r : \text{Proj}(A) \to \text{Proj}(A')$, and the diagram is cartesian. Moreover there are canonical isomorphisms $\theta : r^*\mathcal{O}_{\text{Proj}(A')}(n) \to \mathcal{O}_{\text{Proj}(A)}(n)$ compatible with multiplication maps.
\[ \xymatrix{ \text{Proj}(A) \ar[r] \ar[d] & \text{Proj}(A') \ar[d] \\ U \ar[r] & U' } \]
Proof. Let $R = \mathcal{O}_ S(U)$ and $R' = \mathcal{O}_ S(U')$. Note that the map $R \otimes _{R'} A' \to A$ is an isomorphism as $\mathcal{A}$ is quasi-coherent (see Schemes, Lemma 26.7.3 for example). Hence the lemma follows from Lemma 27.11.6. $\square$
In particular the morphism $\text{Proj}(A) \to \text{Proj}(A')$ of the lemma is an open immersion.
Lemma 27.15.3. In Situation 27.15.1. Suppose $U \subset U' \subset U'' \subset S$ are affine opens. Let $A = \mathcal{A}(U)$, $A' = \mathcal{A}(U')$ and $A'' = \mathcal{A}(U'')$. The composition of the morphisms $r : \text{Proj}(A) \to \text{Proj}(A')$, and $r' : \text{Proj}(A') \to \text{Proj}(A'')$ of Lemma 27.15.2 gives the morphism $r'' : \text{Proj}(A) \to \text{Proj}(A'')$ of Lemma 27.15.2. A similar statement holds for the isomorphisms $\theta $.
Proof. This follows from Lemma 27.11.2 since the map $A'' \to A$ is the composition of $A'' \to A'$ and $A' \to A$. $\square$
Lemma 27.15.4. In Situation 27.15.1. There exists a morphism of schemes with the following properties:
for every affine open $U \subset S$ there exists an isomorphism $i_ U : \pi ^{-1}(U) \to \text{Proj}(A)$ with $A = \mathcal{A}(U)$, and
for $U \subset U' \subset S$ affine open the composition
with $A = \mathcal{A}(U)$, $A' = \mathcal{A}(U')$ is the open immersion of Lemma 27.15.2 above.
\[ \pi : \underline{\text{Proj}}_ S(\mathcal{A}) \longrightarrow S \]
\[ \xymatrix{ \text{Proj}(A) \ar[r]^{i_ U^{-1}} & \pi ^{-1}(U) \ar[rr]^{inclusion} & & \pi ^{-1}(U') \ar[r]^{i_{U'}} & \text{Proj}(A') } \]
Proof. Follows immediately from Lemmas 27.2.1, 27.15.2, and 27.15.3. $\square$
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 uniquely determined by the following property: For every affine open $U \subset S$ with $A = \mathcal{A}(U)$ there is an isomorphism of $\mathbf{Z}$-graded $\mathcal{O}_{\pi ^{-1}(U)}$-algebras such that is commutative.
\[ \psi : \mathcal{A} \longrightarrow \bigoplus \nolimits _{n \geq 0} \pi _*\left(\mathcal{O}_{\underline{\text{Proj}}_ S(\mathcal{A})}(n)\right) \]
\[ \theta _ U : i_ U^*\left( \bigoplus \nolimits _{n \in \mathbf{Z}} \mathcal{O}_{\text{Proj}(A)}(n) \right) \longrightarrow \left( \bigoplus \nolimits _{n \in \mathbf{Z}} \mathcal{O}_{\underline{\text{Proj}}_ S(\mathcal{A})}(n) \right)|_{\pi ^{-1}(U)} \]
\[ \xymatrix{ A_ n \ar[rr]_\psi \ar[dr]_-{(01MP)} & & \Gamma (\pi ^{-1}(U), \mathcal{O}_{\underline{\text{Proj}}_ S(\mathcal{A})}(n)) \\ & \Gamma (\text{Proj}(A), \mathcal{O}_{\text{Proj}(A)}(n)) \ar[ru]_-{\theta _ U} & } \]
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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