Lemma 27.15.4. In Situation 27.15.1. There exists a morphism of schemes

$\pi : \underline{\text{Proj}}_ S(\mathcal{A}) \longrightarrow S$

with the following properties:

1. 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

2. for $U \subset U' \subset S$ affine open the composition

$\xymatrix{ \text{Proj}(A) \ar[r]^{i_ U^{-1}} & \pi ^{-1}(U) \ar[rr]^{inclusion} & & \pi ^{-1}(U') \ar[r]^{i_{U'}} & \text{Proj}(A') }$

with $A = \mathcal{A}(U)$, $A' = \mathcal{A}(U')$ is the open immersion of Lemma 27.15.2 above.

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