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
\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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