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The Stacks project

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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