Situation 27.3.1. Here S is a scheme, and \mathcal{A} is a quasi-coherent \mathcal{O}_ S-algebra. This means that \mathcal{A} is a sheaf of \mathcal{O}_ S-algebras which is quasi-coherent as an \mathcal{O}_ S-module.
27.3 Relative spectrum via glueing
In this section we outline how to construct a morphism of schemes
\underline{\mathop{\mathrm{Spec}}}_ S(\mathcal{A}) \longrightarrow S
by glueing the spectra \mathop{\mathrm{Spec}}(\Gamma (U, \mathcal{A})) where U ranges over the affine opens of S. We first show that the spectra of the values of \mathcal{A} over affines form a suitable collection of schemes, as in Lemma 27.2.1.
Lemma 27.3.2. In Situation 27.3.1. Suppose U \subset U' \subset S are affine opens. Let A = \mathcal{A}(U) and A' = \mathcal{A}(U'). The map of rings A' \to A induces a morphism \mathop{\mathrm{Spec}}(A) \to \mathop{\mathrm{Spec}}(A'), and the diagram
\xymatrix{ \mathop{\mathrm{Spec}}(A) \ar[r] \ar[d] & \mathop{\mathrm{Spec}}(A') \ar[d] \\ U \ar[r] & U' }
is cartesian.
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). The result follows from the description of the fibre product of affine schemes in Schemes, Lemma 26.6.7. \square
In particular the morphism \mathop{\mathrm{Spec}}(A) \to \mathop{\mathrm{Spec}}(A') of the lemma is an open immersion.
Lemma 27.3.3. In Situation 27.3.1. 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 \mathop{\mathrm{Spec}}(A) \to \mathop{\mathrm{Spec}}(A'), and \mathop{\mathrm{Spec}}(A') \to \mathop{\mathrm{Spec}}(A'') of Lemma 27.3.2 gives the morphism \mathop{\mathrm{Spec}}(A) \to \mathop{\mathrm{Spec}}(A'') of Lemma 27.3.2.
Proof. This follows as the map A'' \to A is the composition of A'' \to A' and A' \to A (because \mathcal{A} is a sheaf). \square
Lemma 27.3.4. In Situation 27.3.1. There exists a morphism of schemes
\pi : \underline{\mathop{\mathrm{Spec}}}_ S(\mathcal{A}) \longrightarrow S
with the following properties:
for every affine open U \subset S there exists an isomorphism i_ U : \pi ^{-1}(U) \to \mathop{\mathrm{Spec}}(\mathcal{A}(U)) over U, and
for U \subset U' \subset S affine open the composition
\xymatrix{ \mathop{\mathrm{Spec}}(\mathcal{A}(U)) \ar[r]^{i_ U^{-1}} & \pi ^{-1}(U) \ar[rr]^{inclusion} & & \pi ^{-1}(U') \ar[r]^{i_{U'}} & \mathop{\mathrm{Spec}}(\mathcal{A}(U')) }is the open immersion of Lemma 27.3.2 above.
Moreover, \underline{\mathop{\mathrm{Spec}}}_ S(\mathcal{A}) is unique up to unique isomorphism over S.
Proof. Follows immediately from Lemmas 27.2.1, 27.3.2, and 27.3.3. Uniqueness is stated in the last sentence of Lemma 27.2.1. \square
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