The Stacks project

27.3 Relative spectrum via glueing

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.

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:

  1. 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

  2. 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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