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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