The Stacks project

Lemma 27.4.6. Let $S$ be a scheme. Let $\mathcal{A}$ be a quasi-coherent sheaf of $\mathcal{O}_ S$-algebras. Let $\pi : \underline{\mathop{\mathrm{Spec}}}_ S(\mathcal{A}) \to S$ be the relative spectrum of $\mathcal{A}$ over $S$.

  1. For every affine open $U \subset S$ the inverse image $\pi ^{-1}(U)$ is affine.

  2. For every morphism $g : S' \to S$ we have $S' \times _ S \underline{\mathop{\mathrm{Spec}}}_ S(\mathcal{A}) = \underline{\mathop{\mathrm{Spec}}}_{S'}(g^*\mathcal{A})$.

  3. The universal map

    \[ \mathcal{A} \longrightarrow \pi _*\mathcal{O}_{\underline{\mathop{\mathrm{Spec}}}_ S(\mathcal{A})} \]

    is an isomorphism of $\mathcal{O}_ S$-algebras.

Proof. Part (1) comes from the description of the relative spectrum by glueing, see Lemma 27.3.4. Part (2) follows immediately from Lemma 27.4.1. Part (3) follows because it is local on $S$ and it is clear in case $S$ is affine by Lemma 27.4.2 for example. $\square$

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