The Stacks project

Lemma 27.4.7. Let $f : X \to S$ be a quasi-compact and quasi-separated morphism of schemes. By Schemes, Lemma 26.24.1 the sheaf $f_*\mathcal{O}_ X$ is a quasi-coherent sheaf of $\mathcal{O}_ S$-algebras. There is a canonical morphism

\[ can : X \longrightarrow \underline{\mathop{\mathrm{Spec}}}_ S(f_*\mathcal{O}_ X) \]

of schemes over $S$. For any affine open $U \subset S$ the restriction $can|_{f^{-1}(U)}$ is identified with the canonical morphism

\[ f^{-1}(U) \longrightarrow \mathop{\mathrm{Spec}}(\Gamma (f^{-1}(U), \mathcal{O}_ X)) \]

coming from Schemes, Lemma 26.6.4.

Proof. The morphism comes, via the definition of $\underline{\mathop{\mathrm{Spec}}}$ as the scheme representing the functor $F$, from the canonical map $\varphi : f^*f_*\mathcal{O}_ X \to \mathcal{O}_ X$ (which by adjointness of push and pull corresponds to $\text{id} : f_*\mathcal{O}_ X \to f_*\mathcal{O}_ X$). The statement on the restriction to $f^{-1}(U)$ follows from the description of the relative spectrum over affines, see Lemma 27.4.2. $\square$


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