Subsection 2.5.2. (00N1) · ScalingStacks

2.5.2. Global situation

One can analytify a scheme of finite type defined over $k$ by glueing local constructions.

Definition 2.93. Let $X$ be a finite type scheme over $\spec k$ , and write $X$ as $\bigcup X_{i}$ where $X_{i}=\spec A_{X_{i}}$ are affine charts. The Berkovich analytification of $(X,\mathscr{O}_{X})$ is the locally ringed space obtained by gluing the Berkovich analytification $((X_{i})^{an},\mathscr{O}_{(X_{i})^{an}})$ of each $(X_{i},\mathscr{O}_{X_{i}})$ .

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Proposition 2.94. Let $\phi:X\rightarrow Y$ be a morphism of schemes of locally finite type over $\spec k$ . Then it induces a continuous map $\phi^{\mathrm{an}}:X^{\mathrm{an}}\rightarrow Y^{\mathrm{an}}$ . And $\phi$ is (1) separated, (2) injective, (3) surjective, (4) an open immersion and (5) an isomorphism if and only if $\phi^{\mathrm{an}}$ has the same property. ([Ber, Proposition 3.4.6])

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Theorem 2.95. If $X$ is proper, then $X^{\mathrm{an}}$ is Hausdorff and compact. ([Ber, Theorem 3.4.8])