ScalingStacks

Example 2.24 . [02J6]

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Example 2.24.

Let X=ℙK0=Spec⁡(K)X=\mathbb{P}^{0}_{K}=\operatorname{Spec}(K). A line bundle LL on XX is necessarily trivial, that is, L≃KL\simeq K. Consider the model (𝒳,ℒ,e)({\mathcal{X}},{\mathcal{L}},e) of (X,L)(X,L) given by 𝒳=Spec⁡(K∘){\mathcal{X}}=\operatorname{Spec}(K^{\circ}), e≥1e\geq 1, and ℒ{\mathcal{L}} a free K∘K^{\circ}-submodule of L⊗eL^{\otimes e} of rank one. Let v∈L⊗ev\in L^{\otimes e} be a basis of ℒ{\mathcal{L}}. For a section ss of LL we can write s⊗e=α​vs^{\otimes e}=\alpha v with α∈K\alpha\in K. Hence,

‖s‖=|α|1/e.\|s\|=|\alpha|^{1/e}.

All algebraic metrics on LanL^{{\text{\rm an}}} can be obtained in this way.

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