ScalingStacks

Subsection [04WS]

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(1.5) We fix an algebraically closed field kk of characteristic 00 and we set R=k⁡[[t]]R=k[\negthinspace[t]\negthinspace] and K=k⁡((t))K=k(\negthinspace(t)\negthinspace). We also fix an algebraic closure KaK^{a} of KK. We denote by ordt\mathrm{ord}_{t} the tt-adic valuation on KK and we define an absolute value |⋅||\cdot| on KK by setting |a|=exp⁡(−ordt​a)|a|=\exp(-\mathrm{ord}_{t}a) for every a∈K×a\in K^{\times}. This turns KK into a complete non-archimedean field. We denote by (⋅)an(\cdot)^{\mathrm{an}} the analytification functor from the category of KK-schemes of finite type to Berkovich’s category of KK-analytic spaces. For every RR-scheme 𝒳\mathscr{X}, we will denote by 𝒳k=𝒳×Rk\mathscr{X}_{k}=\mathscr{X}\times_{R}k and 𝒳K=𝒳×RK\mathscr{X}_{K}=\mathscr{X}\times_{R}K its special and generic fiber.

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