4.1 [035J]
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4.1
We recall first the construction of the analytification of . Let be an open affine subset of , then is the set of multiplicative seminorms on extending the given absolute value on . This set is endowed with the topology generated by the functions with ranging over . By glueing, we get a topological space which is connected locally compact and Hausdorff. We can endow it with a sheaf of analytic functions leading to a Berkovich analytic space over which we call the analytification of . For a morphism of algebraic varieties over , we get an analytic morphism induced by composing the multiplicative semiorms with on suitable affine open subsets. We refer to [Be90] for details, or to [BPS11], §1.2, for a neat description of the analytification.