Morphisms of Berkovich curves and the different function

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• 作者：Adina Cohen ^{adina.cohen@mail.huji.ac.il" class="auth_mail" title="E-mail the corresponding author} ; Michael Temkin ; ^{temkin@math.huji.ac.il" class="auth_mail" title="E-mail the corresponding author} ; Dmitri Trushin ^{trushindima@yandex.ru" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Berkovich analytic spaces ; The different ; Topological ramification
• 刊名：Advances in Mathematics
• 出版年：2016
• 出版时间：5 November 2016
• 年：2016
• 卷：303
• 期：Complete
• 页码：800-858
• 全文大小：1002 K

文摘

Given a generically étale morphism f:Y→X$f:Y\to X$ of quasi-smooth Berkovich curves, we define a different function δ_f:Y→[0,1]${\delta }_f:Y\to [0,1]$ that measures the wildness of the topological ramification locus of f. This provides a new invariant for studying f  , which cannot be obtained by the usual reduction techniques. We prove that δ_f${\delta }_f$ is a piecewise monomial function satisfying a balancing condition at type 2 points analogous to the classical Riemann–Hurwitz formula, and show that δ_f${\delta }_f$ can be used to explicitly construct the simultaneous skeletons of X and Y. As another application, we use our results to completely describe the topological ramification locus of f when its degree equals to the residue characteristic p.