普遍热带化与布氏分析
The universal tropicalization and the Berkovich analytification
摘要：
给定一个非阿基米德值域k上的整概形X，我们构造了X到一个k-概型中的泛闭嵌入，该概型在该域上具有一个单元素的模型（复曲面簇的推广）.嵌入到这样一个环境空间确定了热带化的X的早期工作的作者，我们表明，集理论热带化的X关于这个普遍的嵌入是Berkovich分析的X。此外，利用模式理论热带化，我们得到了一个热带模式$Trop_{univ}(X)$，它的T-点给出了分析，并正则地映射到X的所有其他模式理论热带化。这就明确了布氏分析是普遍的热带化的观点。当X = spec A是仿射的时，我们证明了$Trop_{univ}(X)$是X关于仿射空间中所有嵌入的热带化的极限，从而给出了Payne的一个著名结果的一个图式理论的丰富.最后，我们证明了$Trop_{univ}(X)$表示X上半赋值的模函子，并且当X = spec A是仿射的时，A上存在取值于泛热带化的正则函数的幂等半环中的泛半赋值.
英文摘要：
Given an integral scheme X over a non-archimedean valued field k , we construct acuniversal closed embedding of X into a k-scheme equipped with a model over the field with one element (a generalization of a toric variety). An embedding into such an ambient space determines a tropicalization of X by earlier work of the authors, and we show that the set-theoretic tropicalization of X with respect to this universal embedding is the Berkovich analytification of X. Moreover, using the scheme-theoretic tropicalization, we obtain a tropical scheme $Trop_{univ}(X)$ whose T-points give the analytification and which canonically maps to all other scheme-theoretic tropicalizations of X. This makes precise the idea that the Berkovich analytification is the universal tropicalization. When X = spec A is affine, we show that $Trop_{univ}(X)$ is the limit of the tropicalizations of X with respect to all embeddings in affine space, thus giving a scheme-theoretic enrichment of a well-known result of Payne. Finally, we show that $Trop_{univ}(X)$ represents the moduli functor of semivaluations on X, and when X = spec A is affine there is a universal semivaluation on A taking values in the idempotent semiring of regular functions on the universal tropicalization.
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