The universal tropicalization and the Berkovich analytification

Abstract

Given an integral scheme X over a non-archimedean valued field k, we construct a universal closed embedding of X into a k-scheme equipped with a model over the field with one element F 1 (a generalization of a toric variety). An embedding into such an ambient space determines a tropicalization of X by [GG13], and we show that the set-theoretic tropicalization of X with respect to this universal embedding is the Berkovich analytification X an . Moreover, using the scheme-theoretic tropicalization of [GG13], we obtain a tropical scheme Trop univ (X) whose T-points give the analytification and that 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 valuations on X, and when X = Spec A is affine there is a universal valuation on A taking values in the semiring of regular functions on the universal tropicalization.

Vector bundles on tropical schemes

“…There is a canonical way to impose a topology on X (k), called the fine Zariski topology. In tropical geometry, the fine Zariski topology has been used to give a homeomorphism between Berkovich analytification and a set of rational points of a scheme over some "generalized algebraic structures", for instance, see [GG14], [Lor15], and [Jun17b].…”

“…The extra structure φ allows one to perform the scheme-theoretic tropicalization for Spec A with respect to the map φ . We note that the idea of labelled algebras is not new; a similar idea has been implemented in tropical geometry as a tool to study scheme-theoretic tropicalization, for instance, see [Lor15] and [GG14].…”

Equations of tropical varieties

“…For instance, as we see in the following example, the T-points of a tropical variety more closely reflect familiar geometry than its prime spectrum. Moreover, there is a natural topology on the T-points of a T-scheme such that closed subschemes, as defined above, induce closed subsets of the T-points; see [GG14,§3.4] where this notion is introduced and used to show that the Berkovich analytification of a scheme is homeomorphic to the T-points of a certain tropicalization of the scheme. T = Spec T[x] is clearly T itself, but the ideal-theoretic kernels of the corresponding homomorphisms T[x] → T are all trivial except for the point x → −∞ for which the ideal is maximal.…”

“…If k is a non-archimedean field and R is a k-algebra, then the set of valuations on R extending the valuation on k is the Berkovich analytification of Spec R relative to k [Ber90]. The Tpoints of our moduli space Val (R) can thus be viewed as an absolute version of the analytification of Spec R. This is explored further in [GG14].…”

Čech cohomology of semiring schemes

“…In [1], J.Giansiracusa and N.Giansiracusa proved that one can associate a semiring scheme X to a tropical variety Y in such a way that Y can be identified with X(R max ), the set of 'R max -rational points' of X, where R max = (R ∪ {−∞}, max, +) is a tropical semifield with a maximum convention (also, see [2] for the further development in connection to the Berkovich analytification). This opens the door to approach tropical geometry by means of semiring schemes and, to this end, one needs to better comprehend semiring schemes in perspective of both F 1 -geometry and tropical geometry.…”