Local tropicalization

Abstract

In this paper we propose a general functorial definition of the operation of local tropicalization in commutative algebra. Let R be a commutative ring, Γ a finitely generated subsemigroup of a lattice, γ : Γ → R/R * a morphism of semigroups, and V(R) the topological space of valuations on R taking values in R ∪ ∞. Then we may tropicalize with respect to γ any subset W of the space of valuations V(R). By definition, we get a subset of a rational polyhedral cone canonically associated to Γ, enriched with strata at infinity. In particular, when R is a local ring, γ is a local morphism of semigroups, and W is the space of valuations which are either positive or non-negative on R, we call these processes local tropicalizations. They depend only on the ambient toroidal structure, which in turn allows to define tropicalizations of subvarieties of toroidal embeddings. We prove that with suitable hypothesis, these local tropicalizations are the supports of finite rational polyhedral fans enriched with strata at infinity and we compare the global and local tropicalizations of a subvariety of a toric variety.

Functorial tropicalization of logarithmic schemes: the case of constant coefficients

“…Our approach to the construction of is a global version of the local tropicalization map defined by Popescu‐Pampu and Stepanov in [, Section 6]. Fix a morphism from a monoid into the multiplicative monoid of an algebra of finite type over (or more generally into the quotient ) and set .…”

“…The natural continuous tropicalization map is then formally defined as the ‘analytification’ of the characteristic morphism . In the special case that both and are affine, we have and the tropicalization map is nothing but the local tropicalization map of Popescu‐Pampu and Stepanov .…”

Universal valued fields and lifting points in local tropical varieties

“…Since n ∩ R/p = m is the maximal ideal of R/p, the ideal n is also maximal. Then, consider the localization (R/q) n and its completion R ′ with respect to n. The valuation v extends canonically to a local valuation v ′ of R ′ (see, e.g., [PPS,Lemma 5.16]). Finally, passing to the quotient of R ′ by home(v ′ ), we can assume that R ′ is a domain and home(v ′ ) = {0}.…”

“…where v runs over all local valuations of R that take value +∞ on the ideal I, and R = R ∪ {+∞}. This is a particular case of [PPS,Definition 6.6]; see also Section 4. Let us introduce two special fields.…”

“…Let k be a field and consider all possible formal series m∈Γ a m χ m , where a m ∈ k. Any two such series can be added and multiplied in a natural way, in particular, χ m · χ m ′ = χ m+m ′ . These series form a complete Noetherian local ring k[[Γ]] called the ring of formal power series over Γ, see[PPS, Section 8]. There is a natural semigroup homomorphism Γ → k[[Γ]] defined by Γ ∋ m → χ m , the elements χ m ∈ k[[Γ]] being called the monomials.…”