1204.6154Apr 27, 201267 pages, 6 figures. With respect to the previous version, it incorporates the changes asked by the referees. In particular, there are more explanations, examples and figures. It will appear in "Tropical Geometry", Proceedings Castro Urdiales 2011, E. Brugallé, M.A. Cueto, A. Dickenstein, E.M. Feichtner and I. Itenberg editors, Contemporary Mathematics, AMS, 2013
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 (R) the topological space of valuations on R
Nearby in the stack
taking values in
R∪∞
. Then we may tropicalize with respect to
γ
any subset
of the space of valuations
(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
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.