Local tropicalization
Patrick Popescu‐Pampu, D. A. Stepanov · Contemporary mathematics - American Mathematical Society · 2013
In this paper we propose a general functorial definition of the operation of local tropicalization in commutative algebra. Let R R be a commutative ring, Γ \Gamma a finitely generated subsemigroup of a lattice, γ : Γ → R / R ∗ \gamma : \Gamma \rightarrow R/ R^* a morphism of semigroups, and V ( R ) \mathcal {V}(R) the topological space of valuations on R R taking values in R ∪ ∞ \mathbb {R} \cup \infty . Then we may tropicalize with respect to γ \gamma any subset W \mathcal {W} of the space of valuations V ( R ) \mathcal {V}(R) . By definition, we get a subset of a rational polyhedral cone canonically associated to Γ \Gamma , enriched with strata at infinity. In particular, when R R is a local ring, γ \gamma is a local morphism of semigroups, and W </