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Abstract. It is well known that the exceptional set in a resolution of a rational surface singularity is a tree of rational curves. We generalize the combinatoric part of this statement to higher dimensions and show that the highest cohomologies of the dual complex associated to a resolution of an isolated rational singularity vanish. We also prove that the dual complex associated to a resolution of an isolated hypersurface singularity is simply connected. As a consequence, we show that the dual complex associated to a resolution of a 3-dimensional Gorenstein terminal singularity has the homotopy type of a point.

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.