  The aim of this paper is to study homological properties of tropical fans and
to propose a notion of smoothness in tropical geometry, which goes beyond
matroids and their Bergman fans and which leads to an enrichment of the
category of smooth tropical varieties.
  Among the resulting applications, we prove the Hodge isomorphism theorem
which asserts that the Chow rings of smooth unimodular tropical fans are
isomorphic to the tropical cohomology rings of their corresponding canonical
compactifications, and prove a slightly weaker statement for any unimodular
fan. We furthermore introduce a notion of shellability for tropical fans and
show that shellable tropical fans are smooth and thus enjoy all the nice
homological properties of smooth tropical fans. Several other interesting
properties for tropical fans are shown to be shellable. Finally, we obtain a
generalization, both in the tropical and in the classical setting, of the
pioneering work of Feichtner-Yuzvinsky and De Concini-Procesi on the cohomology
ring of wonderful compactifications of complements of hyperplane arrangements.
  The results in this paper form the basis for our subsequent works on Hodge
theory for tropical and non-Archimedean varieties.
