We study toric varieties over an arbitrary field with an emphasis on toric surfaces in
the Merkurjev–Panin motivic category of “K-motives”. We explore the decomposition
of certain toric varieties as K-motives into products of central simple algebras, the
geometric and topological information encoded in these central simple algebras, and
the relationship between the decomposition of the K-motives and the semiorthogonal
decomposition of the derived categories. We obtain the information mentioned above
for toric surfaces by explicitly classifying all minimal smooth projective toric surfaces
using toric geometry.
