Intersection Theory on Linear Subvarieties of Toric Varieties

We give a complete description of the cohomology ring $A^{\ast }(\bar{Z})$ of a compactification of a linear subvariety $Z$ of a torus in a smooth toric variety whose fan $\mathrm{\Sigma }$ is supported on the tropicalization of $Z$. It turns out that cocycles on $\bar{Z}$ canonically correspond to Minkowski weights on $\mathrm{\Sigma }$ and that the cup product is described by the intersection product on the tropical matroid variety $\mathrm{Trop}(Z)$.