On the integral cohomology of smooth toric varieties

Let $X_\Sigma$ be a smooth, not necessarily compact toric variety. We show that a certain complex, defined in terms of the fan $\Sigma$, computes the integral cohomology of $X_\Sigma$, including the module structure over the homology of the torus. In some cases we can also give the product. As a corollary we obtain that the cycle map from Chow groups to integral Borel-Moore homology is split injective for smooth toric varieties. Another result is that the differential algebra of singular cochains on the Borel construction of $X_\Sigma$ is formal.