On the integral cohomology of smooth toric varieties

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Let XΣ be a smooth, not necessarily compact toric variety. We show that a certain complex, defined in terms of the fan Σ, computes the integral cohomology of XΣ, 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 XΣ is formal.