Koszul duality and equivariant cohomology for tori

Koszul duality (in its simplest form) refers to the equivalence of derived categories of differential modules over symmetric and exterior algebras. Goresky, Kottwitz, and MacPherson have shown that one can use Koszul duality to compute the real equivariant cohomology of a $G$-space as $H^*(BG)$-module from the non-equivariant cochain complex. Similarly, the equivariant cochain complex determines the ordinary cohomology as $H_*(G)$-module. For the case of torus actions and singular cohomology, I will present a different approach which extends to arbitrary coefficients. It permits moreover to recover the product structure in equivariant cohomology. As application I will describe the integral cohomology of smooth toric varieties. This complements a result of Buchstaber and Panov.