Derived category of equivariant coherent sheaves on a smooth toric variety and Koszul duality

Let X be a smooth toric variety defined by the fan. We consider as a finite set with topology and define a natural sheaf of graded algebras A_ on. The category of modules over A_ is studied (together with other related categories). This leads to a certain combinatorial Koszul duality equivalence. We describe the equivariant category of coherent sheaves cohₗ, ₓ and a related (slightly bigger) equivariant category Oₗ, ₓ-mod in terms of sheaves of modules over the sheaf of algebras A_. Eventually (for a complete X) the combinatorial Koszul duality is interpreted in terms of the Serre functor on Dᵇ (cohₗ, ₓ)