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 $\Sigma$. We consider $\Sigma$ as a finite set with topology and define a natural sheaf of graded algebras $\mathcal{A}_\Sigma$ on $\Sigma$. The category of modules over $\mathcal{A}_\Sigma$ is studied (together with other related categories). This leads to a certain combinatorial Koszul duality equivalence.
We describe the equivariant category of coherent sheaves $\mathrm{coh}_{X,T}$ and a related (slightly bigger) equivariant category $\mathcal{O}_{X,T}\text{-}\mathrm{mod}$ in terms of sheaves of modules over the sheaf of algebras $\mathcal{A}_\Sigma$. Eventually (for a complete $X$), the combinatorial Koszul duality is interpreted in terms of the Serre functor on $D^b(\mathrm{coh}_{X,T})$.