Abstract
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 \(D^b(\mathrm{coh}_{X,T})\) .
