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

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 A_{\Sigma} on {\Sigma} . The category of modules over 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 coh_{X,T} and a related (slightly bigger) equivariant category O_{X,T}-mod in terms of sheaves of modules over the sheaf of algebras A_{\Sigma} . Eventually (for a complete X ) the combinatorial Koszul duality is interpreted in terms of the Serre functor on D^b(coh_{X,T})