Quasicoherent sheaves on toric schemes

Abstract

Let X be the toric scheme over a ring R associated with a fan Σ. It is shown that there are a group B, a B-graded R-algebra S and a graded ideal I⊆S such that there is an essentially surjective, exact functor •˜ from the category of B-graded S-modules to the category of quasicoherent 풪X-modules that vanishes on I-torsion modules and that induces for every B-graded S-module F a surjection ΞF from the set of I-saturated graded sub-S-modules of F onto the set of quasicoherent sub-풪X-modules of F˜. If Σ is simplicial, the above data can be chosen such that •˜ vanishes precisely on I-torsion modules and that ΞF is bijective for every F. In case R is noetherian, a toric version of the Serre–Grothendieck correspondence is proven, relating sheaf cohomology on X with B-graded local cohomology with support in I.