Quasicoherent sheaves on toric schemes

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