Quasicoherent sheaves on toric schemes

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Let X be the toric scheme over a ring R associated with a fan Sigma. It is shown that there are a group B, a B-graded R-algebra S and a graded ideal I of S such that there is an essentially surjective, exact functor from the category of B-graded S-modules to the category of quasicoherent Oₓ-modules that vanishes on I-torsion modules and that induces for every B-graded S-module F a surjection XiF from the set of I-saturated graded sub-S-modules of F onto the set of quasicoherent sub-Oₓ-modules of F. If Sigma is simplicial, the above data can be chosen such that vanishes precisely on I-torsion modules and that XiF 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.