Quasicoherent sheaves on toric schemes

TL;DR

This paper develops a framework for understanding quasicoherent sheaves on toric schemes using graded algebraic structures, establishing a correspondence with local cohomology, especially when the fan is simplicial or the base ring is noetherian.

Contribution

It constructs a functorial correspondence between graded modules and quasicoherent sheaves on toric schemes, extending Serre-Grothendieck correspondence to the toric setting.

Findings

Established a functor from graded modules to quasicoherent sheaves

Proved bijectivity of the correspondence for simplicial fans

Derived a toric version of Serre-Grothendieck correspondence

Abstract

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 O_X-modules that vanishes on I-torsion modules and that induces for every B-graded S-module F a surjection Xi_F from the set of I-saturated graded sub-S-modules of F onto the set of quasicoherent sub-O_X-modules of ~F. If Sigma is simplicial, the above data can be chosen such that ~ vanishes precisely on I-torsion modules and that Xi_F 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.