On toric schemes

TL;DR

This paper explores the extension of toric varieties to schemes over arbitrary base rings, analyzing how the base affects geometry and describing sheaves via graded modules, with a focus on cohomology relations.

Contribution

It generalizes results by Cox and Mustata to describe quasicoherent sheaves on toric schemes over arbitrary bases and establishes a toric Serre-Grothendieck correspondence.

Findings

Base ring influences the geometry of toric schemes.

Quasicoherent sheaves can be described using graded modules.

Cohomology of sheaves relates to local cohomology of graded modules.

Abstract

Studying toric varieties from a scheme-theoretical point of view leads to toric schemes, i.e. "toric varieties over arbitrary base rings". It is shown how the base ring affects the geometry of a toric scheme. Moreover, generalisations of results by Cox and Mustata allow to describe quasicoherent sheaves on toric schemes in terms of graded modules. Finally, a toric version of the Serre-Grothendieck correspondence relates cohomology of quasicoherent sheaves on toric schemes to local cohomology of graded modules.