Purely Algebraic Method to Construct Toric Schemes

TL;DR

This paper introduces an algebraic approach to constructing toric schemes over arbitrary rings from fans, analyzing their geometric properties and morphisms, and establishing criteria for separation, properness, and regularity.

Contribution

It provides a purely algebraic method for constructing and analyzing toric schemes over arbitrary rings, including criteria for their geometric properties.

Findings

Constructs toric schemes over arbitrary rings from fans.

Proves the scheme is separated and characterizes when it is proper.

Studies regularity, logarithmic regularity, and morphisms of toric schemes.

Abstract

In this article, we first give some elementary proprieties of monoids and fans, then construct a toric scheme over an arbitrary ring, from a given fan. Using Valuative Criterion, we prove that this scheme is separated and give the sufficient and necessary condition when it is proper. We also study the regularity and logarithmic regularity of it. Finally we study the morphisms of toric schemes induced by the homomorphisms of fans.