The Explicit Construction of Orders on Surfaces

TL;DR

This paper develops explicit methods for constructing noncommutative orders on algebraic surfaces, including Calabi-Yau and elliptically fibred surfaces, expanding the toolkit for noncommutative algebraic geometry.

Contribution

It introduces a noncommutative cyclic covering trick and applies Ogg-Shafarevich theory to explicitly build various noncommutative orders on different types of surfaces.

Findings

Constructed numerous numerically Calabi-Yau orders on surfaces.

Explicitly built maximal orders on rational and ruled surfaces.

Used Ogg-Shafarevich theory to create Azumaya algebras on elliptic surfaces.

Abstract

We implement a noncommutative analogue of the well-known commutative cyclic covering trick and implement it to explicitly construct a vast collection of numerically Calabi-Yau orders, noncommutative analogues of surfaces of Kodaira dimension 0. We construct maximal orders, noncommutative analogues of normal schemes, on rational surfaces and ruled surfaces. We also use Ogg-Shafarevich theory to construct Azumaya algebras and, more generally, maximal orders on elliptically fibred surfaces.