Terminal orders on arithmetic surfaces

TL;DR

This paper extends the classification of terminal Brauer classes to arithmetic surfaces, providing explicit local structure theorems for prime degree p > 5, and shows these orders have global dimension two, generalizing geometric results.

Contribution

It offers explicit etale local structure theorems for terminal orders on arithmetic surfaces of prime degree p > 5, extending known geometric classifications.

Findings

All terminal orders can be explicitly constructed as matrix algebras over symbols.

Such orders have global dimension two, indicating homological regularity.

The results generalize the structure theorem from geometric to arithmetic surfaces.

Abstract

The local structure of terminal Brauer classes on arithmetic surfaces were classified in [CI21] generalising the classification on geometric surfaces carried out in [CI05]. Part of the interest in these classifications is that it enables the minimal model program to be applied to the noncommutative setting of orders on surfaces. In this paper, we give etale local structure theorems for terminal orders on arithemtic surfaces, at least when the degree is a prime p >5. This generalises the structure theorem given in the geometric case. They can all be explicitly constructed as algebras of matrices over symbols. From this description one sees that such terminal orders all have global dimension two, thus generalising the fact that terminal (commutative) surfaces are smooth and hence homologically regular.