Terminal orders on arithmetic surfaces

Chan, Daniel; Ingalls, Colin

doi:10.2140/ant.2024.18.2027

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1944-7833 (online)

ISSN 1937-0652 (print)

Author index

To appear

Other MSP journals

Terminal orders on arithmetic surfaces

Daniel Chan and Colin Ingalls

Vol. 18 (2024), No. 11, 2027–2046

DOI: 10.2140/ant.2024.18.2027

Abstract

The local structure of terminal Brauer classes on arithmetic surfaces was classified (2021), generalising the classification on geometric surfaces (2005). 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. We give étale local structure theorems for terminal orders on arithmetic 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.

Keywords

orders, arithmetic surfaces, minimal model program

Mathematical Subject Classification

Primary: 16H10, 16S38

Milestones

Received: 24 November 2022

Revised: 5 July 2023

Accepted: 27 November 2023

Published: 18 October 2024

Authors

Daniel Chan
School of Mathematics and Statistics University of New South Wales Sydney Sydney Australia

Colin Ingalls
School of Mathematics and Statistics Carleton University Ottawa, ON Canada

Open Access made possible by participating institutions via Subscribe to Open.

Vol. 18, No. 11, 2024