Low dimensional orders of finite representation type

TL;DR

This paper classifies noncommutative surface singularities with finite representation type, connecting two existing classifications, and studies noncommutative plane curves of finite representation type, providing explicit AR-quivers.

Contribution

It links Artin's and Reiten-Van den Bergh's classifications of log terminal orders via group cohomology, and classifies finite type noncommutative plane curves with their AR-quivers.

Findings

Connected classifications via group cohomology and ramification data.

Extended canonical orders to log terminal orders.

Classified noncommutative plane curves of finite representation type.

Abstract

In this paper, we study noncommutative surface singularities arising from orders. The singularities we study are mild in the sense that they have finite representation type or, equivalently, are log terminal in the sense of the Mori minimal model program for orders [CI05]. These were classified independently by Artin (in terms of ramification data) and Reiten-Van den Bergh (in terms of their AR-quivers). The first main goal of this paper is to connect these two classifications, by going through the finite subgroups $G \subset \mathrm{GL}_2$, explicitly computing $H^2(G,k^*)$, and then matching these up with Artin's list of ramification data and Reiten-Van den Bergh's AR-quivers. This provides a semi-independent proof of their classifications and extends the study of canonical orders in [CHI09] to the case of log terminal orders. A secondary goal of this paper is to study noncommutative analogues of plane curves which arise as follows. Let $B = k_{\zeta} [[ x,y ]]$ be the skew power series ring where $\zeta$ is a root of unity, or more generally a terminal order over a complete local ring. We consider rings of the form $A = B/(f)$ where $f \in Z(B)$ which we interpret to be the ring of functions on a noncommutative plane curve. We classify those noncommutative plane curves which are of finite representation type and compute their AR-quivers.