Global dimensions of local geodesic ghor algebras

TL;DR

This paper investigates the global dimension of cyclic localizations of geodesic ghor algebras on surfaces, establishing bounds related to the genus and the center's Krull dimension, with conditions for equality.

Contribution

It provides a precise upper bound for the global dimension of localizations of geodesic ghor algebras on higher genus surfaces and characterizes when this bound is achieved.

Findings

Global dimension is bounded above by 2g+1 for genus g surfaces.

The bound matches the Krull dimension of the algebra's center.

Equality holds over the noetherian locus of the center.

Abstract

A ghor algebra is a path algebra with relations of a dimer quiver in a compact surface. We show that the global dimension of any cyclic localization of a geodesic ghor algebra on a genus $g \geq 1$ surface is bounded above by $2g+1$.This number coincides with the Krull dimension of the center of the ghor algebra. We further show that the bound is an equality if and only if the point of localization sits over the noetherian locus of the center.