Dimension theory of noncommutative curves

TL;DR

This paper computes various dimensions of derived categories of orbifold curves, completing the noncommutative curve dimension theory, and constructs stability conditions, providing new insights into their structure and properties.

Contribution

It advances the understanding of noncommutative curves by calculating their dimensions and constructing stability conditions, including a characterization of orbifold curves with hereditary tilting bundles.

Findings

Computed several types of dimension for derived categories of orbifold curves.

Constructed stability conditions for orbifold curves.

Characterized orbifold curves with hereditary tilting bundles via diagonal dimension.

Abstract

We compute several types of dimension for the bounded derived categories of coherent sheaves of orbifold curves. This completes the calculation of these dimensions for derived categories of noncommutative curves in the sense of Reiten-van den Bergh. Along the way we construct stability conditions for orbifold curves. We also obtain a characterisation of orbifold curves with hereditary tilting bundle in terms of diagonal dimension.