Derived categories of skew-gentle algebras and orbifolds

TL;DR

This paper studies skew-gentle algebras, showing they are Koszul, their duals are skew-gentle, and providing a geometric model of their derived categories via orbifold surface dissections, linking algebraic and geometric properties.

Contribution

It introduces a geometric model for the derived categories of skew-gentle algebras using orbifold surface dissections, connecting algebraic invariants with geometric structures.

Findings

Skew-gentle algebras are Koszul and their duals are skew-gentle.

Orbifold dissections encode homological properties like singularity categories.

The geometric model relates algebraic invariants to orbifold surface structures.

Abstract

Skew-gentle algebras are a generalisation of the well-known class of gentle algebras with which they share many common properties. In this work, using non-commutative Gr\"obner basis theory, we show that these algebras are Koszul and that the Koszul dual is again skew-gentle. We give a geometric model of their bounded derived categories in terms of polygonal dissections of orbifold surfaces establishing a correspondence between curves in the orbifold and indecomposable objects. Moreover, we show that the orbifold dissections encode homological properties of skew-gentle algebras such as their singularity categories, their Gorenstein dimensions and derived invariants such as the determinant of their q-Cartan matrices.