Indecomposable objects in the derived category of a skew-gentle algebra using orbifolds

TL;DR

This paper provides a topological classification of indecomposable objects in the derived category of skew-gentle algebras using orbifold surfaces, extending previous work on gentle algebras.

Contribution

It offers a complete topological description of indecomposable objects in skew-gentle algebras' derived categories, avoiding combinatorial methods.

Findings

Complete classification of indecomposable objects

Use of orbifold surface curves for description

Avoidance of combinatorial string and band methods

Abstract

Skew-gentle algebras are skew-group algebras of certain gentle algebras endowed with a Z 2-action. Using the topological description of Opper, Plamondon and Schroll in [OPS] for the indecomposable objects of the derived category of any gentle algebra, one obtains here a complete description of indecomposable objects in the derived category of any skew-gentle algebras in terms of curves on an orbifold surface.The results presented here are complementary to the ones in [LSV]. First, we obtain a complete classification of indecomposable objects and not of ''homotopy strings'' and ''homotopy bands'' which are not always indecomposable. Second, the classification obtained here does not use the combinatorial description of [BMM03], but topological arguments coming from the double cover of the orbifold surface constructed in [AB].