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

Abstract

Skew-gentle algebras over a field of characteristic $\ne 2$ are skew-group algebras of certain gentle algebras endowed with a ${\mathbb{Z}}_2$-action. Using the topological description of Opper, Plamondon and Schroll (2018) 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 algebra in terms of curves on an orbifold surface. The results presented here are complementary to the ones by Labardini-Fragoso, Schroll and Valdivieso (2022). 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 Bekkert, Marcos and Merklen (2003), but topological arguments coming from the double cover of the orbifold surface constructed by Amiot and Brüstle (2022).