Exceptional sequences in the derived category of a gentle algebra

TL;DR

This paper explores the structure of full exceptional sequences in the derived category of gentle algebras, linking their existence and properties to surface topology and braid group actions, with implications for algebraic and geometric representation theory.

Contribution

It characterizes when full exceptional sequences exist in the derived category of gentle algebras based on surface topology and describes their symmetry and transitivity properties.

Findings

Full exceptional sequences exist only for surfaces without punctures and with at least two boundary points.

The action of the braid group is transitive on full exceptional sequences for genus zero surfaces.

Transitivity in higher genus surfaces depends on specific sequences of exceptional object pairs.

Abstract

In this paper, using the correspondence of gentle algebras and dissections of marked surfaces, we study full exceptional sequences in the perfect derived category $\mathsf{K^b(A)}$ of a gentle algebra $\mathsf{A}$. We show that full exceptional sequences in $\mathsf{K^b(A)}$ exist if and only if the associated marked surface has no punctures and has at least two marked points on the boundary. Furthermore, by using induction on cuts of surfaces, we characterise when an exceptional sequence can be completed to a full exceptional sequence. If the genus of the associated surface is zero then we show that the action of the braid group together with the grading shift on full exceptional sequences in $\mathsf{K^b(A)}$ is transitive. For the case of surfaces of higher genus, we reduce the problem of transitivity to the problem of the existence of certain sequences of pairs of exceptional objects. Finally, we interpret the duality of a full exceptional sequence induced by the longest element in the associated symmetric group using Koszul duality.