A geometric model for the derived category of gentle algebras

TL;DR

This paper introduces a geometric model for the derived category of graded gentle algebras using ribbon graphs and surfaces, linking algebraic objects to curves and intersections to morphisms.

Contribution

It constructs a geometric framework for the derived categories of gentle algebras, connecting curves on surfaces to indecomposable objects and morphisms.

Findings

Homotopy classes of curves correspond to indecomposable objects.

Curve intersections represent morphisms.

Surface encodes derived invariants.

Abstract

In this paper we construct a geometric model for the triangulated category generated by the simple modules of any graded gentle algebra. This leads to a geometric model of their perfect derived categories and by a recent paper of Booth, Goodbody and the first author also of their derived categories of objects with finite-dimensional cohomology. The construction is based on the ribbon graph associated to a gentle algebra in the work of the third author, and is linked to partially wrapped Fukaya categories by the work of Haiden, Katzarkov and Kontsevich and to derived categories of coherent sheaves on nodal stacky curves by the work of Lekili and Polishchuk. The ribbon graph gives rise to an oriented surface with boundary and marked points in the boundary. We show that the homotopy classes of curves connecting marked points and of closed curves are in bijection with the isomorphism classes of indecomposable objects in the derived category of the graded gentle algebra. Intersections of curves correspond to morphisms and resolving the crossings of curves gives rise to mapping cones. The Auslander-Reiten translate corresponds to rotating endpoints of curves along the boundary. Furthermore, we show that the surface encodes the derived invariant of Avella-Alaminos and Geiss.