A geometric model for the derived category of gentle algebras

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.

v7: Paper rewritten to apply to the `bounded' derived category of any finite dimensional graded gentle algebra. The numberings of sections and results of older versions have been kept identical. Appendices containing new material for the graded case have been added as well as recent references

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• Cambrian combinatorics on quiver representations (type A)
• On auto-equivalences and complete derived invariants of gentle algebras
• On extensions for gentle algebras
• Derived equivalence classification of Brauer graph algebras
• Pairwise Compatibility for 2-Simple Minded Collections
• Schemes of modules over gentle algebras and laminations of surfaces
• Tame algebras have dense $\mathbf{g}$-vector fans
• Graded decorated marked surfaces: Calabi-Yau-$\mathbb{X}$ categories of gentle algebras
• A geometric model for the module category of a skew-gentle algebra
• Contractible flow of stability conditions via global dimension function
• Higher gentle algebras
• A geometric realization of silting theory for gentle algebras
• Jordan recoverability of some subcategories of modules over gentle algebras
• Calabi-Yau properties of ribbon graph orders
• A multiplication formula for cluster characters in gentle algebras
• Generalized Kauer moves and derived equivalences of Brauer graph algebras
• A geometric realization of socle-projective categories for posets of type $\mathbb{A}$