A geometric model for the derived category of gentle algebras

By Sebastian Opper, Pierre-Guy Plamondon, Sibylle Schroll. 2018

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In this paper we construct a geometric model for the bounded derived category of a gentle algebra. The construction is based on the ribbon graph associated to a gentle algebra by the third author. This 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 bounded derived category of the gentle algebra up to the action of the shift functor. 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.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/5e46c58dd56661585148b21a1f3799b62a782214

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Citations in the corpus

A complete derived invariant for gentle algebras via winding numbers and Arf invariants (2019, methodology): “ows i → j and j → k is a relation. Example 2.2. Below is the quiver with relations of the ◦-dissection of Example 1.10. 1 5 6 4 7 3 2 8 The dotted lines in the ﬁgure represent relations. Theorem 2.3 ([OPS18][PPP18]). The assignment (S,M,P),∆) → A(∆) deﬁnes a bijection from the set of homeomorphism c”