Exceptional sequences in the derived category of a gentle algebra

Abstract

AbstractIn this paper, using the correspondence of gentle algebras and dissections of marked surfaces, we study full exceptional sequences in the perfect derived category $${{\,\mathrm{\mathsf {K^b(A)}}\,}}$$ K b ( A ) of a gentle algebra $${{\,\mathrm{\textsf{A}}\,}}$$ A . We show that full exceptional sequences in $${{\,\mathrm{\mathsf {K^b(A)}}\,}}$$ 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 characterize 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 $${{\,\mathrm{\mathsf {K^b(A)}}\,}}$$ 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.