Abstract
AbstractSkew-gentle algebras are a generalisation of the well-known class of gentle algebras with which they share many common properties. In this work, using non-commutative Gröbner basis theory, we show that these algebras are strong Koszul and that the Koszul dual is again skew-gentle. We give a geometric model of their bounded derived categories in terms of polygonal dissections of surfaces with orbifold points, establishing a correspondence between curves in the orbifold and indecomposable objects. Moreover, we show that the orbifold dissections encode homological properties of skew-gentle algebras such as their singularity categories, their Gorenstein dimensions and derived invariants such as the determinant of theirq-Cartan matrices.
Highlights
Derived categories play an important role in many branches of mathematics such as algebraic geometry and representation theory, where they provide the proper setting for tilting theory [9, 10, 25].In general, giving a concrete description of the derived category of a finite dimensional algebra is not easy to achieve
In this work, using non-commutative Gröbner basis theory, we show that these algebras are strong Koszul and that the Koszul dual is again skew-gentle
The derived categories of gentle algebras have been gaining relevance in several branches of mathematics; for example, recently these categories have been linked to homological mirror symmetry, a homological framework developed by Kontsevich [29] to explain the similarities between the symplectic geometry of the so-called A-model, and the algebraic geometry of the so-called B-model of certain Calabi-Yau manifolds
Summary
Derived categories play an important role in many branches of mathematics such as algebraic geometry and representation theory, where they provide the proper setting for tilting theory [9, 10, 25]. We realise the indecomposable objects of the bounded derived category of a skew-gentle algebra as curves on a surface with orbifold points of order two. According to the classification in [5], indecomposable objects in the bounded derived category of a skew-gentle algebra fall within two classes, the so-called string objects and band objects, the latter coming in infinite families. Using this fact, we show the following. A is a strong Koszul algebra and its Koszul dual A! is (locally) skew-gentle, and A and A! give rise to dual orbifold dissections on the same surface with orbifold points
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