Abstract

In this article we describe the triangulated structure of the bounded derived category of a gentle algebra by describing the triangles induced by the morphisms between indecomposable objects in a basis of their Hom-space.

Highlights

  • Derived categories provide a common framework for homological algebra in subjects such as algebra, geometry and mathematical physics

  • In mathematical physics, in the context of homological mirror symmetry, they are the natural setting for Bridgeland’s stability conditions [11]. They arise in the study of non-commutative crepant resolutions which are often studied via an algebra whose derived category is equivalent to the derived category of the smooth variety resolving the singularity [47]

  • In representation theory, derived categories are the natural setting for tilting theory of finite dimensional algebras, see for example [34]

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Summary

Introduction

Derived categories provide a common framework for homological algebra in subjects such as algebra, geometry and mathematical physics. In mathematical physics, in the context of homological mirror symmetry, they are the natural setting for Bridgeland’s stability conditions [11] In algebraic geometry, they arise in the study of non-commutative crepant resolutions which are often studied via an algebra whose derived category is equivalent to the derived category of the smooth variety resolving the singularity [47]. Gentle algebras first arose in the setting of tilting theory in the classification of iterated tilted algebras of type A and type A in [5] and [7] respectively They play an important role in many areas of mathematics: in algebra, they occur in cluster theory as Jacobian algebras associated to surface triangulations [6, 28, 39], and in recent advances in invariant theory [22]. Throughout this article, all modules will be left modules and all maps will be composed from left to right

Background
Mapping cones of graph maps
Mapping cones of single maps
Mapping cones of double maps
Mapping cones and quasi-graph maps
Examples
Full Text
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