Indecomposables in derived categories of gentle algebras
The authors parameterize the indecomposable objects of the bounded derived category of modules over a gentle algebra. Gentle algebras were introduced by Assem and Skowroński as a class of tame algebras by some restricted conditions on the quiver and the relations defining the algebra. The authors identify the problem of determining the indecomposable objects with a matrix problem presented by Bondarenko. Though the result of the classification is not really complicated it is still too technical to be stated here. As an application the authors characterize when such a gentle algebra is derived discrete (as defined by Vossieck) or derived finite (meaning that up to shift there are only finitely many isomorphism classes of indecomposable objects). They also obtain as a corollary that gentle algebras are derived tame. That the bounded derived category of a gentle algebra is tame is usually shown using the repetitive algebra and Happel's embedding. Recently Igor Burban showed in his 2003 thesis that also the right bounded derived category is tame.
scientific article; zbMATH DE number 1993342