For any gentle algebra, the BD-gentle algebra and the CM-Auslander algebra have the same representation-finiteness and derived-discreteness as the original algebra, and this remains true under repeated applications of either construction.
On the derived categories of gentle and skew-gentle algebras: homological algebra and matrix problems
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In this paper, we investigate properties of the bounded derived category of finite dimensional modules over a gentle or skew-gentle algebra. We show that the Rouquier dimension of the derived category of such an algebra is at most one. Using this result, we prove that the Rouquier dimension of an arbitrary tame projective curve is equal to one, too. Finally, we elaborate the classification of indecomposable objects of the (possibly unbounded) homotopy category of projective modules of a gentle algebra.
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On BD-algebra and CM-Auslander algebra for a gentle algebra and their representation types
For any gentle algebra, the BD-gentle algebra and the CM-Auslander algebra have the same representation-finiteness and derived-discreteness as the original algebra, and this remains true under repeated applications of either construction.