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By Arnold, John Charles

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Amiot, On the structure of triangulated categories with finitely many indecomposables. Bull. Soc. Math. France 135 (2007), 435–474. [2] C. Amiot, Sur les petites catégories triangulées. D. thesis (2008). pdf [3] C. Amiot, Cluster categories for algebras of global dimension 2 and quivers with potential. Ann. Inst. Fourier 59 (2009), 2525–2590. Amiot, O. Iyama and I. Reiten, Stable categories of Cohen-Macaulay modules and cluster categories. In preparation. [5] C. Amiot, O. Iyama, I. Reiten and G.

Then the algebra ƒ is derived equivalent to kQ. 14 can also be used to describe algebras of a given cluster type and to classify these algebras up to derived equivalence (see example below). 16. As shown in [8], there are 11 algebras of cluster type Az2;2 (up to isomorphism). These algebras belong to two derived equivalence classes. In the first one we find the 8 piecewise hereditary algebras: . . . . . . . . . . . . . . . . 45 On generalized cluster categories The second equivalence class contains the 3 algebras which are not piecewise hereditary: .

6, one deduces that the following algebras of global dimension at most 2 (presented by a quiver with relations) are in the same derived equivalence class. In this picture, a dotted line between two vertices means that a path between these two vertices is zero: 2 1 2 3 1 2 3 1 3. 3 Application to cluster equivalence. In this subsection, we describe another application of graded mutation to the notion of cluster equivalence. 8. ƒ0 / between their cluster categories. 10 below). This notion is more general than derived equivalence: two derived equivalent algebras of global dimension at most two are cluster equivalent, the converse is not true, as shown in the following example.

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