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Abstract

This paper shows a new phenomenon in higher cluster tilting theory. For each positive integer d, we exhibit a triangulated category 𝖢 with the following properties.

On the one hand, the d-cluster tilting subcategories of 𝖢 have very simple mutation behaviour: Each indecomposable object has exactly d mutations. On the other hand, the weakly d-cluster tilting subcategories of 𝖢 which lack functorial finiteness can have much more complicated mutation behaviour: For each 0 ≤ ℓ ≤ d - 1, we show a weakly d-cluster tilting subcategory 𝖳 which has an indecomposable object with precisely ℓ mutations.

The category 𝖢 is the algebraic triangulated category generated by a (d + 1)-spherical object and can be thought of as a higher cluster category of Dynkin type A .

an n-angulated category for any ðn 2Þ-representation fi- nite algebra L. 6.3. n-angulated categories in Calabi–Yau categories. Assume T is a triangulated d-Calabi–Yau-category with Serre functor S. By our standard construction, an ðn 2Þ- cluster tilting subcategory FHT which is closed under Sn2 has a structure of an n-angulated category with suspension Sn :¼ Sn2. By the remarks in Section 4.2, we can conclude that d ¼ d 0ðn 2Þ for some integer d 0 A Z, and in particular Sd 0n ¼ S. By Remark 5.4, we conclude that modF is ðd þ 2d 0 1Þ-Calabi–Yau. By arguments in [29

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