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
which has an indecomposable object with precisely
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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