Indecomposable objects determined by their index in higher homological algebra

Reid, Joseph

doi:10.1090/proc/14924

Indecomposable objects determined by their index in higher homological algebra
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Proc. Amer. Math. Soc. 148 (2020), 2331-2343 Request permission

Abstract:

Let $\mathscr {C}$ be a 2-Calabi-Yau triangulated category, and let $\mathscr {T}$ be a cluster tilting subcategory of $\mathscr {C}$. An important result from Dehy and Keller tells us that a rigid object $c \in \mathscr {C}$ is uniquely defined by its index with respect to $\mathscr {T}$.

The notion of triangulated categories extends to the notion of $(d+2)$-angulated categories. Thanks to a paper by Oppermann and Thomas, we now have a definition for cluster tilting subcategories in higher dimensions. This paper proves that under a technical assumption, an indecomposable object in a $(d+2)$-angulated category is uniquely defined by its index with respect to a higher dimensional cluster tilting subcategory. We also demonstrate that this result applies to higher dimensional cluster categories of Dynkin-type $A$.

References

Maurice Auslander and Idun Reiten, Modules determined by their composition factors, Illinois J. Math. 29 (1985), no. 2, 280–301. MR 784524
Maurice Auslander, Idun Reiten, and Sverre O. Smalø, Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics, vol. 36, Cambridge University Press, Cambridge, 1997. Corrected reprint of the 1995 original. MR 1476671
P. Caldero, F. Chapoton, and R. Schiffler, Quivers with relations arising from clusters ($A_n$ case), Trans. Amer. Math. Soc. 358 (2006), no. 3, 1347–1364. MR 2187656, DOI 10.1090/S0002-9947-05-03753-0
Raika Dehy and Bernhard Keller, On the combinatorics of rigid objects in 2-Calabi-Yau categories, Int. Math. Res. Not. IMRN 11 (2008), Art. ID rnn029, 17. MR 2428855, DOI 10.1093/imrn/rnn029
Christof Geiss, Bernhard Keller, and Steffen Oppermann, $n$-angulated categories, J. Reine Angew. Math. 675 (2013), 101–120. MR 3021448, DOI 10.1515/crelle.2011.177
Osamu Iyama, Higher-dimensional Auslander-Reiten theory on maximal orthogonal subcategories, Adv. Math. 210 (2007), no. 1, 22–50. MR 2298819, DOI 10.1016/j.aim.2006.06.002
Karin M. Jacobsen and Peter Jørgensen, $d$-abelian quotients of $(d+2)$-angulated categories, J. Algebra 521 (2019), 114–136. MR 3884695, DOI 10.1016/j.jalgebra.2018.11.019
Gustavo Jasso, $n$-abelian and $n$-exact categories, Math. Z. 283 (2016), no. 3-4, 703–759. MR 3519980, DOI 10.1007/s00209-016-1619-8
P. Jørgensen, Tropical friezes and the index in higher homological algebra, preprint, arXiv:1806.02108 (2018).
Steffen Koenig and Bin Zhu, From triangulated categories to abelian categories: cluster tilting in a general framework, Math. Z. 258 (2008), no. 1, 143–160. MR 2350040, DOI 10.1007/s00209-007-0165-9
Steffen Oppermann and Hugh Thomas, Higher-dimensional cluster combinatorics and representation theory, J. Eur. Math. Soc. (JEMS) 14 (2012), no. 6, 1679–1737. MR 2984586, DOI 10.4171/JEMS/345
Yann Palu, Cluster characters for 2-Calabi-Yau triangulated categories, Ann. Inst. Fourier (Grenoble) 58 (2008), no. 6, 2221–2248 (English, with English and French summaries). MR 2473635