On the structure of triangulated categories with finitely many indecomposables

Claire Amiot

Bulletin de la Société Mathématique de France (2007)

  • Volume: 135, Issue: 3, page 435-474
  • ISSN: 0037-9484

Abstract

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We study the problem of classifying triangulated categories with finite-dimensional morphism spaces and finitely many indecomposables over an algebraically closed field k . We obtain a new proof of the following result due to Xiao and Zhu: the Auslander-Reiten quiver of such a category 𝒯 is of the form ℤ Δ / G where Δ is a disjoint union of simply-laced Dynkin diagrams and G a weakly admissible group of automorphisms of ℤ Δ . Then we prove that for ‘most’ groups G , the category 𝒯 is standard,i.e. k -linearly equivalent to an orbit category 𝒟 b ( m o d k Δ ) / Φ . This happens in particular when 𝒯 is maximal d -Calabi-Yau with d ≥ 2 . Moreover, if 𝒯 is standard and algebraic, we can even construct a triangle equivalence between 𝒯 and the corresponding orbit category. Finally we give a sufficient condition for the category of projectives of a Frobenius category to be triangulated. This allows us to construct non standard 1 -Calabi-Yau categories using deformed preprojective algebras of generalized Dynkin type.

How to cite

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Amiot, Claire. "On the structure of triangulated categories with finitely many indecomposables." Bulletin de la Société Mathématique de France 135.3 (2007): 435-474. <http://eudml.org/doc/272357>.

@article{Amiot2007,
abstract = {We study the problem of classifying triangulated categories with finite-dimensional morphism spaces and finitely many indecomposables over an algebraically closed field $k$. We obtain a new proof of the following result due to Xiao and Zhu: the Auslander-Reiten quiver of such a category $\mathcal \{T\}$ is of the form $\mathbb \{Z\}\Delta /G$ where $\Delta $ is a disjoint union of simply-laced Dynkin diagrams and $G$ a weakly admissible group of automorphisms of $\mathbb \{Z\}\Delta $. Then we prove that for ‘most’ groups $G$, the category $\mathcal \{T\}$ is standard,i.e.$k$-linearly equivalent to an orbit category $\mathcal \{D\}^b(~mod \;k\Delta )/\Phi $. This happens in particular when $\mathcal \{T\}$ is maximal $d$-Calabi-Yau with $d\ge 2$. Moreover, if $\mathcal \{T\}$ is standard and algebraic, we can even construct a triangle equivalence between $\mathcal \{T\}$ and the corresponding orbit category. Finally we give a sufficient condition for the category of projectives of a Frobenius category to be triangulated. This allows us to construct non standard $1$-Calabi-Yau categories using deformed preprojective algebras of generalized Dynkin type.},
author = {Amiot, Claire},
journal = {Bulletin de la Société Mathématique de France},
keywords = {locally finite triangulated category; Calabi-Yau category; Dynkin diagram; Auslander-Reiten quiver; orbit category},
language = {eng},
number = {3},
pages = {435-474},
publisher = {Société mathématique de France},
title = {On the structure of triangulated categories with finitely many indecomposables},
url = {http://eudml.org/doc/272357},
volume = {135},
year = {2007},
}

TY - JOUR
AU - Amiot, Claire
TI - On the structure of triangulated categories with finitely many indecomposables
JO - Bulletin de la Société Mathématique de France
PY - 2007
PB - Société mathématique de France
VL - 135
IS - 3
SP - 435
EP - 474
AB - We study the problem of classifying triangulated categories with finite-dimensional morphism spaces and finitely many indecomposables over an algebraically closed field $k$. We obtain a new proof of the following result due to Xiao and Zhu: the Auslander-Reiten quiver of such a category $\mathcal {T}$ is of the form $\mathbb {Z}\Delta /G$ where $\Delta $ is a disjoint union of simply-laced Dynkin diagrams and $G$ a weakly admissible group of automorphisms of $\mathbb {Z}\Delta $. Then we prove that for ‘most’ groups $G$, the category $\mathcal {T}$ is standard,i.e.$k$-linearly equivalent to an orbit category $\mathcal {D}^b(~mod \;k\Delta )/\Phi $. This happens in particular when $\mathcal {T}$ is maximal $d$-Calabi-Yau with $d\ge 2$. Moreover, if $\mathcal {T}$ is standard and algebraic, we can even construct a triangle equivalence between $\mathcal {T}$ and the corresponding orbit category. Finally we give a sufficient condition for the category of projectives of a Frobenius category to be triangulated. This allows us to construct non standard $1$-Calabi-Yau categories using deformed preprojective algebras of generalized Dynkin type.
LA - eng
KW - locally finite triangulated category; Calabi-Yau category; Dynkin diagram; Auslander-Reiten quiver; orbit category
UR - http://eudml.org/doc/272357
ER -

References

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  1. [1] H. Asashiba – « The derived equivalence classification of representation-finite selfinjective algebras », J. Algebra214 (1999), p. 182–221. Zbl0949.16013MR1684880
  2. [2] M. Auslander – « Functors and morphisms determined by objects », in Representation theory of algebras (Proc. Conf., Temple Univ., Philadelphia, Pa., 1976), Lecture Notes in Pure Appl. Math., vol. 37, Dekker, 1978, p. 1–244. Lecture Notes in Pure Appl. Math., Vol. 37. Zbl0383.16015MR480688
  3. [3] —, « Isolated singularities and existence of almost split sequences », in Representation theory, II (Ottawa, Ont., 1984), Lecture Notes in Math., vol. 1178, Springer, 1986, p. 194–242. Zbl0633.13007MR842486
  4. [4] M. Auslander & I. Reiten – « McKay quivers and extended Dynkin diagrams », Trans. Amer. Math. Soc.293 (1986), p. 293–301. Zbl0588.20001MR814923
  5. [5] —, « Cohen-Macaulay and Gorenstein Artin algebras », in Representation theory of finite groups and finite-dimensional algebras (Bielefeld, 1991), Progr. Math., vol. 95, Birkhäuser, 1991, p. 221–245. Zbl0759.16007MR1112162
  6. [6] R. Bautista, P. Gabriel, A. V. Roĭter & L. Salmerón – « Representation-finite algebras and multiplicative bases », Invent. Math.81 (1985), p. 217–285. Zbl0575.16012MR799266
  7. [7] A. A. Beĭlinson, J. Bernstein & P. Deligne – « Faisceaux pervers », in Analysis and topology on singular spaces, I (Luminy, 1981), Astérisque, vol. 100, Soc. Math. France, 1982, p. 5–171. Zbl0536.14011MR751966
  8. [8] J. Białkowski, K. Erdmann & A. Skowroński – « Deformed preprojective algebras of generalized Dynkin type », Trans. Amer. Math. Soc.359 (2007), p. 2625–2650. Zbl1117.16005MR2286048
  9. [9] J. Białkowski & A. Skowroński – « Calabi-Yau stable module categories of finite type », preprint http://www.mat.uni.torun.pl/preprints/, 2006. Zbl1168.16003
  10. [10] —, « Nonstandard additively finite triangulated categories of Calabi-Yau dimension one in characteristic 3 », to appear in Algebra and Discrete Mathematics. Zbl1164.16006
  11. [11] A. B. Buan, R. Marsh, M. Reineke, I. Reiten & G. Todorov – « Tilting theory and cluster combinatorics », Adv. Math.204 (2006), p. 572–618. Zbl1127.16011MR2249625
  12. [12] P. Caldero, F. Chapoton & R. Schiffler – « Quivers with relations arising from clusters ( A n case) », Trans. Amer. Math. Soc.358 (2006), p. 1347–1364. Zbl1137.16020MR2187656
  13. [13] P. Caldero & B. Keller – « From triangulated categories to cluster algebras », preprint arXiv:math.RT/0506018, 2005. Zbl1141.18012MR2385670
  14. [14] E. Dieterich – « The Auslander-Reiten quiver of an isolated singularity », in Singularities, representation of algebras, and vector bundles (Lambrecht, 1985), Lecture Notes in Math., vol. 1273, Springer, 1987, p. 244–264. Zbl0632.14004MR915179
  15. [15] K. Erdmann & N. Snashall – « On Hochschild cohomology of preprojective algebras. I, II », J. Algebra 205 (1998), p. 391–412, 413–434. Zbl0937.16013MR1632808
  16. [16] —, « Preprojective algebras of Dynkin type, periodicity and the second Hochschild cohomology », in Algebras and modules, II (Geiranger, 1996), CMS Conf. Proc., vol. 24, Amer. Math. Soc., 1998, p. 183–193. Zbl1034.16501MR1648626
  17. [17] P. Gabriel & A. V. Roĭter – « Representations of finite-dimensional algebras », in Algebra, VIII, Encyclopaedia Math. Sci., vol. 73, Springer, 1992, With a chapter by B. Keller, p. 1–177. Zbl0839.16001MR1239447
  18. [18] C. Geiß, B. Leclerc & J. Schröer – « Auslander algebras and initial seeds for cluster algebras », preprint arXiv:math.RT/0506405, 2005. Zbl1135.17007MR2352732
  19. [19] —, « Partial flag varieties and preprojective algebras », arXiv:math.RT/0609138, 2006. 
  20. [20] C. Geiß, B. Leclerc & J. Schröer – « Rigid modules over preprojective algebras », Invent. Math.165 (2006), p. 589–632. Zbl1167.16009MR2242628
  21. [21] D. Happel – « On the derived category of a finite-dimensional algebra », Comment. Math. Helv.62 (1987), p. 339–389. Zbl0626.16008MR910167
  22. [22] —, Triangulated categories in the representation theory of finite-dimensional algebras, London Mathematical Society Lecture Note Series, vol. 119, Cambridge University Press, 1988. Zbl0635.16017MR935124
  23. [23] D. Happel, U. Preiser & C. M. Ringel – « Binary polyhedral groups and Euclidean diagrams », Manuscripta Math.31 (1980), p. 317–329. Zbl0436.20005MR576503
  24. [24] —, « Vinberg’s characterization of Dynkin diagrams using subadditive functions with application to D Tr -periodic modules », in Representation theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979), Lecture Notes in Math., vol. 832, Springer, 1980, p. 280–294. Zbl0446.16032MR607159
  25. [25] A. Heller – « Stable homotopy categories », Bull. Amer. Math. Soc.74 (1968), p. 28–63. Zbl0177.25605MR224090
  26. [26] T. Holm & P. Jørgensen – « Cluster categories and selfinjective algebras: type A », preprint arXiv:math.RT/0610728, 2006. 
  27. [27] —, « Cluster categories and selfinjective algebras: type D », preprint arXiv:math.RT/0612451, 2006. 
  28. [28] O. Iyama & Y. Yoshino – « Mutations in triangulated categories and rigid Cohen-Macaulay modules », preprint arXiv:math.RT/0607736, 2006. Zbl1140.18007MR2385669
  29. [29] B. Keller – « Derived categories and universal problems », Comm. Algebra19 (1991), p. 699–747. Zbl0722.18002MR1102982
  30. [30] —, « On triangulated orbit categories », Doc. Math.10 (2005), p. 551–581. Zbl1086.18006MR2184464
  31. [31] —, « On differential graded categories », preprint arXiv:math.KT/0601185, 2006. MR2275593
  32. [32] B. Keller & I. Reiten – « Cluster-tilted algebras are Gorenstein and stably Calabi-Yau », preprint arXiv:math.RT/0512471, 2005. Zbl1128.18007MR2313531
  33. [33] —, « Acyclic Calabi-Yau categories », preprint arXiv:math.RT/0610594, 2006. 
  34. [34] B. Keller & D. Vossieck – « Sous les catégories dérivées », C. R. Acad. Sci. Paris Sér. I Math.305 (1987), p. 225–228. Zbl0628.18003MR907948
  35. [35] R. Marsh, M. Reineke & A. Zelevinsky – « Generalized associahedra via quiver representations », Trans. Amer. Math. Soc.355 (2003), p. 4171–4186. Zbl1042.52007MR1990581
  36. [36] A. Neeman – Triangulated categories, Annals of Mathematics Studies, vol. 148, Princeton University Press, 2001. Zbl0974.18008MR1812507
  37. [37] Y. Palu – « On algebraic Calabi-Yau categories », Thèse, in preparation. Zbl0191.38001
  38. [38] I. Reiten & M. Van den Bergh – « Noetherian hereditary abelian categories satisfying Serre duality », J. Amer. Math. Soc.15 (2002), p. 295–366. Zbl0991.18009MR1887637
  39. [39] C. Riedtmann – « Algebren, Darstellungsköcher, Überlagerungen und zurück », Comment. Math. Helv.55 (1980), p. 199–224. Zbl0444.16018MR576602
  40. [40] —, « Representation-finite self-injective algebras of class A n », in Representation theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979), Lecture Notes in Math., vol. 832, Springer, 1980, p. 449–520. Zbl0455.16014MR607169
  41. [41] —, « Many algebras with the same Auslander-Reiten quiver », Bull. London Math. Soc.15 (1983), p. 43–47. Zbl0487.16021MR686347
  42. [42] —, « Representation-finite self-injective algebras of class D n », Compositio Math.49 (1983), p. 231–282. MR704393
  43. [43] C. M. Ringel – Tame algebras and integral quadratic forms, Lecture Notes in Mathematics, vol. 1099, Springer, 1984. Zbl0546.16013MR774589
  44. [44] —, « Hereditary triangulated categories », to appear in Comp. Math., 2006. 
  45. [45] G. Tabuada – « On the structure of Calabi-Yau categories with a cluster tilting subcategory », preprint arXiv:math.RT/0607394, 2006. Zbl1122.18007MR2302527
  46. [46] J.-L. Verdier – « Catégories dérivées. Quelques résultats », Lect. Notes Math.569 (1977), p. 262–311. Zbl0407.18008
  47. [47] —, « Des catégories dérivées des catégories abéliennes », Astérisque (1996), p. 253 pp. (1997), With a preface by Luc Illusie, Edited and with a note by Georges Maltsiniotis. Zbl0882.18010MR1453167
  48. [48] J. Xiao & B. Zhu – « Relations for the Grothendieck groups of triangulated categories », J. Algebra257 (2002), p. 37–50. Zbl1021.18004MR1942270
  49. [49] —, « Locally finite triangulated categories », J. Algebra290 (2005), p. 473–490. Zbl1110.16013MR2153264
  50. [50] Y. Yoshino – Cohen-Macaulay modules over Cohen-Macaulay rings, London Mathematical Society Lecture Note Series, vol. 146, Cambridge University Press, 1990. Zbl0745.13003MR1079937

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