Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Numerically finite hereditary categories with Serre duality


Author: Adam-Christiaan van Roosmalen
Journal: Trans. Amer. Math. Soc. 368 (2016), 7189-7238
MSC (2010): Primary 18E10, 18E30, 18G20; Secondary 16G20, 14F05
DOI: https://doi.org/10.1090/tran/6569
Published electronically: February 11, 2016
MathSciNet review: 3471089
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \mathcal {A}$ be an abelian hereditary category with Serre duality. We provide a classification of such categories up to derived equivalence under the additional condition that the Grothendieck group modulo the radical of the Euler form is a free abelian group of finite rank. Such categories are called numerically finite and this condition is satisfied by the category of coherent sheaves on a smooth projective variety.


References [Enhancements On Off] (What's this?)

  • [1] Ibrahim Assem, María José Souto Salorio, and Sonia Trepode, Ext-projectives in suspended subcategories, J. Pure Appl. Algebra 212 (2008), no. 2, 423-434. MR 2357343 (2009d:18016), https://doi.org/10.1016/j.jpaa.2007.06.002
  • [2] M. F. Atiyah, Vector bundles over an elliptic curve, Proc. London Math. Soc. (3) 7 (1957), 414-452. MR 0131423 (24 #A1274)
  • [3] Maurice Auslander and Idun Reiten, Stable equivalence of Artin algebras, Proceedings of the Conference on Orders, Group Rings and Related Topics (Ohio State Univ., Columbus, Ohio, 1972) Springer, Berlin, 1973, pp. 8-71. Lecture Notes in Math., Vol. 353. MR 0335575 (49 #356)
  • [4] Maurice Auslander, Idun Reiten, and Sverre O. Smalø, Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics, vol. 36, Cambridge University Press, Cambridge, 1995. MR 1314422 (96c:16015)
  • [5] Paul Balmer and Marco Schlichting, Idempotent completion of triangulated categories, J. Algebra 236 (2001), no. 2, 819-834. MR 1813503 (2002a:18013), https://doi.org/10.1006/jabr.2000.8529
  • [6] A. A. Beĭlinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981) Astérisque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5-171 (French). MR 751966 (86g:32015)
  • [7] Carl Fredrik Berg and Adam-Christiaan van Roosmalen, Hereditary categories with Serre duality which are generated by preprojectives, J. Algebra 335 (2011), 220-257. MR 2792575 (2012e:18018), https://doi.org/10.1016/j.jalgebra.2010.11.021
  • [8] Carl Fredrik Berg and Adam-Christiaan van Roosmalen, Representations of thread quivers, Proc. Lond. Math. Soc. (3) 108 (2014), no. 2, 253-290. MR 3166353, https://doi.org/10.1112/plms/pdt021
  • [9] A. I. Bondal, Representations of associative algebras and coherent sheaves, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 1, 25-44 (Russian); English transl., Math. USSR-Izv. 34 (1990), no. 1, 23-42. MR 992977 (90i:14017)
  • [10] A. I. Bondal and M. M. Kapranov, Representable functors, Serre functors, and reconstructions, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 6, 1183-1205, 1337 (Russian); English transl., Math. USSR-Izv. 35 (1990), no. 3, 519-541. MR 1039961 (91b:14013)
  • [11] Tom Bridgeland, Equivalences of triangulated categories and Fourier-Mukai transforms, Bull. London Math. Soc. 31 (1999), no. 1, 25-34. MR 1651025 (99k:18014), https://doi.org/10.1112/S0024609398004998
  • [12] Tom Bridgeland, Stability conditions on triangulated categories, Ann. of Math. (2) 166 (2007), no. 2, 317-345. MR 2373143 (2009c:14026), https://doi.org/10.4007/annals.2007.166.317
  • [13] Tom Bridgeland, Alastair King, and Miles Reid, The McKay correspondence as an equivalence of derived categories, J. Amer. Math. Soc. 14 (2001), no. 3, 535-554 (electronic). MR 1824990 (2002f:14023), https://doi.org/10.1090/S0894-0347-01-00368-X
  • [14] Kristian Brüning and Igor Burban, Coherent sheaves on an elliptic curve, Interactions between homotopy theory and algebra, Contemp. Math., vol. 436, Amer. Math. Soc., Providence, RI, 2007, pp. 297-315. MR 2355779 (2008k:14039), https://doi.org/10.1090/conm/436/08414
  • [15] Ī. Ī. Burban and Ī. M. Burban, Twist functors and D-branes, Ukraïn. Mat. Zh. 57 (2005), no. 1, 18-31 (Ukrainian, with English and Ukrainian summaries); English transl., Ukrainian Math. J. 57 (2005), no. 1, 18-34. MR 2190952 (2006k:14023), https://doi.org/10.1007/s11253-005-0169-8
  • [16] Xiao-Wu Chen, Yu Ye, and Pu Zhang, Algebras of derived dimension zero, Comm. Algebra 36 (2008), no. 1, 1-10. MR 2378361 (2009a:16020), https://doi.org/10.1080/00927870701649184
  • [17] Claude Cibils and Pu Zhang, Calabi-Yau objects in triangulated categories, Trans. Amer. Math. Soc. 361 (2009), no. 12, 6501-6519. MR 2538602 (2010h:18022), https://doi.org/10.1090/S0002-9947-09-04682-0
  • [18] William Crawley-Boevey, Exceptional sequences of representations of quivers [ MR1206935 (94c:16017)], Representations of algebras (Ottawa, ON, 1992) CMS Conf. Proc., vol. 14, Amer. Math. Soc., Providence, RI, 1993, pp. 117-124. MR 1265279
  • [19] Pierre Gabriel, Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323-448 (French). MR 0232821 (38 #1144)
  • [20] Werner Geigle and Helmut Lenzing, A class of weighted projective curves arising in representation theory of finite-dimensional algebras, Singularities, representation of algebras, and vector bundles (Lambrecht, 1985) Lecture Notes in Math., vol. 1273, Springer, Berlin, 1987, pp. 265-297. MR 915180 (89b:14049), https://doi.org/10.1007/BFb0078849
  • [21] Werner Geigle and Helmut Lenzing, Perpendicular categories with applications to representations and sheaves, J. Algebra 144 (1991), no. 2, 273-343. MR 1140607 (93b:16011), https://doi.org/10.1016/0021-8693(91)90107-J
  • [22] Sergei I. Gelfand and Yuri I. Manin, Methods of homological algebra, 2nd ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. MR 1950475 (2003m:18001)
  • [23] A. L. Gorodentsev and A. N. Rudakov, Exceptional vector bundles on projective spaces, Duke Math. J. 54 (1987), no. 1, 115-130. MR 885779 (88e:14018), https://doi.org/10.1215/S0012-7094-87-05409-3
  • [24] Dieter Happel, Triangulated categories in the representation theory of finite-dimensional algebras, London Mathematical Society Lecture Note Series, vol. 119, Cambridge University Press, Cambridge, 1988. MR 935124 (89e:16035)
  • [25] Dieter Happel, A characterization of hereditary categories with tilting object, Invent. Math. 144 (2001), no. 2, 381-398. MR 1827736 (2002a:18014), https://doi.org/10.1007/s002220100135
  • [26] Dieter Happel and Claus Michael Ringel, Tilted algebras, Trans. Amer. Math. Soc. 274 (1982), no. 2, 399-443. MR 675063 (84d:16027), https://doi.org/10.2307/1999116
  • [27] Dieter Happel and Dan Zacharia, A homological characterization of piecewise hereditary algebras, Math. Z. 260 (2008), no. 1, 177-185. MR 2413349 (2009g:16011), https://doi.org/10.1007/s00209-007-0268-3
  • [28] Robin Hartshorne, Residues and duality, Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64. With an appendix by P. Deligne. Lecture Notes in Mathematics, No. 20, Springer-Verlag, Berlin-New York, 1966. MR 0222093 (36 #5145)
  • [29] T. Hübner, Exzeptionelle Vektorbündel und Reflektionen an Kippgarben über projectiven gewichteten Kurven, (1996).
  • [30] Bernhard Keller, On triangulated orbit categories, Doc. Math. 10 (2005), 551-581. MR 2184464 (2007c:18006)
  • [31] Bernhard Keller, Derived categories and tilting, Handbook of Tilting Theory (Lidia Angeleri Hügel, Dieter Happel, and Henning Krause, eds.), London Mathematical Society Lecture Notes Series, vol. 332, Cambridge University Press, Cambridge, 2007, pp. 49-104.
  • [32] Helmut Lenzing, A $ K$-theoretic study of canonical algebras, Representation theory of algebras (Cocoyoc, 1994) CMS Conf. Proc., vol. 18, Amer. Math. Soc., Providence, RI, 1996, pp. 433-454. MR 1388066 (97e:16020)
  • [33] Helmut Lenzing, Hereditary categories, Handbook of Tilting Theory (Lidia Angeleri Hügel, Dieter Happel, and Henning Krause, eds.), London Mathematical Society Lecture Notes Series, vol. 332, Cambridge University Press, Cambridge, 2007, pp. 105-146.
  • [34] Hagen Meltzer, Tubular mutations, Colloq. Math. 74 (1997), no. 2, 267-274. MR 1477569 (99a:18004)
  • [35] Hagen Meltzer, Exceptional vector bundles, tilting sheaves and tilting complexes for weighted projective lines, Mem. Amer. Math. Soc. 171 (2004), no. 808, viii+139. MR 2074151 (2005k:14033), https://doi.org/10.1090/memo/0808
  • [36] Idun Reiten and Michel Van den Bergh, Grothendieck groups and tilting objects, Algebr. Represent. Theory 4 (2001), no. 1, 1-23. Special issue dedicated to Klaus Roggenkamp on the occasion of his 60th birthday. MR 1825805 (2002c:18007), https://doi.org/10.1023/A:1009902810813
  • [37] I. Reiten and M. Van den Bergh, Noetherian hereditary abelian categories satisfying Serre duality, J. Amer. Math. Soc. 15 (2002), no. 2, 295-366. MR 1887637 (2003a:18011), https://doi.org/10.1090/S0894-0347-02-00387-9
  • [38] Claus Michael Ringel, Tame algebras and integral quadratic forms, Lecture Notes in Mathematics, vol. 1099, Springer-Verlag, Berlin, 1984. MR 774589 (87f:16027)
  • [39] Claus Michael Ringel, The diamond category of a locally discrete ordered set, Representations of algebra. Vol. I, II, Beijing Norm. Univ. Press, Beijing, 2002, pp. 387-395. MR 2067391 (2005i:16025)
  • [40] Claus Michael Ringel, Hereditary triangulated categories, Compositio Math. (2005).
  • [41] Paul Seidel and Richard Thomas, Braid group actions on derived categories of coherent sheaves, Duke Math. J. 108 (2001), no. 1, 37-108. MR 1831820 (2002e:14030), https://doi.org/10.1215/S0012-7094-01-10812-0
  • [42] Donald Stanley and Adam-Christiaan van Roosmalen, $ t$-Structures for hereditary categories, preprint.
  • [43] Adam-Christiaan van Roosmalen, Abelian 1-Calabi-Yau categories, Int. Math. Res. Not. IMRN 6 (2008), Art. ID rnn003, 20. MR 2427460 (2009g:18016), https://doi.org/10.1093/imrn/rnn003
  • [44] Adam-Christiaan van Roosmalen, Abelian hereditary fractionally Calabi-Yau categories, Int. Math. Res. Not. IMRN (2012), no. 12, 2708-2750.
  • [45] Adam-Christiaan van Roosmalen, Hereditary uniserial categories with Serre duality, Algebr. Represent. Theory 15 (2012), no. 6, 1291-1322. MR 2994026, https://doi.org/10.1007/s10468-011-9289-z

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 18E10, 18E30, 18G20, 16G20, 14F05

Retrieve articles in all journals with MSC (2010): 18E10, 18E30, 18G20, 16G20, 14F05


Additional Information

Adam-Christiaan van Roosmalen
Affiliation: Department of Algebra, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Prague 8, Czech Republic
Address at time of publication: Department of Mathematics and Statistics, Hasselt University, B-3590 Diepenbeek, Belgium
Email: vanroosmalen@karlin.mff.cuni.cz, adamchristiaan.vanroosmalen@uhasselt.be

DOI: https://doi.org/10.1090/tran/6569
Received by editor(s): April 21, 2014
Received by editor(s) in revised form: September 11, 2014
Published electronically: February 11, 2016
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society