Numerically finite hereditary categories with Serre duality

van Roosmalen, Adam-Christiaan

doi:10.1090/tran/6569

Numerically finite hereditary categories with Serre duality
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Trans. Amer. Math. Soc. 368 (2016), 7189-7238 Request permission

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

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, DOI 10.1016/j.jpaa.2007.06.002
M. F. Atiyah, Vector bundles over an elliptic curve, Proc. London Math. Soc. (3) 7 (1957), 414–452. MR 131423, DOI 10.1112/plms/s3-7.1.414
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) Lecture Notes in Math., Vol. 353, Springer, Berlin, 1973, pp. 8–71. MR 0335575
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, DOI 10.1017/CBO9780511623608
Paul Balmer and Marco Schlichting, Idempotent completion of triangulated categories, J. Algebra 236 (2001), no. 2, 819–834. MR 1813503, DOI 10.1006/jabr.2000.8529
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
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, DOI 10.1016/j.jalgebra.2010.11.021
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, DOI 10.1112/plms/pdt021
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, DOI 10.1070/IM1990v034n01ABEH000583
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, DOI 10.1070/IM1990v035n03ABEH000716
Tom Bridgeland, Equivalences of triangulated categories and Fourier-Mukai transforms, Bull. London Math. Soc. 31 (1999), no. 1, 25–34. MR 1651025, DOI 10.1112/S0024609398004998
Tom Bridgeland, Stability conditions on triangulated categories, Ann. of Math. (2) 166 (2007), no. 2, 317–345. MR 2373143, DOI 10.4007/annals.2007.166.317
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. MR 1824990, DOI 10.1090/S0894-0347-01-00368-X
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, DOI 10.1090/conm/436/08414
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, DOI 10.1007/s11253-005-0169-8
Xiao-Wu Chen, Yu Ye, and Pu Zhang, Algebras of derived dimension zero, Comm. Algebra 36 (2008), no. 1, 1–10. MR 2378361, DOI 10.1080/00927870701649184
Claude Cibils and Pu Zhang, Calabi-Yau objects in triangulated categories, Trans. Amer. Math. Soc. 361 (2009), no. 12, 6501–6519. MR 2538602, DOI 10.1090/S0002-9947-09-04682-0
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
Pierre Gabriel, Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323–448 (French). MR 232821
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, DOI 10.1007/BFb0078849
Werner Geigle and Helmut Lenzing, Perpendicular categories with applications to representations and sheaves, J. Algebra 144 (1991), no. 2, 273–343. MR 1140607, DOI 10.1016/0021-8693(91)90107-J
Sergei I. Gelfand and Yuri I. Manin, Methods of homological algebra, 2nd ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. MR 1950475, DOI 10.1007/978-3-662-12492-5
A. L. Gorodentsev and A. N. Rudakov, Exceptional vector bundles on projective spaces, Duke Math. J. 54 (1987), no. 1, 115–130. MR 885779, DOI 10.1215/S0012-7094-87-05409-3
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, DOI 10.1017/CBO9780511629228
Dieter Happel, A characterization of hereditary categories with tilting object, Invent. Math. 144 (2001), no. 2, 381–398. MR 1827736, DOI 10.1007/s002220100135
Dieter Happel and Claus Michael Ringel, Tilted algebras, Trans. Amer. Math. Soc. 274 (1982), no. 2, 399–443. MR 675063, DOI 10.1090/S0002-9947-1982-0675063-2
Dieter Happel and Dan Zacharia, A homological characterization of piecewise hereditary algebras, Math. Z. 260 (2008), no. 1, 177–185. MR 2413349, DOI 10.1007/s00209-007-0268-3
Robin Hartshorne, Residues and duality, Lecture Notes in Mathematics, No. 20, Springer-Verlag, Berlin-New York, 1966. Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64; With an appendix by P. Deligne. MR 0222093
T. Hübner, Exzeptionelle Vektorbündel und Reflektionen an Kippgarben über projectiven gewichteten Kurven, (1996).
Bernhard Keller, On triangulated orbit categories, Doc. Math. 10 (2005), 551–581. MR 2184464
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.
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, DOI 10.4064/cm-71-2-161-181
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.
Hagen Meltzer, Tubular mutations, Colloq. Math. 74 (1997), no. 2, 267–274. MR 1477569, DOI 10.4064/cm-74-2-267-274
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, DOI 10.1090/memo/0808
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, DOI 10.1023/A:1009902810813
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, DOI 10.1090/S0894-0347-02-00387-9
Claus Michael Ringel, Tame algebras and integral quadratic forms, Lecture Notes in Mathematics, vol. 1099, Springer-Verlag, Berlin, 1984. MR 774589, DOI 10.1007/BFb0072870
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
Claus Michael Ringel, Hereditary triangulated categories, Compositio Math. (2005).
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, DOI 10.1215/S0012-7094-01-10812-0
Donald Stanley and Adam-Christiaan van Roosmalen, $t$-Structures for hereditary categories, preprint.
Adam-Christiaan van Roosmalen, Abelian 1-Calabi-Yau categories, Int. Math. Res. Not. IMRN 6 (2008), Art. ID rnn003, 20. MR 2427460, DOI 10.1093/imrn/rnn003
Adam-Christiaan van Roosmalen, Abelian hereditary fractionally Calabi-Yau categories, Int. Math. Res. Not. IMRN (2012), no. 12, 2708–2750.
Adam-Christiaan van Roosmalen, Hereditary uniserial categories with Serre duality, Algebr. Represent. Theory 15 (2012), no. 6, 1291–1322. MR 2994026, DOI 10.1007/s10468-011-9289-z