Noetherian hereditary abelian categories satisfying Serre duality

Reiten, I.¹ ; Van den Bergh, M.²

¹ Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491 Trondheim, Norway
² Department WNI, Limburgs Universitair Centrum, Universitaire Campus, Building D, 3590 Diepenbeek, Belgium

Journal of the American Mathematical Society, Tome 15 (2002) no. 2, pp. 295-366

Résumé

In this paper we classify $\operatorname {Ext}$-finite noetherian hereditary abelian categories over an algebraically closed field $k$ satisfying Serre duality in the sense of Bondal and Kapranov. As a consequence we obtain a classification of saturated noetherian hereditary abelian categories. As a side result we show that when our hereditary abelian categories have no non-zero projectives or injectives, then the Serre duality property is equivalent to the existence of almost split sequences.

Détail

DOI : 10.1090/S0894-0347-02-00387-9

Bibliographie

[1] Artin, M., Stafford, J. T. Noncommutative graded domains with quadratic growth Invent. Math. 1995 231 276

[2] Artin, M., Zhang, J. J. Noncommutative projective schemes Adv. Math. 1994 228 287

[3] Auslander, M., Reiten, I. Almost split sequences in dimension two Adv. in Math. 1987 88 118

[4] Auslander, Maurice, Reiten, Idun, Smalã¸, Sverre O. Representation theory of Artin algebras 1995

[5] Beä­Linson, A. A. On the derived category of perverse sheaves 1987 27 41

[6] Beä­Linson, A. A., Bernstein, J., Deligne, P. Faisceaux pervers 1982 5 171

[7] Bondal, A. I., Kapranov, M. M. Representable functors, Serre functors, and reconstructions Izv. Akad. Nauk SSSR Ser. Mat. 1989

[8] Gabriel, Pierre Des catégories abéliennes Bull. Soc. Math. France 1962 323 448

[9] Geigle, Werner, Lenzing, Helmut A class of weighted projective curves arising in representation theory of finite-dimensional algebras 1987 265 297

[10] Grothendieck, Alexander Sur quelques points d’algèbre homologique Tohoku Math. J. (2) 1957 119 221

[11] Happel, Dieter Triangulated categories in the representation theory of finite-dimensional algebras 1988

[12] Happel, Dieter A characterization of hereditary categories with tilting object Invent. Math. 2001 381 398

[13] Happel, Dieter, Reiten, Idun, Smalã¸, Sverre O. Tilting in abelian categories and quasitilted algebras Mem. Amer. Math. Soc. 1996

[14] Hartshorne, Robin Algebraic geometry 1977

[15] Kashiwara, Masaki, Schapira, Pierre Sheaves on manifolds 1990

[16] Lenzing, Helmut Hereditary Noetherian categories with a tilting complex Proc. Amer. Math. Soc. 1997 1893 1901

[17] Mcconnell, J. C., Robson, J. C. Noncommutative Noetherian rings 1987

[18] Reiner, I. Maximal orders 1975

[19] Reiten, Idun, Riedtmann, Christine Skew group algebras in the representation theory of Artin algebras J. Algebra 1985 224 282

[20] Robson, J. C., Small, Lance W. Hereditary prime P.I. rings are classical hereditary orders J. London Math. Soc. (2) 1974 499 503

[21] Small, L. W., Stafford, J. T., Warfield, R. B., Jr. Affine algebras of Gel′fand-Kirillov dimension one are PI Math. Proc. Cambridge Philos. Soc. 1985 407 414

[22] Smalã¸, Sverre O. Almost split sequences in categories of representations of quivers Proc. Amer. Math. Soc. 2001 695 698

[23] Smith, S. Paul, Zhang, James J. Curves on quasi-schemes Algebr. Represent. Theory 1998 311 351

[24] Stenstrã¶M, Bo Rings of quotients 1975

[25] Van Gastel, Martine, Van Den Bergh, Michel Graded modules of Gelfand-Kirillov dimension one over three-dimensional Artin-Schelter regular algebras J. Algebra 1997 251 282

[26] Verdier, Jean-Louis Des catégories dérivées des catégories abéliennes Astérisque 1996