The relationship between homological properties and representation theoretic realization of artin algebras

Iyama, Osamu

doi:10.1090/S0002-9947-04-03482-8

The relationship between homological properties and representation theoretic realization of artin algebras
HTML articles powered by AMS MathViewer

by Osamu Iyama PDF

Trans. Amer. Math. Soc. 357 (2005), 709-734 Request permission

Abstract:

We will study the relationship of quite different objects in the theory of artin algebras, namely Auslander-regular rings of global dimension two, torsion theories, $\tau$-categories and almost abelian categories. We will apply our results to characterization problems of Auslander-Reiten quivers.

References

Maurice Auslander, Representation theory of Artin algebras. I, II, Comm. Algebra 1 (1974), 177–268; ibid. 1 (1974), 269–310. MR 349747, DOI 10.1080/00927877408548230
Maurice Auslander and Mark Bridger, Stable module theory, Memoirs of the American Mathematical Society, No. 94, American Mathematical Society, Providence, R.I., 1969. MR 0269685
Maurice Auslander and Idun Reiten, Representation theory of Artin algebras. III. Almost split sequences, Comm. Algebra 3 (1975), 239–294. MR 379599, DOI 10.1080/00927877508822046
Maurice Auslander and Idun Reiten, $k$-Gorenstein algebras and syzygy modules, J. Pure Appl. Algebra 92 (1994), no. 1, 1–27. MR 1259667, DOI 10.1016/0022-4049(94)90044-2
Maurice Auslander and Idun Reiten, Syzygy modules for Noetherian rings, J. Algebra 183 (1996), no. 1, 167–185. MR 1397392, DOI 10.1006/jabr.1996.0212
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
M. Auslander and Sverre O. Smalø, Almost split sequences in subcategories, J. Algebra 69 (1981), no. 2, 426–454. MR 617088, DOI 10.1016/0021-8693(81)90214-3
Ibrahim Assem, Tilting theory—an introduction, Topics in algebra, Part 1 (Warsaw, 1988) Banach Center Publ., vol. 26, PWN, Warsaw, 1990, pp. 127–180. MR 1171230
Hyman Bass, On the ubiquity of Gorenstein rings, Math. Z. 82 (1963), 8–28. MR 153708, DOI 10.1007/BF01112819
Jan-Erik Björk, The Auslander condition on Noetherian rings, Séminaire d’Algèbre Paul Dubreil et Marie-Paul Malliavin, 39ème Année (Paris, 1987/1988) Lecture Notes in Math., vol. 1404, Springer, Berlin, 1989, pp. 137–173. MR 1035224, DOI 10.1007/BFb0084075
K. Bongartz and P. Gabriel, Covering spaces in representation-theory, Invent. Math. 65 (1981/82), no. 3, 331–378. MR 643558, DOI 10.1007/BF01396624
Sheila Brenner, A combinatorial characterisation of finite Auslander-Reiten quivers, Representation theory, I (Ottawa, Ont., 1984) Lecture Notes in Math., vol. 1177, Springer, Berlin, 1986, pp. 13–49. MR 842457, DOI 10.1007/BFb0075256
John Clark, Auslander-Gorenstein rings for beginners, International Symposium on Ring Theory (Kyongju, 1999) Trends Math., Birkhäuser Boston, Boston, MA, 2001, pp. 95–115. MR 1851195
Charles W. Curtis and Irving Reiner, Methods of representation theory. Vol. I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1990. With applications to finite groups and orders; Reprint of the 1981 original; A Wiley-Interscience Publication. MR 1038525
Ju. A. Drozd and V. V. Kiričenko, The quasi-Bass orders, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 328–370 (Russian). MR 0304411
Ju. A. Drozd, V. V. Kiričenko, and A. V. Roĭter, Hereditary and Bass orders, Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967), 1415–1436 (Russian). MR 0219527
Robert M. Fossum, Phillip A. Griffith, and Idun Reiten, Trivial extensions of abelian categories, Lecture Notes in Mathematics, Vol. 456, Springer-Verlag, Berlin-New York, 1975. Homological algebra of trivial extensions of abelian categories with applications to ring theory. MR 0389981, DOI 10.1007/BFb0065404
Edward L. Green and Roberto Martínez Villa, Koszul and Yoneda algebras, Representation theory of algebras (Cocoyoc, 1994) CMS Conf. Proc., vol. 18, Amer. Math. Soc., Providence, RI, 1996, pp. 247–297. MR 1388055
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
Mitsuo Hoshino, Tilting modules and torsion theories, Bull. London Math. Soc. 14 (1982), no. 4, 334–336. MR 663483, DOI 10.1112/blms/14.4.334
Mitsuo Hoshino, On dominant dimension of Noetherian rings, Osaka J. Math. 26 (1989), no. 2, 275–280. MR 1017586
Dieter Happel, Udo Preiser, and Claus Michael Ringel, Vinberg’s characterization of Dynkin diagrams using subadditive functions with application to $D\textrm {Tr}$-periodic modules, Representation theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979) Lecture Notes in Math., vol. 832, Springer, Berlin, 1980, pp. 280–294. MR 607159
H. Hijikata and K. Nishida, Bass orders in nonsemisimple algebras, J. Math. Kyoto Univ. 34 (1994), no. 4, 797–837. MR 1311621, DOI 10.1215/kjm/1250518887
Osamu Iyama, A generalization of rejection lemma of Drozd-Kirichenko, J. Math. Soc. Japan 50 (1998), no. 3, 697–718. MR 1625391, DOI 10.2969/jmsj/05030697
Osamu Iyama, Some categories of lattices associated to a central idempotent, J. Math. Kyoto Univ. 38 (1998), no. 3, 487–501. MR 1661224, DOI 10.1215/kjm/1250518062
O. Iyama: $\tau$-categories I: Ladders, to appear in Algebras and Representation Theory.
O. Iyama: $\tau$-categories II: Nakayama pairs and rejective subcategories, to appear in Algebras and Representation Theory.
O. Iyama: $\tau$-categories III: Auslander orders and Auslander-Reiten quivers, to appear in Algebras and Representation Theory.
Osamu Iyama, Representation theory of orders, Algebra—representation theory (Constanta, 2000) NATO Sci. Ser. II Math. Phys. Chem., vol. 28, Kluwer Acad. Publ., Dordrecht, 2001, pp. 63–96. MR 1858032
Osamu Iyama, A proof of Solomon’s second conjecture on local zeta functions of orders, J. Algebra 259 (2003), no. 1, 119–126. MR 1953711, DOI 10.1016/S0021-8693(02)00548-3
Osamu Iyama, Finiteness of representation dimension, Proc. Amer. Math. Soc. 131 (2003), no. 4, 1011–1014. MR 1948089, DOI 10.1090/S0002-9939-02-06616-9
O. Iyama: Symmetry and Duality on $n$-Gorenstein rings, J. Algebra 269 (2003), no. 2, 528–535.
Kiyoshi Igusa and Gordana Todorov, Radical layers of representable functors, J. Algebra 89 (1984), no. 1, 105–147. MR 748231, DOI 10.1016/0021-8693(84)90238-2
Kiyoshi Igusa and Gordana Todorov, Radical layers of representable functors, J. Algebra 89 (1984), no. 1, 105–147. MR 748231, DOI 10.1016/0021-8693(84)90238-2
Kiyoshi Igusa and Gordana Todorov, A numerical characterization of finite Auslander-Reiten quivers, Representation theory, I (Ottawa, Ont., 1984) Lecture Notes in Math., vol. 1177, Springer, Berlin, 1986, pp. 181–198. MR 842466, DOI 10.1007/BFb0075265
Klaus W. Roggenkamp and Verena Huber-Dyson, Lattices over orders. I, Lecture Notes in Mathematics, Vol. 115, Springer-Verlag, Berlin-New York, 1970. MR 0283013
Wolfgang Rump, Almost abelian categories, Cahiers Topologie Géom. Différentielle Catég. 42 (2001), no. 3, 163–225 (English, with French summary). MR 1856638
Wolfgang Rump, $\ast$-modules, tilting, and almost abelian categories, Comm. Algebra 29 (2001), no. 8, 3293–3325. MR 1849488, DOI 10.1081/AGB-100105023
Wolfgang Rump, Ladder functors with an application to representation-finite Artinian rings, An. Ştiinţ. Univ. Ovidius Constanţa Ser. Mat. 9 (2001), no. 1, 107–123. To Mirela Ştefănescu, at her 60’s. MR 1946161
W. Rump: Lattice-finite rings, to appear in Algebras and Representation Theory.
W. Rump: The category of lattices over a lattice-finite ring, to appear in Algebras and Representation Theory.
Idun Reiten and Michel Van den Bergh, Two-dimensional tame and maximal orders of finite representation type, Mem. Amer. Math. Soc. 80 (1989), no. 408, viii+72. MR 978602, DOI 10.1090/memo/0408
Claus Michael Ringel and Dieter Vossieck, Hammocks, Proc. London Math. Soc. (3) 54 (1987), no. 2, 216–246. MR 872806, DOI 10.1112/plms/s3-54.2.216
Daniel Simson, Linear representations of partially ordered sets and vector space categories, Algebra, Logic and Applications, vol. 4, Gordon and Breach Science Publishers, Montreux, 1992. MR 1241646
Hiroyuki Tachikawa, Quasi-Frobenius rings and generalizations. $\textrm {QF}-3$ and $\textrm {QF}-1$ rings, Lecture Notes in Mathematics, Vol. 351, Springer-Verlag, Berlin-New York, 1973. Notes by Claus Michael Ringel. MR 0349740, DOI 10.1007/BFb0059997