Telescope conjecture, idempotent ideals, and the transfinite radical

Šťovíček, Jan

doi:10.1090/S0002-9947-09-04812-0

Telescope conjecture, idempotent ideals, and the transfinite radical
HTML articles powered by AMS MathViewer

by Jan Šťovíček

Trans. Amer. Math. Soc. 362 (2010), 1475-1489

DOI: https://doi.org/10.1090/S0002-9947-09-04812-0

Published electronically: October 9, 2009

PDF | Request permission

Abstract:

We show that for an artin algebra $\Lambda$, the telescope conjecture for module categories is equivalent to certain idempotent ideals of $\operatorname {mod}\Lambda$ being generated by identity morphisms. As a consequence, we prove the conjecture for domestic standard selfinjective algebras and domestic special biserial algebras. We achieve this by showing that in any Krull-Schmidt category with local d.c.c. on ideals, any idempotent ideal is generated by identity maps and maps from the transfinite radical.

References

Jiří Adámek and Jiří Rosický, Locally presentable and accessible categories, London Mathematical Society Lecture Note Series, vol. 189, Cambridge University Press, Cambridge, 1994. MR 1294136, DOI 10.1017/CBO9780511600579
Lidia Angeleri Hügel, Jan Šaroch, and Jan Trlifaj, On the telescope conjecture for module categories, J. Pure Appl. Algebra 212 (2008), no. 2, 297–310. MR 2357346, DOI 10.1016/j.jpaa.2007.05.027
Lidia Angeleri Hügel and Jan Trlifaj, Direct limits of modules of finite projective dimension, Rings, modules, algebras, and abelian groups, Lecture Notes in Pure and Appl. Math., vol. 236, Dekker, New York, 2004, pp. 27–44. MR 2050699
D. Benson, S. Iyengar and H. Krause, Stratifying modular representations of finite groups, preprint, arXiv:0810.1339v1.
Flávio U. Coelho, Eduardo N. Marcos, Héctor A. Merklen, and Andrzej Skowroński, Domestic semiregular branch enlargements of tame concealed algebras, Representation theory of algebras (Cocoyoc, 1994) CMS Conf. Proc., vol. 18, Amer. Math. Soc., Providence, RI, 1996, pp. 141–154. MR 1388047, DOI 10.1080/00927879408825084
Flávio U. Coelho, Eduardo N. Marcos, Héctor A. Merklen, and Andrzej Skowroński, Module categories with infinite radical cube zero, J. Algebra 183 (1996), no. 1, 1–23. MR 1397384, DOI 10.1006/jabr.1996.0204
W. W. Crawley-Boevey, On tame algebras and bocses, Proc. London Math. Soc. (3) 56 (1988), no. 3, 451–483. MR 931510, DOI 10.1112/plms/s3-56.3.451
Laszlo Fuchs and Sang Bum Lee, From a single chain to a large family of submodules, Port. Math. (N.S.) 61 (2004), no. 2, 193–205. MR 2066674
Werner Geigle, The Krull-Gabriel dimension of the representation theory of a tame hereditary Artin algebra and applications to the structure of exact sequences, Manuscripta Math. 54 (1985), no. 1-2, 83–106. MR 808682, DOI 10.1007/BF01171701
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
I. M. Gel′fand and V. A. Ponomarev, Indecomposable representations of the Lorentz group, Uspehi Mat. Nauk 23 (1968), no. 2 (140), 3–60 (Russian). MR 0229751
Rüdiger Göbel and Jan Trlifaj, Approximations and endomorphism algebras of modules, De Gruyter Expositions in Mathematics, vol. 41, Walter de Gruyter GmbH & Co. KG, Berlin, 2006. MR 2251271, DOI 10.1515/9783110199727
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
Bernhard Keller, A remark on the generalized smashing conjecture, Manuscripta Math. 84 (1994), no. 2, 193–198. MR 1285956, DOI 10.1007/BF02567453
Otto Kerner and Andrzej Skowroński, On module categories with nilpotent infinite radical, Compositio Math. 77 (1991), no. 3, 313–333. MR 1092772
Henning Krause, Cohomological quotients and smashing localizations, Amer. J. Math. 127 (2005), no. 6, 1191–1246. MR 2183523, DOI 10.1353/ajm.2005.0041
H. Krause, Derived categories, resolutions, and Brown representability, lecture notes for the summer school “Interactions between Homotopy Theory and Algebra” at the University of Chicago, 2004.
Henning Krause, Smashing subcategories and the telescope conjecture—an algebraic approach, Invent. Math. 139 (2000), no. 1, 99–133. MR 1728877, DOI 10.1007/s002229900022
Henning Krause, The spectrum of a module category, Mem. Amer. Math. Soc. 149 (2001), no. 707, x+125. MR 1803703, DOI 10.1090/memo/0707
Henning Krause and Øyvind Solberg, Applications of cotorsion pairs, J. London Math. Soc. (2) 68 (2003), no. 3, 631–650. MR 2009441, DOI 10.1112/S0024610703004757
Amnon Neeman, The connection between the $K$-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel, Ann. Sci. École Norm. Sup. (4) 25 (1992), no. 5, 547–566. MR 1191736, DOI 10.24033/asens.1659
Mike Prest, Model theory and modules, London Mathematical Society Lecture Note Series, vol. 130, Cambridge University Press, Cambridge, 1988. MR 933092, DOI 10.1017/CBO9780511600562
Mike Prest, Morphisms between finitely presented modules and infinite-dimensional representations, Algebras and modules, II (Geiranger, 1996) CMS Conf. Proc., vol. 24, Amer. Math. Soc., Providence, RI, 1998, pp. 447–455. MR 1648645, DOI 10.2307/2586842
Mike Prest and Jan Schröer, Serial functors, Jacobson radical and representation type, J. Pure Appl. Algebra 170 (2002), no. 2-3, 295–307. MR 1904849, DOI 10.1016/S0022-4049(01)00133-5
Jan Šaroch and Jan Šťovíček, The countable telescope conjecture for module categories, Adv. Math. 219 (2008), no. 3, 1002–1036. MR 2442060, DOI 10.1016/j.aim.2008.05.012
Jan Šaroch and Jan Trlifaj, Completeness of cotorsion pairs, Forum Math. 19 (2007), no. 4, 749–760. MR 2336972, DOI 10.1515/FORUM.2007.029
Jan Schröer, On the infinite radical of a module category, Proc. London Math. Soc. (3) 81 (2000), no. 3, 651–674. MR 1781151, DOI 10.1112/S0024611500012600
Andrzej Skowroński, Cycle-finite algebras, J. Pure Appl. Algebra 103 (1995), no. 1, 105–116. MR 1354071, DOI 10.1016/0022-4049(94)00094-Y
Andrzej Skowroński, Selfinjective algebras of polynomial growth, Math. Ann. 285 (1989), no. 2, 177–199. MR 1016089, DOI 10.1007/BF01443513
Andrzej Skowroński and Josef Waschbüsch, Representation-finite biserial algebras, J. Reine Angew. Math. 345 (1983), 172–181. MR 717892
Jan Šťovíček and Jan Trlifaj, Generalized Hill lemma, Kaplansky theorem for cotorsion pairs and some applications, Rocky Mountain J. Math. 39 (2009), no. 1, 305–324. MR 2476814, DOI 10.1216/RMJ-2009-39-1-305
Jan Trlifaj, Cotorsion theories induced by tilting and cotilting modules, Abelian groups, rings and modules (Perth, 2000) Contemp. Math., vol. 273, Amer. Math. Soc., Providence, RI, 2001, pp. 285–300. MR 1817171, DOI 10.1090/conm/273/04443