All tilting modules are of finite type
Authors:
Silvana Bazzoni and Jan Šťovíček
Journal:
Proc. Amer. Math. Soc. 135 (2007), 3771-3781
MSC (2000):
Primary 16D90, 16D30; Secondary 03E75, 16G99.
DOI:
https://doi.org/10.1090/S0002-9939-07-08911-3
Published electronically:
August 30, 2007
MathSciNet review:
2341926
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: We prove that any infinitely generated tilting module is of finite type, namely that its associated tilting class is the Ext-orthogonal of a set of modules possessing a projective resolution consisting of finitely generated projective modules.
- Lidia Angeleri Hügel and Flávio Ulhoa Coelho, Infinitely generated tilting modules of finite projective dimension, Forum Math. 13 (2001), no. 2, 239–250. MR 1813669, DOI https://doi.org/10.1515/form.2001.006
- Lidia Angeleri Hügel, Dolors Herbera, and Jan Trlifaj, Tilting modules and Gorenstein rings, Forum Math. 18 (2006), no. 2, 211–229. MR 2218418, DOI https://doi.org/10.1515/FORUM.2006.013
- Lidia Angeleri Hügel, Alberto Tonolo, and Jan Trlifaj, Tilting preenvelopes and cotilting precovers, Algebr. Represent. Theory 4 (2001), no. 2, 155–170. MR 1834843, DOI https://doi.org/10.1023/A%3A1011485800557
- Maurice Auslander and Idun Reiten, Applications of contravariantly finite subcategories, Adv. Math. 86 (1991), no. 1, 111–152. MR 1097029, DOI https://doi.org/10.1016/0001-8708%2891%2990037-8
- M. Auslander and Sverre O. Smalø, Preprojective modules over Artin algebras, J. Algebra 66 (1980), no. 1, 61–122. MR 591246, DOI https://doi.org/10.1016/0021-8693%2880%2990113-1
- Silvana Bazzoni, A characterization of $n$-cotilting and $n$-tilting modules, J. Algebra 273 (2004), no. 1, 359–372. MR 2032465, DOI https://doi.org/10.1016/S0021-8693%2803%2900432-0
- Silvana Bazzoni, Paul C. Eklof, and Jan Trlifaj, Tilting cotorsion pairs, Bull. London Math. Soc. 37 (2005), no. 5, 683–696. MR 2164830, DOI https://doi.org/10.1112/S0024609305004728
- S. Bazzoni, D. Herbera One dimensional tilting modules are of finite type, Algebr. Represent. Theory, in press 10.1007/s10468-007-9064-3
- Sheila Brenner and M. C. R. Butler, Generalizations of the Bernstein-Gel′fand-Ponomarev reflection functors, Representation theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979) Lecture Notes in Math., vol. 832, Springer, Berlin-New York, 1980, pp. 103–169. MR 607151
- R. R. Colby and K. R. Fuller, Tilting, cotilting, and serially tilted rings, Comm. Algebra 18 (1990), no. 5, 1585–1615. MR 1059750, DOI https://doi.org/10.1080/00927879008823985
- Riccardo Colpi and Jan Trlifaj, Tilting modules and tilting torsion theories, J. Algebra 178 (1995), no. 2, 614–634. MR 1359905, DOI https://doi.org/10.1006/jabr.1995.1368
- William Crawley-Boevey, Infinite-dimensional modules in the representation theory of finite-dimensional algebras, Algebras and modules, I (Trondheim, 1996) CMS Conf. Proc., vol. 23, Amer. Math. Soc., Providence, RI, 1998, pp. 29–54. MR 1648602
- Paul C. Eklof, László Fuchs, and Saharon Shelah, Baer modules over domains, Trans. Amer. Math. Soc. 322 (1990), no. 2, 547–560. MR 974514, DOI https://doi.org/10.1090/S0002-9947-1990-0974514-8
- Paul C. Eklof and Alan H. Mekler, Almost free modules, Revised edition, North-Holland Mathematical Library, vol. 65, North-Holland Publishing Co., Amsterdam, 2002. Set-theoretic methods. MR 1914985
- Paul C. Eklof and Jan Trlifaj, How to make Ext vanish, Bull. London Math. Soc. 33 (2001), no. 1, 41–51. MR 1798574, DOI https://doi.org/10.1112/blms/33.1.41
- Edgar E. Enochs, Injective and flat covers, envelopes and resolvents, Israel J. Math. 39 (1981), no. 3, 189–209. MR 636889, DOI https://doi.org/10.1007/BF02760849
- A. Grothendieck, “Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. I”, Inst. Hautes Études Sci. Publ. Math. 11, 1961.
- Dieter Happel and Claus Michael Ringel, Tilted algebras, Trans. Amer. Math. Soc. 274 (1982), no. 2, 399–443. MR 675063, DOI https://doi.org/10.1090/S0002-9947-1982-0675063-2
- Wilfrid Hodges, In singular cardinality, locally free algebras are free, Algebra Universalis 12 (1981), no. 2, 205–220. MR 608664, DOI https://doi.org/10.1007/BF02483879
- Yoichi Miyashita, Tilting modules of finite projective dimension, Math. Z. 193 (1986), no. 1, 113–146. MR 852914, DOI https://doi.org/10.1007/BF01163359
- Saharon Shelah, A compactness theorem for singular cardinals, free algebras, Whitehead problem and transversals, Israel J. Math. 21 (1975), no. 4, 319–349. MR 389579, DOI https://doi.org/10.1007/BF02757993
- J. Šťovíček, J. Trlifaj, All tilting modules are of countable type, Bull. Lond. Math. Soc. 39 (2007), no. 1, 121–132.
- 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 https://doi.org/10.1090/conm/273/04443
- Jinzhong Xu, Flat covers of modules, Lecture Notes in Mathematics, vol. 1634, Springer-Verlag, Berlin, 1996. MR 1438789
Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 16D90, 16D30, 03E75, 16G99.
Retrieve articles in all journals with MSC (2000): 16D90, 16D30, 03E75, 16G99.
Additional Information
Silvana Bazzoni
Affiliation:
Dipartimento di Matematica Pura e Applicata, Universitá di Padova, Via Trieste 63, 35121 Padova, Italy
MR Author ID:
33015
Email:
bazzoni@math.unipd.it
Jan Šťovíček
Affiliation:
Katedra algebry MFF UK, Sokolovská 83, 186 75 Praha 8, Czech Republic
Email:
stovicek@karlin.mff.cuni.cz
Keywords:
Tilting modules,
cotorsion pairs.
Received by editor(s):
October 1, 2005
Received by editor(s) in revised form:
September 9, 2006
Published electronically:
August 30, 2007
Additional Notes:
The first author was supported by Università di Padova (Progetto di Ateneo CDPA048343 “Decomposition and tilting theory in modules, derived and cluster categories”).
The second author was supported by a grant of the Industrie Club Duesseldorf, GAČR 201/05/H005, and the research project MSM 0021620839.
Communicated by:
Martin Lorenz
Article copyright:
© Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.