Freyd’s generating hypothesis with almost split sequences
HTML articles powered by AMS MathViewer
- by Jon F. Carlson, Sunil K. Chebolu and Ján Mináč PDF
- Proc. Amer. Math. Soc. 137 (2009), 2575-2580 Request permission
Abstract:
Freyd’s generating hypothesis for the stable module category of a non-trivial finite group $G$ is the statement that a map between finitely generated $kG$-modules that belongs to the thick subcategory generated by the field $k$ factors through a projective module if the induced map on Tate cohomology is trivial. In this paper we show that Freyd’s generating hypothesis fails for $kG$ when the Sylow $p$-subgroup of $G$ has order at least $4$ using almost split sequences. By combining this with our earlier work, we obtain a complete answer to Freyd’s generating hypothesis for the stable module category of a finite group. We also derive some consequences of the generating hypothesis.References
- 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
- David J. Benson, Sunil K. Chebolu, J. Daniel Christensen, and Ján Mináč, The generating hypothesis for the stable module category of a $p$-group, J. Algebra 310 (2007), no. 1, 428–433. MR 2307802, DOI 10.1016/j.jalgebra.2006.12.013
- Jon F. Carlson, Modules and group algebras, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1996. Notes by Ruedi Suter. MR 1393196, DOI 10.1007/978-3-0348-9189-9
- Jon F. Carlson, The variety of an indecomposable module is connected, Invent. Math. 77 (1984), no. 2, 291–299. MR 752822, DOI 10.1007/BF01388448
- Sunil K. Chebolu, J. Daniel Christensen, and Ján Mináč, Groups which do not admit ghosts, Proc. Amer. Math. Soc. 136 (2008), no. 4, 1171–1179. MR 2367091, DOI 10.1090/S0002-9939-07-09058-2
- Sunil K. Chebolu, J. Daniel Christensen, and Ján Mináč. Freyd’s generating hypothesis for groups with periodic cohomology. Preprint, 2007. arXiv:0710.3356
- Sunil K. Chebolu, J. Daniel Christensen, and Ján Mináč, Ghosts in modular representation theory, Adv. Math. 217 (2008), no. 6, 2782–2799. MR 2397466, DOI 10.1016/j.aim.2007.11.008
- Peter Freyd, Stable homotopy, Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965) Springer, New York, 1966, pp. 121–172. MR 0211399
- Mark Hovey, Keir Lockridge, and Gena Puninski, The generating hypothesis in the derived category of a ring, Math. Z. 256 (2007), no. 4, 789–800. MR 2308891, DOI 10.1007/s00209-007-0103-x
- Keir H. Lockridge, The generating hypothesis in the derived category of $R$-modules, J. Pure Appl. Algebra 208 (2007), no. 2, 485–495. MR 2277690, DOI 10.1016/j.jpaa.2006.01.018
Additional Information
- Jon F. Carlson
- Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602
- MR Author ID: 45415
- Email: jfc@math.uga.edu
- Sunil K. Chebolu
- Affiliation: Department of Mathematics, Illinois State University, Normal, Illinois 61790
- Email: schebol@ilstu.edu
- Ján Mináč
- Affiliation: Department of Mathematics, University of Western Ontario, London, ON N6A 5B7, Canada
- Email: minac@uwo.ca
- Received by editor(s): June 12, 2008
- Received by editor(s) in revised form: October 21, 2008
- Published electronically: February 6, 2009
- Additional Notes: The first author is partially supported by a grant from the NSF
The third author is supported by the NSERC - Communicated by: Birge Huisgen-Zimmermann
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 137 (2009), 2575-2580
- MSC (2000): Primary 20C20, 20J06; Secondary 55P42
- DOI: https://doi.org/10.1090/S0002-9939-09-09826-8
- MathSciNet review: 2497468