Skip to main content Accessibility help
×
Home
Hostname: page-component-59b7f5684b-n9lxd Total loading time: 0.25 Render date: 2022-10-02T01:33:26.190Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "displayNetworkTab": true, "displayNetworkMapGraph": false, "useSa": true } hasContentIssue true

Freyd's Generating Hypothesis for Groups with Periodic Cohomology

Published online by Cambridge University Press:  20 November 2018

Sunil K. Chebolu
Affiliation:
Department of Mathematics, Illinois State University, Normal, IL 61761, U.S.A. e-mail: schebol@ilstu.edu
J. Daniel Christensen
Affiliation:
Department of Mathematics, University of Western Ontario, London, ON N6A 5B7 e-mail: jdc@uwo.ca minac@uwo.ca
Ján Mináč
Affiliation:
Department of Mathematics, University of Western Ontario, London, ON N6A 5B7 e-mail: jdc@uwo.ca minac@uwo.ca
Rights & Permissions[Opens in a new window]

Abstract

HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let $G$ be a finite group, and let $k$ be a field whose characteristic $p$ divides the order of $G$. Freyd's generating hypothesis for the stable module category of $G$ is the statement that a map between finite-dimensional $kG$-modules in the thick subcategory generated by $k$ factors through a projective if the induced map on Tate cohomology is trivial. We show that if $G$ has periodic cohomology, then the generating hypothesis holds if and only if the Sylow $p$-subgroup of $G$ is ${{C}_{2}}$ or ${{C}_{3}}$. We also give some other conditions that are equivalent to the $\text{GH}$ for groups with periodic cohomology.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2012

References

[1] Alperin, J. L., Local representation theory. Modular representations as an introduction to the local representation theory of finite groups. Cambridge Studies in Advanced Mathematics, 11, Cambridge University Press, Cambridge, 1986.CrossRefGoogle Scholar
[2] Benson, D. J., Cohomology of modules in the principal block of a finite group. New York J. Math. 1(1994/95), 196205, electronic.Google Scholar
[3] Benson, D. J., Representations and cohomology. I. Basic representation theory of finite groups and associative algebras. Second ed., Cambridge Studies in Advanced Mathematics, 30, Cambridge University Press, Cambridge, 1998.Google Scholar
[4] Benson, D. J., Carlson, J. F., and Robinson, G. R., On the vanishing of group cohomology. J. Algebra 131(1990), no. 1, 4073. doi:10.1016/0021-8693(90)90165-KCrossRefGoogle Scholar
[5] Benson, D. J., Chebolu, S. K., Christensen, J. D., and Mináč, Ján, The generating hypothesis for the stable module category of a p-group. J. Algebra 310(2007), no. 1, 428433. doi:10.1016/j.jalgebra.2006.12.013CrossRefGoogle Scholar
[6] Carlson, J. F., Modules and group algebras. Notes by Ruedi Suter. Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1996.CrossRefGoogle Scholar
[7] Carlson, J. F., Chebolu, S. K., and Mináč, J., Freyd's generating hypothesis with almost split sequences. Proc. Amer. Math. Soc. 137(2009), no. 8, 25752580. doi:10.1090/S0002-9939-09-09826-8CrossRefGoogle Scholar
[8] Carlson, J. F., Chebolu, S. K., and Mináč, J., Finite generation of Tate cohomology. Represent. Theory 15(2011), 244257.CrossRefGoogle Scholar
[9] Cartan, H. and Eilenberg, S., Homological algebra. Reprint of the 1956 original. Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1999.Google Scholar
[10] Chebolu, S. K., Christensen, J. D., and Mináč, J., Ghosts in modular representation theory. Adv. Math. 217(2008), no. 6, 27822799. doi:10.1016/j.aim.2007.11.008CrossRefGoogle Scholar
[11] Chebolu, S. K., Christensen, J. D., and Mináč, J., Groups which do not admit ghosts. Proc. Amer. Math. Soc. 136(2008), no. 4, 11711179. doi:10.1090/S0002-9939-07-09058-2CrossRefGoogle Scholar
[12] Curtis, C. W. and Reiner, I., Methods of representation theory. I. With applications to finite groups and orders. Reprint of the 1981 original, Wiley Classics Library, JohnWiley & Sons Inc., New York, 1990.Google Scholar
[13] Curtis, C. W. and Reiner, I., Representation theory of finite groups and associative algebras. Reprint of the 1962 original,Wiley Classics Library, John Wiley & Sons Inc., New York, 1988.Google Scholar
[14] Freyd, P., Stable homotopy. In: 1966 Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965), Springer, New York, 1966, pp. 121172.CrossRefGoogle Scholar
[15] The GAP Group, GAP—Groups, Algorithms, and Programming, version 4.4.9, 2006. http://www.gap-system.org Google Scholar
[16] Hovey, M., Lockridge, K., and Puninski, G., The generating hypothesis in the derived category of a ring. Math. Z. 256(2007), no. 4, 789800. doi:10.1007/s00209-007-0103-xCrossRefGoogle Scholar
[17] Külshammer, B., The principal block idempotent. Arch. Math. 56(1991), no. 4, 313319.CrossRefGoogle Scholar
[18] Lockridge, K. H., The generating hypothesis in the derived category of R-modules. J. Pure Appl. Algebra 208(2007), no. 2, 485495. doi:10.1016/j.jpaa.2006.01.018CrossRefGoogle Scholar
[19] Ringel, C. M. and Tachikawa, H., QF-3 rings. J. Reine Angew. Math. 272(1974), 4972.Google Scholar
[20] Swan, R. G., Groups with periodic cohomology. Bull. Amer. Math. Soc. 65(1959), 368370. doi:10.1090/S0002-9904-1959-10378-5CrossRefGoogle Scholar
[21] Webb, P. J., The Auslander-Reiten quiver of a finite group. Math. Z. 179, no. 1, 97121. doi:10.1007/BF01173918CrossRefGoogle Scholar
[22] Webb, P. J., Reps—a GAP package for modular representation theory, 2007. http://www.math.umn.edu/»webb/GAPfiles/ Google Scholar
You have Access
3
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Freyd's Generating Hypothesis for Groups with Periodic Cohomology
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

Freyd's Generating Hypothesis for Groups with Periodic Cohomology
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

Freyd's Generating Hypothesis for Groups with Periodic Cohomology
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *