Abstract
We construct a differential graded algebra (DGA) modelling certain \(A_{\infty }\) algebras associated with a finite group G with cyclic Sylow subgroups, namely H∗BG and \(H_{*}{\Omega } BG{^{^{\wedge }}_p}\). We use our construction to investigate the singularity and cosingularity categories of these algebras. We give a complete classification of the indecomposables in these categories, and describe the Auslander–Reiten quiver. The theory applies to Brauer tree algebras in arbitrary characteristic, and we end with an example in characteristic zero coming from the Hecke algebras of symmetric groups.
Article PDF
Similar content being viewed by others
References
Adams, J. F.: Lectures on generalised cohomology, Category Theory, Homology Theory and their Applications, III, Lecture Notes in Mathematics, vol. 99, pp 1–138. Springer, New York (1969)
Alperin, J. L.: Local representation theory, Cambridge Studies in Advanced Mathematics, vol. 11. Cambridge University Press, Cambridge (1986)
Amiot, C.: On the structure of triangulated categories with finitely many indecomposables. Bull. Soc. Math. France 135(3), 435–474 (2007)
Amiot, C.: Sur Les Petites Catégories Triangulées. Thèse de Doctorat, Université Paris VII (2008)
Ariki, S.: Representation type for block algebras of Hecke algebras of classical type. Adv. Math. 317, 823–845 (2017)
Auslander, M., Reiten, I.: Representation theory of Artin algebras IV: Invariants given by almost split sequences. Comm. Algebra 5(5), 443–518 (1977)
Benson, D. J.: Representations and Cohomology II: Cohomology of Groups and Modules Cambridge Studies in Advanced Mathematics, 2nd edn., vol. 31. Cambridge University Press, Cambridge (1998)
Benson, D. J., Erdmann, K., Mikaelian, A.: Cohomology of Hecke algebras. Homology. Homotopy & Appl. 12(2), 353–370 (2010)
Benson, D. J., Greenlees, J. P. C.: Massey products in the homology of the loopspace of a p-completed classifying space: finite groups with cyclic Sylow p-subgroups. Proc. Edinb. Math. Soc. 64(4), 908–915 (2021)
Boardman, J.M.: Conditionally convergent specral sequences, Homotopy invariant algebraic strucurs. In: Meyer, J.-P., Morava, J., Wilson, W.S. (eds.) Contemp. Math., vol. 239, pp 49–84. American Math. Society (1999)
Bogdanic, D.: Graded Brauer tree algebras. J. Pure & Applied Algebra 214, 1534–1552 (2010)
Bongartz, K., Gabriel, P.: Covering spaces in representation-theory, Invent. Math. 65(3), 331–378 (1982)
Brauer, R.: Investigations on group characters. Ann. Math. 42(4), 936–958 (1941)
Buchweitz, R.-O., Roberts, C.: The multiplicative structure on Hochschild cohomology of a complete intersection. J. Pure & Applied Algebra 219, 402–428 (2015)
uijs, U. , Moreno-Fernández, J.M., Murillo, A.: \(A_{\infty }\) structures and Massey products. Mediterr. J. Math. 17(1), 15 (2020). Paper No. 31
Dade, E. C.: Blocks with cyclic defect groups. Ann. of Math. 84, 20–48 (1966)
Ene, V., Popescu, D.: On the structure of maximal Cohen-Macaulay modules over the ring k[[x,y]]/(xn). Algebras and Representation Theory 11, 191–205 (2008)
Gabriel, P., Riedtmann, C.: Group representations without groups. Comment. Math. Helvetici 54, 240–287 (1979)
Geck, M.: Brauer trees of Hecke algebras. Comm Algebra 20(10), 2937–2973 (1992)
Getzler, E., Jones, J. D. S.: \(A_{\infty }\)-algebras and the cyclic bar complex. Illinois J. Math. 34(2), 256–283 (1990)
Green, J. A.: Walking around the Brauer tree. J. Austral. Math. Soc. 17, 197–213 (1974)
Greenlees, J. P. C., May, J. P.: Generalized Tate cohomology. Mem. AMS 113, 543 (1995). American Math Society
Greenlees, J. P. C., Stevenson, G.: Morita theory and singularity categories. Adv. in Math. 365(107055), 51 (2020)
Greenlees, J.P.C., Stojanoska, V. : Anderson and Gorenstein duality, Geometric and topological aspects of the representation theory of finite groups. In: Carlson, J.F., Iyengar, S.B., Pevtsova, J. (eds.) Springer Proc. Math. Stat., vol. 242, pp 105–130. Springer, New York (2018)
Happel, D.: On the derived category of a finite-dimensional algebra. Comment Math. Helvetici 62, 339–389 (1987)
Happel, D.: Triangulated categories in the representation theory of finite dimensional algebras, London Math. Soc. Lecture Note Series, vol. 119. Cambridge University Press, Cambridge (1988)
Hovey, M.: Model categories, Mathematical Surveys and Monographs, vol. 63. American Math Society, American (1999)
Kadeishvili, T. V.: The algebraic structure in the homology of an \(a(\infty )\)-algebra. Soobshch. Akad. Nauk Gruzin. SSR 108, 249–252 (1982)
Keller, B.: Derived invariance of higher structures on the Hochschild complex Preprint (2018)
Keller, B.: On the construction of triangle equivalences, Derived equivalences for group rings. In: König, Zimmermann (eds.) Lecture Notes in Mathematics, vol. 1685, pp 155–176. Springer, New York (1998)
Keller, B.: Introduction to A-infinity algebras and modules Homology. Homotopy & Appl. 3, 1–35 (2001). — Addendum, ibid. 4 (2002), 25–28
Keller, B. In: Happel, D., Zhang, Y.B. (eds.) : A-Infinity Algebras in Representation Theory, Representations of Algebras, Proceedings of the Ninth International Conference (Beijing, vol. 2000. Normal University Press, Beijing (2002). Vol. I
Keller, B.: On triangulated orbit categories. Doc. Math. 10, 551–581 (2005)
Keller, B.: A-infinity algebras, modules and functor categories, Trends in Representation Theory of Algebras and Related Topics. In: de la Peña, J.A., Bautista, R. (eds.) Contemp. Math., vol. 406, pp 67–93 (2006)
Keller, B.: On differential graded categories, International Congress of Mathematicians. Eur. Math. Soc. Zürich II, 151–190 (2006)
Keller, B.: A remark on Hochschild cohomology and Koszul duality, Advances in the Representation Theory of Algebras. In: Assem, I., Geiß, C., Trepode, S. (eds.) Contemp. Math., vol. 761, pp 131–136 (2021)
Lefèvre-Hasegawa, K.: Sur Les \(A_{\infty }\)-Catégories, Thèse de doctorat, Université Paris VII (2003)
Lu, D. -M., Palmieri, J. H., Wu, Q. -S., Zhang, J. J.: A-infinity structure on Ext-algebras. J. Pure & Applied Algebra 213(11), 2017–2037 (2009)
Rickard, J.: Derived categories and equivalence stable. J. Pure & Applied Algebra 61, 303–317 (1989)
Riedtmann, C.: Algebren, darstellungsköcher, Ueberlagerungen und zurück. Comment. Math. Helvetici 55, 199–224 (1980)
Roitzheim, C., Whitehouse, S.: Uniqueness of \(a_{\infty }\)-structures and Hochschild cohomology. Algebr. Geom. Topol. 11, 107–143 (2011)
Stasheff, J.D.: H-spaces from a Homotopy Point of View Lecture Notes in Mathematics, vol. 161. Springe, New York (1970)
Xiao, J., Zhu, B.: Relations for the Grothendieck groups of triangulated categories. J. Algebra 257, 37–50 (2002)
Acknowledgements
The authors are grateful to the Engineering and Physical Sciences Research Council (EPSRC): the second author is supported by grant EP/P031080/1, which also enabled the first author to visit Warwick. The authors would also like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for providing an opportunity to work on this project during the simultaneous programmes ‘K-theory, algebraic cycles and motivic homotopy theory’ and ‘Groups, representations and applications: new perspectives’ supported by EPSRC grant EP/R014604/1 (one author was supported by each programme). The authors are also grateful to Bernhard Keller and Greg Stevenson for conversations related to this work.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of Interests
There are no conflicts of interest associated with this work.
Additional information
Presented by: Andrew Mathas
Data Sharing
is not applicable to this article as no datasets were generated or analysed during the current study.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Benson, D., Greenlees, J. The Singularity and Cosingularity Categories of C∗BG for Groups with Cyclic Sylow p-subgroups. Algebr Represent Theor 26, 1181–1216 (2023). https://doi.org/10.1007/s10468-022-10129-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10468-022-10129-2
Keywords
- \(A_{\infty }\) algebras
- Auslander–Reiten quiver
- Auslander–Reiten triangles
- Brauer trees
- Cyclic Sylow subgroups
- Cohomology of groups
- Cosingularity categories
- Derived categories
- DG Hopf algebras
- Hecke algebras
- Hochschild cohomology
- Loop spaces
- Massey products
- p-completed classifying spaces
- Singularity categories
- Spectral sequences