The Singularity and Cosingularity Categories of C ∗ B G for Groups with Cyclic Sylow p-subgroups

David Benson* (Corresponding Author), John Greenlees

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


We construct a differential graded algebra (DGA) modelling certain A∞ algebras associated with a finite group G with cyclic Sylow subgroups, namely H∗BG and H∗BG∧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.
Original languageEnglish
Pages (from-to)1181-1216
Number of pages36
JournalAlgebras and Representation Theory
Issue number4
Early online date14 May 2022
Publication statusPublished - 1 Aug 2023

Bibliographical note

Open Access via the Springer Compact Agreement.

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.

Data Availability Statement

Data Sharing is not applicable to this article as no datasets were generated or analysed during the current study.


  • A∞ 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


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