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

David Benson; John Greenlees

doi:10.1007/s10468-022-10129-2

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

Mathematical Science

Research output: Contribution to journal › Article › peer-review

Abstract

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 language	English
Journal	Algebras and Representation Theory
Early online date	14 May 2022
DOIs	https://doi.org/10.1007/s10468-022-10129-2
Publication status	E-pub ahead of print - 14 May 2022

Keywords

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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10.1007/s10468-022-10129-2Licence: CC BY

Cite this

@article{ec76990d2c494570adc238b06cf0fa59,

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

abstract = "We construct a differential graded algebra (DGA) modelling certain A∞ algebras associatedwith a finite group G with cyclic Sylow subgroups, namely H∗BG and H∗BG∧p. We useour construction to investigate the singularity and cosingularity categories of these algebras.We give a complete classification of the indecomposables in these categories, and describethe 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 algebrasof symmetric groups.",

keywords = "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",

author = "David Benson and John Greenlees",

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 {\textquoteleft}K-theory, algebraic cycles and motivic homotopy theory{\textquoteright} and {\textquoteleft}Groups, representations and applications: new perspectives{\textquoteright} 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. ",

year = "2022",

month = may,

day = "14",

doi = "10.1007/s10468-022-10129-2",

language = "English",

journal = "Algebras and Representation Theory",

issn = "1386-923X",

publisher = "Springer Netherlands",

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TY - JOUR

T1 - The Singularity and Cosingularity Categories of C ∗ B G for Groups with Cyclic Sylow p-subgroups

AU - Benson, David

AU - Greenlees, John

N1 - 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.

PY - 2022/5/14

Y1 - 2022/5/14

N2 - We construct a differential graded algebra (DGA) modelling certain A∞ algebras associatedwith a finite group G with cyclic Sylow subgroups, namely H∗BG and H∗BG∧p. We useour construction to investigate the singularity and cosingularity categories of these algebras.We give a complete classification of the indecomposables in these categories, and describethe 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 algebrasof symmetric groups.

AB - We construct a differential graded algebra (DGA) modelling certain A∞ algebras associatedwith a finite group G with cyclic Sylow subgroups, namely H∗BG and H∗BG∧p. We useour construction to investigate the singularity and cosingularity categories of these algebras.We give a complete classification of the indecomposables in these categories, and describethe 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 algebrasof symmetric groups.

KW - A∞ algebras

KW - Auslander–Reiten quiver

KW - Auslander–Reiten triangles

KW - Brauer trees

KW - Cyclic Sylow subgroups

KW - Cohomology of groups

KW - Cosingularity categories

KW - Derived categories

KW - DG Hopf algebras

KW - Hecke algebras

KW - Hochschild cohomology

KW - Loop spaces

KW - Massey products

KW - p-completed classifying spaces ·

KW - Singularity categories

KW - Spectral sequences

U2 - 10.1007/s10468-022-10129-2

DO - 10.1007/s10468-022-10129-2

M3 - Article

JO - Algebras and Representation Theory

JF - Algebras and Representation Theory

SN - 1386-923X

ER -

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

Abstract

Keywords

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