The singularity and cosingularity categories of $C^*BG$ for groups with cyclic Sylow $p$-subgroups

TL;DR

This paper constructs a differential graded algebra model for certain cohomology and homology algebras of finite groups with cyclic Sylow p-subgroups, analyzing their singularity categories and classifying indecomposables.

Contribution

It introduces a new DGA model for $H^*BG$ and $H_*\, ext{Omega} BG^{^ ext{wedge}}_p$, enabling classification of indecomposables and analysis of singularity categories for these algebras.

Findings

Complete classification of indecomposables in singularity and cosingularity categories.

Description of the Auslander--Reiten quiver for these categories.

Application to Brauer tree algebras and an example from Hecke algebras.

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.