$G$-spectra of cyclic defect

TL;DR

This paper extends Broué's Abelian Defect Conjecture to ring spectrum coefficients, proving it for cyclic defect cases using Rouquier's approach, thereby advancing the understanding of derived equivalences in modular representation theory.

Contribution

The paper generalizes Broué's Conjecture to ring spectrum coefficients and proves it specifically for cyclic defect cases, providing new insights into derived equivalences.

Findings

Proves the generalized conjecture for cyclic defect cases

Extends the scope of Broué's Conjecture to ring spectrum coefficients

Utilizes Rouquier's argument to establish the result

Abstract

Brou\'{e}'s Abelian Defect Conjecture predicts interesting derived equivalences between derived categories of modular representations of finite groups. We investigate a generalization of Brou\'{e}'s Conjecture to ring spectrum coefficients and prove this generalization in the cyclic defect case, following an argument of Rouquier.