$G$-spectra of cyclic defect

Abstract

Broué's Abelian Defect Conjecture predicts interesting derived equivalences between derived categories of modular representations of finite groups. We investigate a generalization of Broué's Conjecture to ring spectrum coefficients and prove this generalization in the cyclic defect case, following an argument of Rouquier.2. Broué's Conjecture over ring spectra 2.1. Review of modular representation theory. This is a paper about finite group actions on p-complete spectra, but we will start this extended introduction by reviewing an example and some of the theory of finite group actions on p-adic abelian groups. Three basic questions in modular representation theory are:(1) What are the simple representations over a p-adic field of characteristic 0? (2) What are the simple representations over a finite field of characteristic p?(3) What are the projective representations in characteristic p, or over a p-adic ring of integers?The answer to these questions, and some information about blocks, defects, and Brauer trees, are given below in the case G = PSL 2 (F 7 ) and p = 7. We will use this example to illustrate some of the concepts of modular representation theory. Conventionally, one chooses the coefficient rings and fields to have sufficiently many roots of unity. We wish to avoid adding pth roots of unity (or 4th roots when p = 2) to our coefficients for algebraic topology reasons we'll come to later §2.4. We describe the representation theory of PSL 2 (F 7 ) over Z 7 and Q 7 .2.1.1. Irreducible Q 7 [PSL 2 (F 7 )]-modules. Let G = PSL 2 (F 7 ) be the simple group of order 168 and let p = 7. There are 5 irreducible representations of G on Q 7 -vector spaces that we will call