Generation of D^b(C^*(BG)) by Hom_{C_*G}(k, X)

Establish that the bounded derived category D^b(C^*(BG)) of the cochains algebra C^*(BG) on the classifying space BG of a compact Lie group G is generated, as a thick subcategory of D(C^*(BG)), by the set of modules Hom_{C_*G}(k, X) with X ranging over the bounded derived category D^b(C_*G) of the chains algebra C_*G.

Background

This conjecture is adapted from Benson and Greenlees (their Conjecture 1.6 for finite groups) to the compact Lie group setting. It asserts a specific set of generators for the bounded derived category of cochains, expressed via Hom from bounded chain modules.

In the paper, the authors show that this conjecture is equivalent to the companion conjecture asserting generation by C^*(BT), where T is a maximal torus. They also provide a sufficient condition under which the conjecture holds (namely, when D^b(C^*(BG)) is generated by formal modules).