G-spectra of cyclic defect

Broue's conjecture predicts that a block of Z_p[G] and of Z_p[N] have equivalent derived categories, when N is the normalizer of the defect group of the block and this defect group is commutative. In case the defect group is cyclic, there are two old proofs of it, one by Rickard and a while later one by Rouquier. Tony Feng, Allen Yuan and I can construct a similar equivalence between blocks of p-complete G-spectra and p-complete N-spectra, that induces Rouquier's equivalence after taking homology.