The equivariant generating hypothesis (Q969675)

Let \(G\) be a compact Lie group. In this paper the Freyd generating hypothesis is formulated in the \(G\)-equivariant context as the statement that if a map \(f\) between finite \(G\)-spectra induces the zero map in equivariant homotopy, then the map \(f\) is (equivariantly) nullhomotopic. Equivalently, the equivariant homotopy functor is faithful when restricted to finite \(G\)-spectra. The original Freyd conjecture is recovered by taking the group \(G\) to be the trivial group. The major achievement of this paper is to establish Freyd's ``faithful implies full'' result for the case where \(G\) is a finite group. In \(\S\)2, the author follows Freyd in showing that ``in any triangulated category where a version of the generating hypothesis holds, the generating hypothesis extends to the abelian envelope of that category.'' The proof that the weak generating hypothesis implies the strong generating hypothesis when \(G\) is a finite group is contained in \(\S\)3. Finally, in \(\S\)4, an explicit counter-example to the Freyd conjecture in the rational \(S^1\)-equivariant setting is given by applying results of Greenlees to free rational \(S^1\)-spectra.