ON FREYD'S GENERATING HYPOTHESIS

Freyd's generating hypothesis in stable homotopy theory is revisited and new consequences and equivalent forms of it are derived. A surprising such consequence is that I, the Brown–Comenetz dual of the sphere and the source of many counterexamples in stable homotopy, is the cofibre of a self-map of a wedge of spheres. It is also shown that a consequence of the generating hypothesis, that the homotopy of a finite spectrum that is not a wedge of spheres can never be finitely generated as a module over π
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S, is in fact true for many finite torsion spectra.