Phantom Maps and Chromatic Phantom Maps

In the first part, we determine conditions on spectra X and Y under which either every map from X to Y is phantom, or no nonzero maps are. We also address the question of whether such all or nothing behaviour is preserved when X is replaced with V ∧ X for V finite. In the second part, we introduce chromatic phantom maps. A map is n-phantom if it is null when restricted to finite spectra of type at least n. We define divisibility and finite type conditions which are suitable for studying n-phantom maps. We show that the duality functor Wn−1 defined by Mahowald and Rezk is the analog of Brown-Comenetz duality for chromatic phantom maps, and give conditions under which the natural map Y → W 2 n−1Y is an isomorphism.