Phantom maps and chromatic phantom maps

Abstract

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 smash 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 W_{n-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_{n-1}^2 Y is an isomorphism.