## Ideals in triangulated categories: Phantoms, ghosts and skeleta (1998)

Venue: | Adv. in Math |

Citations: | 41 - 5 self |

### BibTeX

@ARTICLE{Christensen98idealsin,

author = {J. Daniel Christensen},

title = {Ideals in triangulated categories: Phantoms, ghosts and skeleta},

journal = {Adv. in Math},

year = {1998}

}

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### OpenURL

### Abstract

ABSTRACT. We begin by showing that in a triangulated category, specifying a projective class is equivalent to specifying an ideal I of morphisms with certain properties, and that if I has these properties, then so does each of its powers. We show how a projective class leads to an Adams spectral sequence and give some results on the convergence and collapsing of this spectral sequence. We use this to study various ideals. In the stable homotopy category we examine phantom maps, skeletal phantom maps, superphantom maps, and ghosts. (A ghost is a map which induces the zero map of homotopy groups.) We show that ghosts lead to a stable analogue of the Lusternik–Schnirelmann category of a space, and we calculate this stable analogue for low-dimensional real projective spaces. We also give a relation between ghosts and the Hopf and Kervaire invariant problems. In the case of A ∞ modules over an A ∞ ring spectrum, the ghost spectral sequence is a universal coefficient spectral sequence. From the phantom projective class we derive a generalized Milnor sequence for filtered diagrams of finite spectra, and from this it follows that the group of phantom maps from X to Y can always be described as a lim1 ←− group. The last two sections focus