  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-infinity modules over an A-infinity 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 lim^1 group. The last two
sections focus on algebraic examples. In the derived category of an abelian
category we study the ideal of maps inducing the zero map of homology groups
and find a natural setting for a result of Kelly on the vanishing of composites
of such maps. We also explain how pure exact sequences relate to phantom maps
in the derived category of a ring and give an example showing that phantoms can
compose non-trivially.
