Ideals in triangulated categories: phantoms, ghosts and skeleta. 

Ph. D. Thesis, J.D. Christensen, 1997. 
To appear in Advances in Mathematics. 52 pages. 

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.