PHANTOM MAPS AND HOMOLOGY THEORIES

We study phantom maps and homology theories in a stable homotopy categ ory L via a certain Abelian category A. We express the group P(X, Y) o f phantom maps X --> Y as an Ext group in A, and give conditions on X or Y which guarantee that it vanishes. We also determine P(X, HB). We show that any composite of two phantom maps is zero, and use this to r educe Margolis's axiomatisation conjecture to an extension problem. We show that a certain functor L --> A is the universal example of a hom ology theory with values in an AB 5 category and compare this with som e results of Freyd. (C) 1997 Elsevier Science Ltd. All rights reserved .