Non-existence of a stable homotopy category for p-complete abelian groups

We investigate the existence of a stable homotopy category (SHC) associated to the category of p-complete abelian groups Ab∧p . First we examine Ab∧p and prove Ab∧p satisfies all but one of the axioms of an abelian category. The connections between an SHC and homology functors are then exploited to draw conclusions about possible SHC structures for Ab∧p . In particular, let K&parl0;Ab∧p &parr0; denote the category whose objects are chain complexes of Ab∧p and morphisms are chain homotopy classes of maps. We show that any homology functor from any subcategory of K&parl0;Ab∧p &parr0; containing the p-adic integers and satisfying the axioms of an SHC will not agree with standard homology on free, finitely generated (as modules over the p-adic integers) chain complexes. Explicit examples of common functors are included to highlight troubles that arrise when working with Ab∧p . We make some first attempts at classifying small objects in K&parl0;Ab∧p &parr0; .