Comparing Completions of a Space at a Prime

There are two practical ways to complete an abelian group A at a prime p. One could complete A with respect to the neighborhood base of zero given by the subgroups pA. This completion, called p-completion, is limA/pA. Or one could complete A with respect to the neighborhood base of zero given by subgroups B ⊆ A of finite p-power index. This completion, called p-pro-finite completion, is limAα where Aα runs over the finite p-group quotients of A. They agree for finitely generated groups, but not in general: if V is an Fp vector space it is p-complete, but its p-pro-finite completion is the double dual V ∗∗. Also, the former, p-completion, is easier to define, but it is neither left nor right exact and the category of p-complete groups is not abelian – it is not closed under cokernels. The latter, p-pro-finite completion, is initially less tractable, but it is right exact and the category of p-pro-finite abelian groups is an abelian category. There are two corresponding completions for topological spaces. The analog of pcompletion is Bousfield-Kan completion [3] and the analog of p-pro-finite completion is related to Sullivan’s pro-finite completion and has recently been given a homotopical definition by Morel [13]. The purpose of this note is to compare these two completions; in addition, we seek to give Morel’s completion the same sort of computational footing that the Bousfield-Kan completion enjoys. To underscore the similarities and differences, let me make some remarks on how these completions of spaces are constructed. If X is a space, the mod p homology H∗X = H∗(X,Fp) is a graded coalgebra over Fp and, as such, there is an isomorphism colim α Cα ∼= H∗X, where Cα ⊆ H∗X runs over the filtered system of sub-coalgebras which are finite dimensional in each degree. Thus there is an isomorphism in cohomology, H∗X ∼= lim α C∗ α, where C ∗ α is the algebra dual to Cα. In short, H ∗X is a pro-finite algebra—which is to say the inverse limit equips H∗X with a topology—and if X → Y is a map of spaces, H∗Y → H∗X is a continuous map of algebras. The Bousfield-Kan completion of X may be obtained by the following program. Define a tower of fibrations · · ·−→ Xs qs −→Xs−1−→ · · ·−→ X1 q1 −→X0 The author was partially supported by the National Science Foundation.