For a finite group G, completion at the augmentation ideal I(G) of the Burnside ring A(G) features in cohomological completion theorems such as the Atiyah–Segal completion theorem and Carlsson's theorem (formerly the Segal Conjecture). Completion of G-spectra at ideals of the Burnside ring was introduced by Greenlees–May to obtain topological versions of these theorems, letting the completion take place at the level of spectra rather than cohomology groups. When G is a p-group, May–McClure showed that I(G)-adic completion coincides with p-completion (modulo some basepoint issues) but in general I(G)-adic completion seems very complicated.
In this talk I will describe how, for the purposes of I(G)-completion, one may replace a G-spectrum E by a related G-spectrum E' whose isotropy groups are all groups of prime power order. The I(G)-adic completion of the latter is often more approachable, admitting a description as a colimit of completions at primes. As an application I will give a simplified description of the function spectrum F(BG,BK) for a finite group and a compact Lie group K.
