Stable splittings of classifying spaces of compact Lie group (Q1374869)

Let \(G\) be a compact Lie group. In this paper the author studies the stable splitting of \(BG\) completed at \(p\) into a wedge sum of indecomposable spectra. In section 2 he specializes to the case of the classifying space of a compact Lie group. In section 3 the author proves that when the toral compact Lie group is Roquette, then there exists a semigroup ring which detects all the primitive idempotents corresponding to the original summands (an indecomposable summand of \(BG\) is said to be a \(p\) original, if it is not a summand of \(BH\) for every proper toral subgroup \(H\) of \(G\)). A nilpotence result in the above semigroup allows one to conclude that the primitive idempotents in the semigroup ring actually come from a subring which happens to be a group ring. This is a central result of section 4. In section 5 the author formulates a necessary and sufficient condition for a summand originating in \(BH\) to be a summand of \(BG\) in terms of modular representation theory.