Mod-m and profinite higher K-theory of exact categories, orders, and groupings

Let be a rational prime, C an exact category. In this chapter we define and study for all n ≥ 0, the profinite higher K-theory of C, that is Kprn (C, Zˆ) := [Mn+1∞ , BQ(C)], as well as Kn(C, Zˆ) := lim←−s [M

n+1 ∞ , BQ(C)], where Mn+1∞ :=

S , and Mn+1

S is the (n + 1)-dimensional mod-S Moore space. We

study connections between Kprn (C, Zˆ) and Kn(C, Zˆ) and prove several - completeness results involving these and associated groups, including the cases where C = M(Λ) (resp. P(Λ)) is the category of finitely generated (resp. finitely generated projective) modules over orders Λ in semi-simple algebras over number fields and p-adic fields. We also define and study continuous K-theory Kcn(Λ)(n ≥ 1) of orders Λ in p-adic semi-simple algebras and show some connections between the profinite and continuous K-theory of Λ. The results in this chapter are due to A.O. Kuku (see [117]).