Some results on higher K-theory of orders, grouprings, and modules over ‘EI’ categories

Recall that if R is a Dedekind domain with quotient field F , and Λ is any R-order in a semi-simple F -algebra Σ, then SKn(Λ) := Ker (Kn(Λ)) → Kn(Σ) and SGn(Λ) := Ker (Gn(Λ)) → Gn(Σ). Also, any R-order in a semisimple F -algebra Σ can be embedded in a maximal R-order Γ, which has well-understood arithmetic properties relative to Σ. More precisely, if Σ =∏r

i=1 Mni(Di), then Γ is Morita equivalent to ∏r

i=1 Mni(Γi) where Γi are maximal orders in the division algebra Di, and so, Kn(Γ) ≈ ⊕Kn(Γi) while Kn(Σ) ≈

Π i=1

Kn(Di). So, the study of K-theory of maximal orders in a

semi-simple algebras can be reduced to the K-theory of maximal orders in division algebras.