Wild kernels for higher K-theory of division and semi-simple algebras

Let SIGMA be a semi-simple algebra over a number field F. In this paper, we prove that for all n >= 0, the wild kernel WK sub n (SIGMA):Ker(K sub n (SIGMA) -> PI sub f sub i sub n sub i sub t sub e subupsilon K sub n (SIGMA subupsilon)) is contained in the torsion part of the image of the natural homomorphism K sub n (LAMBDA) -> K sub n (SIGMA), where LAMBDA is a maximal order in SIGMA. In particular, WK sub n (SIGMA) is finite. In the process, we prove that if LAMBDA is a maximal order in a central division algebra D over F, then the kernel of the reduction map K sub 2 sub n sub - sub 1 (LAMBDA) -> suppi sup subupsilon PI sub f sub i sub n sub i sub t sub e subupsilon K sub 2 sub n sub - sub 1 (d subupsilon) is finite. In paragraph 3 we investigate the connections between WK sub n (D) and div(K sub n (D)) and prove that divK sub 2 (SIGMA) is a subset of WK sub 2 (SIGMA); if the index of D is square free, then div(K sub 2 (D)) approx = div(K sub 2 (F)), WK sub 2 (F) approx = WK sub 2 (D) and vertical bar WK sub 2 (D)/div(K sub 2 (D)) vertical bar <= 2. Finally we prove that if D is a central division algebra over F with [D : F] = m sup 2 , then (1) div(K sub n (D)) sub l = WK sub n (D) sub l for all odd primes I and n <= 2; (2) if I does not divide m, then div(K sub 3 (D)) sub l = WK sub 3 (D) sub l = 0; (3) if F = Q and I does not divide m, then div(K sub n (D)) sub l is a subset of WK sub n (D) sub l for all n.