By Neittaanmaeki P., Mäkelä M.M.

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Homomorphisms defined on as a map on nrA and nr2[ while K1(-), are used here to denote the nrA,,F denotes the reduced norm * e. The following lemma will be useful when computing Ker(nrA). 2 A and let Let be any E F-algebra. Ker[K1(i): (where i(x) = 1®x) Proof be any finite field extension of degree F Then : is central in trf o K1(i) A If v of IR 0vK A K n. 18, the composite Kl(E ®F A) ) K1(A) K1(A) is induced by tensoring with F ) K1(E ®F A)] K1(A) has exponent dividing trf o K1(i) Since n, A, is multiplication by E ®F A, regarded as an E ®F A = An n.

Im,In] U(I) _ Z a 2p submodule of fn E nfR R. for all n, / If I C fR, (5) C [%,IQ], m,n>1 where f E Z(R) and fP E pER, then and m+n \ U(I) = ([r, n s] : m,n> 1, fmrE lm, fnsE ln, fr,fsE I) C [R,I]. So congruences (1) and (2) will both follow, once we have shown the STRUCTURE THEOREMS FOR K1 OF ORDERS CHAPTER 2. 53 relation (mod Log((l+x)(1+y)) = Log(l+x) + Log(l+y) U(I)) (6) for any I and any x,y E I. For ,each n> 1, length n let w C(w) = orbit of be the set of formal (ordered) monomials of W.

For some bimodule 0S, MORN=S and : as bimodules. 9, Mn(D) are inverse For any ring mn R (Mn(R),R)-bimodule. are precisely Ki(Mn(R)) = Ki(R) with CL those induced by identifying GLm(S) In Theorem N ®S M=R K1(S). The simplest example of this is a matrix algebra. and any n > 1, A is an "invertible" S (R). that any maximal 2P order in a simple a division algebra) is conjugate to a matrix (D algebra over the maximal order in D. This is not the case for maximal Z-orders in simple Q-algebras; but a result which is almost as good can be stated in terms of Morita equivalence.

### Nonsmooth Optimization by Neittaanmaeki P., Mäkelä M.M.

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