process

106 F. Muro and A. TonksWe generalize (R9 0 ) in the following lemma.Lemma 4.9. Let C be a Waldhausen category with a functorial coproduct and letA 1 ;:::;A n be objects in C. Given a permutation 2 Sym.n/ of n elements we denotebyŠ A1 ;:::;A nW A 1 CCA n ! A1 CCA nthe isomorphism permuting the factors of the coproduct. Then the following formulaholds in D C C:Œ A1 ;:::;A n D X i>j i < jhŒA i ; ŒA j i:Proof. The result holds for n D 1 by (R4). Suppose 2 Sym.n/ for n 2, and notethat the isomorphism A1 ;:::;A nfactors naturally as. 0 A 1 ;:::;A n 1C 1/.1 C An ;B/W A 1 CCA n ! A 01CCA 0n 1C A n! A 1 CCA n 1 C A nwhere . 0 1 ;:::;0 n 1 ;n/ D . 1;:::; b k ;:::; n ; k / and B D A kC1 C C A n .Therefore by (R4), (R6) and Lemma 4.8,Œ A1 ;:::;A n D ŒA 0 1 ;:::;A n 1C 1 C Œ1 C An ;BD .ŒA 0 1 ;:::;A n 1 ŒAn C 0/ C .0 C Œ An ;B/:By induction, (R9 0 ) and Remark 4.1 this is equal toXhŒA 0 p; ŒA 0 qiChŒB; ŒA n iDX hŒA i ; ŒA j iCi>j i < j q 0 p < 0 qas required.5 Proof of Theorem 2.1X i>j i < j DnhŒA i ; ŒA j iThe morphism of stable quadratic modules D F W D C ! D D induced by an exactfunctor F W C ! D, see [4], takes pairs of weak cofiber sequences to pairs of weakcofiber sequences,8ˆ= ˆ;C 2ˆ:F .A/ F.B/9F.C 1 / >=F.C/ : F.C 2 />;By [4, Theorem 3.2] exact equivalences of Waldhausen categories induce homotopyequivalences of stable quadratic modules, and hence isomorphisms in K 1 . Therefore