Exercise 11, § 11

Take the assumptions and notation of Exercise 8. Denote by $f': D\rightarrow B$ and $g': D\rightarrow A$ the homomorphisms deduced from the two projections and by

$\varphi_1: K_1(D)\rightarrow K_1(A)\times K_1(B)$ and $\psi_1: K_1(A)\times K_1(B)\rightarrow K_1(C)$

the mappings defined by $\varphi_1(z) = (g^{'*}(z), f^{'*}(z))$ and $\psi_1(x, y) =f^*(x)-g^*(y)$. a) Prove that the sequence

is exact (observe that $f(E(A)) = E(C)$). Prove that if, moreover, there exists a ring homomorphism $s: C\rightarrow A$ such that $f\circ s=$ Id$_C$, then $\psi_1$ is surjective.

b) For any integer $n$ and any element $g$ of $\mathbf{G}\mathbf{L}_n$(C), denote by $P_g$ the projective D-module associated with the modules $A^n$ and $B^n$ and the isomorphism from $C\otimes_AA^n$ to $C\otimes_BB^n$ with matrix $g$ (Exercise 8). Construct a homomorphism $\partial : K_1(C)\rightarrow K_0(D)$ such that $\partial \circ \iota_n(g) = [P_g]-[D^n_s]$ for every integer $n$ and every element $g$ of $\mathbf{G}\mathbf{L}_n(C)$.

c) Prove that the sequence

is exact.