Exercise 10, § 11

Let A be a ring. Denote by $\mathbf{G}\mathbf{L}_{\infty}(A)$ the subgroup of $\mathbf{G}\mathbf{L}(A^{(\mathbf{N})})$ consisting of the matrices X such that $X-I$ has only finitely many nonzero entries, and denote by E(A) the subgroup of $\mathbf{G}\mathbf{L}_{\infty}(A)$ generated by the matrices I $+\lambda E_{ij}$ for $i, j$ in $\mathbf{N}$ with $i\not=j$ and $\lambda \in A$; it is the derived group of $\mathbf{G}\mathbf{L}_{\infty}(A)$ (II, §10, p. 422, Exercise 15). Denote the quotient group $\mathbf{G}\mathbf{L}_{\infty}(A)/E(A)$ by $K_1(A)$; its group law is written additively. Every ring homomorphism $f: A\rightarrow B$ induces a group homomorphism $f^*: K_1(A)\rightarrow K_1(B)$.

a) For any integer $n\geqslant 1$, let $\iota_n:\mathbf{G}\mathbf{L}_n(A)\rightarrow K_1(A)$ be the homomorphism that sends $X\in \mathbf{G}\mathbf{L}_n(A)$ to the class of the matrix $\begin{pmatrix} X & 0 \\ 0 & I \end{pmatrix}$. Prove that if the ring A is local, then $\iota_n$ is surjective for every $n$ (use Exercise 17 of VIII, p. 42).

b) Suppose that the ring A is commutative. Construct a surjective homomorphism $d_A: K_1(A)\rightarrow A^*$ such that $d_A\circ \iota_n=$ det for every $n$. If, moreover, A is local or Euclidean, then the homomorphisms $d_A$ and $\iota_1$ are inverse bijections.

c) Let I be a right directed set, $(A_i, f_{ji})$ a direct system of rings, and A its limit. Prove that $K_1(A)$ is the limit of the direct system of groups $(K_1(A_i), f_{ji}^*)$.