Exercise 10, § 5

Let A be a ring and $E(A)$ the set of classes of finitely generated projective A-modules; let G be the Z-module of formal linear combinations of elements of $E(A)$ and N the submodule of G consisting of the elements $\xi - \xi' - \xi''$, where $\xi, \xi', \xi''$ are the classes of three projective A-modules M, M', M'' such that there exists an exact sequence

(it amounts to the same to say that M is isomorphic to the direct sum $M' \oplus M''$ by virtue of Algebra, Chapter II, § 2, no. 2). Let $K(A)$ be the quotient $\mathbf{Z}$-module $G/N$ and, for every finitely generated projective A-module, let $\kappa(M)$ (or $\kappa_A(M)$) be its class in $K(A)$.

(a) Show that there exists on $K(A)$ a unique commutative ring structure whose addition is that the $\mathbf{Z}$-module $K(A)$ and multiplication is such that $\kappa(M)\kappa(N) = \kappa(M \otimes_A N)$ for two finitely generated projective A-modules M, N; $\kappa(A)$ is the unit element of $K(A)$.

(b) For every finitely generated projective $A$-module $M$ and every integer $p \geqslant 0$ let $\lambda^p$ denote the element $\kappa(\bigwedge^p M)$ of $K(A)$. Show that

Let $\lambda_T(M)$ denote the element $\sum_{p=0}^\infty \lambda^p(M) T^p$ in the ring $K(A)[[T]]$ of formal power series in one indeterminate over the ring $K(A)$; show that derive a homomorphism also written $x \mapsto \lambda_T(x)$ of $K(A)$ to the multiplicative group of formal power series of $K(A)[[T]]$ whose constant term is equal to 1. Let $\lambda^p(x)$ denote the coefficient of $T^p$ in $\lambda_T(x)$; then for all $x, y$ in $K(A)$.

(c) If $A$ is a local ring, the ring $K(A)$ is canonically identified with $\mathbf{Z}$ and $\lambda^p(n) = \binom{n}{p}$. Deduce that, if $A$ is the direct composition of $m$ local rings, $K(A)$ is isomorphic to $\mathbf{Z}^m$.

(d) Let $\phi : A \to B$ be a ring homomorphism. Show that there exists a unique ring homomorphism $\phi' : K(A) \to K(B)$ such that $\phi'(\kappa_A(M)) = \kappa_B(M_{(B)})$ for every finitely generated projective $A$-module $M$; then $\phi'(\lambda^p(x)) = \lambda^p(\phi'(x))$ for all $x \in K(A)$; if $\psi : B \to C$ is another ring homomorphism, $(\psi \circ \phi)' = \psi' \circ \phi'$.

(e) Let $\phi : A \to B$ be a ring homomorphism for which $B$ is a finitely generated projective $A$-module; for every finitely generated projective $B$-module $N$, $\phi_*(N)$ is then a finitely generated projective $A$-module; deduce that there exists a $\mathbf{Z}$-module homomorphism $\phi_1 : K(B) \to K(A)$ such that

Show that, for all $x \in K(A)$ and $y \in K(B)$, $\phi_1(y \cdot \phi'(x)) = \phi_1(y) \cdot x$. If $\psi : B \to C$ is another ring homomorphism such that $C$ is a finitely generated projective $B$-module, then $(\psi \circ \phi)_1 = \phi_1 \circ \psi_1$.