Exercise 10, § 5

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

(it amounts to the same thing to say that M is isomorphic to the direct sum $M' \oplus M''$ by virtue of Alg., chap. II, 3rd ed., § 2, no 2). Let $K(A)$ be the $\mathbf{Z}$-module quotient G/N, and for every finitely generated projective A-module M, let $\chi(M)$ (or $\chi_A(M)$) be its class in $K(A)$.

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

b) For every finitely generated projective A-module M and every integer $p \geqslant 0$ denote by $\lambda^p(M)$ the element $\chi(\bigwedge^p M)$ of $K(A)$. Now prove that one has $\lambda^p(M \oplus N) = \sum_{r+s=p} \lambda^r(M).\lambda^s(N)$. Denote by $\lambda_T(M)$ the element $\sum_{p=0}^\infty \lambda^p(M)T^p$ in the ring $K(A)[[T]]$ of formal series in one indeterminate over the ring $K(A)$; now prove that $\lambda_T(M \oplus N) = \lambda_T(M)\lambda_T(N)$; deduce a homomorphism again denoted $x \to \lambda_T(x)$ from $K(A)$ into the multiplicative group of formal series of $K(A)[[T]]$ whose constant term is equal to 1. The coefficient of $T^p$ in $\lambda_T(x)$ will be denoted by $\lambda^p(x)$; one has

for $x, y$ in $K(A)$.

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

d) Let $\varphi : A \to B$ be a homomorphism of rings. Now prove that there exists one and only one homomorphism of rings $\varphi^! : K(A) \to K(B)$ such that $\varphi^!(\chi_A(M)) = \chi_B(M_{(B)})$ for every finitely generated projective A-module M; one has $\varphi^!(\lambda^p(x)) = \lambda^p(\varphi^!(x))$ for every $x \in K(A)$; if $\psi : B \to C$ is a second homomorphism of rings, one has $(\psi \circ \varphi)^! = \psi^! \circ \varphi^!$.

e) Let $\varphi : A \to B$ be a homomorphism of rings, for which B is a finitely generated projective A-module; for every finitely generated projective B-module, $\varphi_*(N)$ is then a finitely generated projective A-module; deduce that there exists a homomorphism of $\mathbf{Z}$-modules $\varphi_! : K(B) \to K(A)$ such that $\varphi_!(\chi_B(N)) = \chi_A(\varphi_*(N))$. Now prove that one has, for $x \in K(A)$ and $y \in K(B)$, $\varphi_!(y.\varphi^!(x)) = \varphi_!(y).x$. If $\psi : B \to C$ is a second homomorphism of rings for which C is a finitely generated projective B-module, then one has $(\psi \circ \varphi)_! = \varphi_! \circ \psi_!$.