Proposition 17

Let G be a commutative group written additively and $\phi$ a mapping from the set $F(A)$ of classes of finitely generated A-modules to G; for every finitely generated A-module M we also write, by an abuse of language, $\phi(M) = \phi(\mathrm{cl}(M))$. Suppose the following conditions are satisfied: (1) If $0 \to M_1 \to M_2 \to M_3 \to 0$ is an exact sequence of finitely generated A-modules, then $\phi(M_2) = \phi(M_1) + \phi(M_3)$. (2) If T is pseudo-zero, then $\phi(T) = 0$. Then there exists a unique homomorphism $\theta : Z \times C \to G$ such that $\phi = 8 \circ y$.

By virtue of Proposition 16 (iv), every element of $Z \times C$ is of the form $(r(M), c(M))$ for some suitable finitely generated A-module M; whence the uniqueness of $\theta$. We apply Proposition 11 of no. 5 to the restriction of $-\phi$ to $T(A)$: then there exists a homomorphism $\theta_0 : D \to G$ such that

for every finitely generated torsion A-module T. Let x be a non-zero element of A; applying property (1) to the exact sequence:

where $h_x$ is multiplication by x, we obtain $\phi(A/xA) = 0$, whence $\theta_0(\mathrm{div}(x)) = 0$. Taking quotients, $\theta_0$ therefore defines a homomorphism $\theta_1 : C \to G$ and $\phi(T) = \theta_1(c(T))$ for every torsion A-module T. We show now that the homomorphism $\theta$ defined by $\theta(n, z) = n.\phi(A) + \theta_1(z)$ solves the problem. For this, we write $\phi'(M) = \phi(M) - \theta(\gamma(M))$ for every finitely generated A-module M; clearly condition (1) is still satisfied if $\phi$ is replaced by $\phi'$. Moreover, $\phi'(M) = 0$ when M is a torsion module or a free module; but as for every finitely generated A-module M, there exists a free sub-module L of M such that M/L is a torsion module (Proposition 15), property (1) shows that $\phi'(M) = 0$ for every finitely generated A-module M.

\* In the language of Abelian categories, Proposition 17 shows that $\mathbf{Z} \times C(A)$ is canonically isomorphic to the Grothendieck group of the quotient category $\mathscr{F}/\mathscr{F}'$, where $\mathscr{F}$ is the category of finitely generated A-modules and $\mathscr{F}'$ the full sub-category of $\mathscr{F}$ consisting of the pseudo-zero modules. \*