Proposition 11

*Let G be a commutative group written additively and $\phi : T(A) \to G$ a mapping; for every finitely generated torsion A-module T, we also write, by an abuse of language, $\phi(T) = \phi(\mathrm{cl}(T))$. Suppose that the following conditions are satisfied: (1) If $0 \to T_1 \to T_2 \to T_3 \to 0$ is an exact sequence of finitely generated torsion A-modules, then $\phi(T_2) = \phi(T_1) + \phi(T_3)$. (2) If T is pseudo-zero, then $\phi(T) = 0$. Then there exists a unique homomorphism $\theta : D(A) \to G$ such that $\phi = \theta \circ \chi$.*

As $\chi(A/p) = p$ for all p, necessarily $\theta(p) = \phi(A/p)$ for all $p \in P$, which proves the uniqueness of $\theta$, since the elements of P form a basis of $D(A)$. Conversely, let $\theta$ be the homomorphism from $D(A)$ to G such that $\theta(p) = \phi(A/p)$ for all $p \in P$ and let us show that it solves the problem. For this, we write

for every finitely generated torsion A-module T; clearly conditions (1) and (2) are also satisfied if $\phi$ is replaced by $\psi$. On the other hand, $\psi(A/p) = 0$ if $p \in P$; if p is a prime ideal $\neq 0$ and not in P, the annihilator of $A/p$ is contained in no ideal of P, hence (no. 4, Theorem 5) $A/p$ is pseudo-zero and therefore $\psi(A/p) = 0$. This being so, every finitely generated torsion A-module T admits a decomposition series whose factors are isomorphic to A-modules of the form $A/p$, where $p \in \mathrm{Supp}(T)$ (Chapter IV, § 1, no. 4, Theorems 1 and 2), and hence $p \neq 0$ since T is a torsion module. By induction on the length of this decomposition series, we deduce (in view of property (1) for $\psi$) that $\psi(T) = 0$.

\* We may, as in no. 4, consider the quotient category $\mathscr{T}/\mathscr{T}'$ of the category $\mathscr{T}$ of finitely generated torsion A-modules by the full sub-category $\mathscr{T}'$ of pseudo-zero finitely generated torsion A-modules. In the language of Abelian categories, Proposition 11 then expresses the fact that the Grothendieck group of the Abelian category $\mathscr{T}/\mathscr{T}'$ is canonically isomorphic to $D(A)$. \*