Proposition 4

a) The mapping $E\mapsto [E]_{\mathscr{C}}$ from $\mathscr{C}$ to $K(\mathscr{C})$ is additive.

b) Let G be an abelian group, and let $\varphi :\mathscr{C}\rightarrow G$ be an additive function of modules. There exists a unique homomorphism $u: K(\mathscr{C})\rightarrow G$ such that we have $\varphi (E) =u([E]_{\mathscr{C}})$ for every module E of type $\mathscr{C}$.

Assertion a) is obvious. Let us prove b). There exists a homomorphism $u':\mathbf{Z}^{(\mathscr{C})}\rightarrow G$ that extends $\varphi$. Since $\varphi$ is additive, we have $u'(r_{\mathscr{E}}) = 0$ for every exact sequence $(\mathscr{E})$ of modules of type $\mathscr{C}$; therefore, R is contained in the kernel of $u'$. Consequently, when passing to the quotient, $u'$ defines a homomorphism $u$ from $K(\mathscr{C})$ to G, and it is clear that we have $\varphi (E) =u([E]_{\mathscr{C}})$ for every A-module of type $\mathscr{C}$. The group $K(\mathscr{C})$ is generated by the set of elements $[E]_{\mathscr{C}}$ for E running through $\mathscr{C}$; the uniqueness of $u$ follows.

Let $\mathscr{C}$ and $\mathscr{D}$ be additive sets of classes of A-modules such that $\mathscr{C}\subset \mathscr{D}$. Since the mapping $E\mapsto [E]_{\mathscr{D}}$ fromn$\mathscr{C}$ to the Grothendieck groupnK($\mathscr{D}$) is additive, there exists a homomorphism $\gamma_{\mathscr{D},\mathscr{C}}: K(\mathscr{C})\rightarrow K(\mathscr{D})$, called canonical, characterized by the formula $\gamma_{\mathscr{D},\mathscr{C}}([E]_{\mathscr{C}}) = [E]_{\mathscr{D}}$ for every module E of type $\mathscr{C}$. It is not always injective (VIII, p. 210, Exercise 13).