Lemma 5

Consider a commutative diagram

where $(P, p)$ (resp. $(P', p')$) is a left resolution of $M$ (resp. $M'$), and where the bottom horizontal line is an exact sequence. There exists a homologism $p'' : \operatorname{Con}(u) \to M''$.

In fact, the exact sequence (X, p. 37, prop. 7)

gives an exact homology sequence

By X, p. 38, lemma 3 a), we have $\partial = - H_0(u)$. Since $H_n(P) = 0 = H_n(P')$ for $n > 0$ and $H_0(u) : H_0(P') \to H_0(P)$ is identified with $f : M' \to M$, we conclude that $H_n(\operatorname{Con}(u)) = 0$ for $n > 0$ and that $H_0(\operatorname{Con}(u))$ is isomorphic to $M''$, whence the lemma.

Let us now prove the theorem.

a) Let M be an A-module of type $\mathscr{C}$. For every bounded left resolution $(P, p)$ of M of type $\mathscr{C}_0$, the element $\chi_{\mathscr{C}_0}(P)$ of $K(\mathscr{C}_0)$ depends only on M. In fact, let $(P_1, p_1)$ and $(P_2, p_2)$ be two resolutions of this type. Consider the resolution

of the A-module $M \times M$ and the homomorphism $\Delta : x \mapsto (x, x)$ of M into $M \times M$. By lemma 4, there exists a bounded left resolution $(Q, q)$ of M of type $\mathscr{C}_0$ and a commutative diagram we deduce from this a commutative diagram By lemma 5, Con $(u \circ pr_i)$ has zero homology, hence $u \circ pr_i$ is a homologism and $\chi_{\mathscr{C}_0}(Q) = \chi_{\mathscr{C}_0}(P_i)$ (X, p. 41, prop. 10); it follows that $\chi_{\mathscr{C}_0}(P_1) = \chi_{\mathscr{C}_0}(P_2)$ as announced.

b) For every A-module M of type $\mathscr{C}$, let $\varphi(M) \in K(\mathscr{C}_0)$ be the common value of the $\chi_{\mathscr{C}_0}(P)$ for all bounded left resolutions P of M of type $\mathscr{C}_0$. Let us show that the function $\varphi : \mathscr{C} \to K(\mathscr{C}_0)$ is additive. Let therefore

be an exact sequence of A-modules of type $\mathscr{C}$. By Lemma 4, there exists a commutative diagram where $(P, p)$ and $(P', p')$ are bounded left resolutions of type $\mathscr{C}_0$. Then we have and by Lemma 5 which was to be proved.

c) Let then $\beta : K(\mathscr{C}) \to K(\mathscr{C}_0)$ be the homomorphism such that, with the preceding notations, we have $\beta([M]_{\mathscr{C}})=\chi_{\mathscr{C}_0}(P)$. Since $p$ is a homomorphism, we have $\chi_{\mathscr{C}}(P)=[M]_{\mathscr{C}}$, hence $\alpha\circ\beta([M]_{\mathscr{C}})=\alpha(\chi_{\mathscr{C}_0}(P))=\chi_{\mathscr{C}}(P)=[M]_{\mathscr{C}}$ and $\alpha\circ\beta=1_{K(\mathscr{C})}$. If $M$ is of type $\mathscr{C}_0$, then $(M,1_M)$ is a resolution of $M$, hence $\varphi(M)=[M]_{\mathscr{C}_0}$ and $\beta\circ\alpha=1_{K(\mathscr{C}_0)}$, which completes the proof.

We shall apply this theorem to modules of "finite projective dimension" in § 8 (X, p. 137).