Proposition 5

Let E and F be modules of type $\mathscr{C}$. The equality $[E]_{\mathscr{C}}= [F]_{\mathscr{C}}$ holds if and only if there exist exact sequences of modules of type $\mathscr{C}$

such that $E\oplus Q$ is isomorphic to $F\oplus P$.

The stated condition is sufficient because it implies the relations

which give $[E]_{\mathscr{C}}= [F]_{\mathscr{C}}$.

Now suppose that we have $[E]_{\mathscr{C}}= [F]_{\mathscr{C}}$. By the construction of the group $K(\mathscr{C})$, there exist two finite families of exact sequences of modules of type $\mathscr{C}$

for $i\in I$ and

for $j\in J$ such that in $\mathbf{Z}^{(\mathscr{C})}$, we have

cl(E) $-$ cl(F) $=\sum_{j\in J}r_{\mathscr{H}_j}-\sum_{i\in I}r_{\mathscr{G}_i}$.

More explicitly, this relation can be written as

(7) cl(E) $+\sum_{i\in I}$ cl(G$_i$) $+\sum_{j\in J}$ cl(H$'_j$) $+\sum_{j\in J}$ cl(H$''_j$)

= cl(F) $+\sum_{i\in I}$ cl(G$'_i$) $+\sum_{i\in I}$ cl(G$''_i$) $+\sum_{j\in J}$ cl(H$_j$).

Set $G =\bigoplus_{i\in I}G_i, G'=\bigoplus_{i\in I}G'_i$, etc. By passing to the direct sums, we obtain the exact sequences

$(\mathscr{G})$ 0 // ${G'}^p$ // G $^q$ // $G''$ // $0$,

$(\mathscr{H})$ 0 // ${H'}^r$ // H $^s$ // $H''$ // 0

consisting of modules of type $\mathscr{C}$.

Next, let $M_1, . . . ,M_m,N_1, . . . ,N_n$ be A-modules of type $\mathscr{C}$. If we have $\sum^m_{i=1}$ cl(M$_i$) $=\sum^n_{j=1}$ cl(N$_j$) in the group $\mathbf{Z}^{(\mathscr{C})}$, then we have $m=n$ and there exists a permutation $\sigma \in \mathfrak{S}_m$ such that cl(M$_i$) $=$ cl(N$_{\sigma(i)}$) for every $1\leqslant i\leqslant m$ (I, §7, No. 9, p. 95, Proposition 11). Consequently, the modules $\bigoplus^m_{i=1}M_i$ and $\bigoplus^n_{j=1}N_j$ are isomorphic. In particular, from (7), we deduce the existence of an isomorphism from $E\oplus Q$ to $F\oplus P$, where we have set

We also set

The modules L, M, P, Q are of type $\mathscr{C}$, and the sequence

$(\mathscr{E})$ 0 // L $^{\lambda}$ // P $^\mu$ // M // $0$,

where we define $\lambda$ and $\mu$ by

is exact. We construct an exact sequence

in the same way; this concludes the proof.

The set $\mathscr{C}$ is a commutative monoid for the law of composition $(E,E')\mapsto$ cl(E$\oplus E'$). We sometimes denote the group of differences of the commutative monoid $\mathscr{C}$ (I, §2, No. 4, p. 20) by $K'(\mathscr{C})$ and call it the Grothendieck group of $\mathscr{C}$ for direct sums. For any module E of type $\mathscr{C}$, we denote the image of cl(E) in $K'(\mathscr{C})$ by $[E]'_{\mathscr{C}}$.