Corollary 3

— *Let L and M be two free $\mathbf{Z}$-modules of rank n, let $(e_i)$ be a basis for L and $(f_i)$ a basis for M ($1 \leq i \leq n$), let u be a homomorphism from L into M, and let U be its matrix with respect to the bases $(e_i)$ and $(f_i)$. Then $\mathrm{Coker}\,u = M/u(L)$ is finite if and only if $\det(U) \neq 0$, and then $\mathrm{Card}(\mathrm{Coker}(u)) = |\det(U)|$. By changing bases in L and M if necessary, we may assume that U has the form described in VII, p. 22, Cor. 1 to Prop. 5 (where the $a_i$ are in this case integers); the corollary then becomes obvious, since the order of a direct sum of $\mathbf{Z}$-modules $\mathbf{Z}/\alpha_i \mathbf{Z}$ ($1 \leq i \leq n$) is infinite if one of the $a_i$ is zero, and is equal to $|\alpha_1 \alpha_2 \ldots a_n|$ otherwise.