Proposition 15

Let M be a finitely generated A-module. There exists a free submodule L of M such that M/L is a torsion module and the element $c(\chi(M/L))$ of C does not depend on the free submodule L chosen.

We write $S = A - \{0\}$ and let $V = S^{-1}M = M \otimes_A K$; if $n$ is the rank of V over K, there exist $n$ elements $e_i$ ($1 \leq i \leq n$) of M whose canonical images in V form a basis of V; these elements are obviously linearly independent in M and hence generate a free submodule L of M such that $S^{-1}(M/L) = S^{-1}M/S^{-1}L = 0$, so that M/L is a torsion module.

Now let L, be another free submodule of M of rank $n$. Since $S^{-1}L = S^{-1}L_1$, there exists $s \in S$ such that $sL_1 \subset L$; we may therefore limit ourselves to proving that, if $L_1 \subset L_2$ are two free submodules of M of rank $n$, then

Now, $\chi(M/L_1) = \chi(M/L_2) + \chi(L_2/L_1)$ and it follows from no. 6, Corollary to Proposition 14 that $\chi(L_2/L_1)$ is a principal divisor and therefore

The element $c(\chi(M/L))$ will be denoted by $-c(M)$ in what follows; we shall say that $c(M)$ is the divisor class attached to M.