Corollary 3

If every divisorial ideal $\neq 0$ of $\mathbf{A}$ admits a finite free resolution, $\mathbf{A}$ is factorial.

This is an immediate consequence of Corollary 2 and § 3, no. 1, Definition 1.

\* We shall see later that a regular local ring satisfies the hypothesis of Corollary 3 and therefore is factorial. \*

If M is a finitely generated A-module, we shall denote its rank by $r(M)$ (recall that it is the rank over K of $M_{(K)} = M \otimes_A K$); if $0 \to M_1 \to M_2 \to M_3 \to 0$ is an exact sequence of finitely generated A-modules, the sequence

is also exact and hence $r(M_2) = r(M_1) + r(M_3)$. We write

$\gamma$ therefore satisfies property (i) of Proposition 16 and, if M is pseudo-zero, $\gamma(M) = 0$ (since M is a torsion module). There exists a unique mapping from $F(A)$ to $\mathbf{Z} \times C(A)$, also denoted by y, such that $\gamma(M) = \gamma(\mathrm{cl}(M))$ for every finitely generated A-module M. We shall see that the above properties essentially characterize y: