Example 2

Suppose A noetherian and integrally closed. Then $Z^1$ is identified with the group $D(A)$ of divisors of A introduced in Chapter VII (§ 1, No. 3, Theorem 2, and No. 6, Theorem 3). The modules of type $\mathscr{C}^{>2}$ are the pseudo-null modules (VII, § 4, No. 4, Def. 2); if M is a torsion module of finite type, then $z^1(M) \in Z^1 = D(A)$ is the content $\chi(M)$ of M (VII, § 4, No. 5, Def. 4). Props. 10 and 11 of loc. cit. are therefore equivalent to the exactness of the sequence $K(\mathscr{C}^{>2}) \to K(\mathscr{C}^{>1}) \to Z^1 \to 0$.