Corollary

Let $0 \to T_n \to T_{n-1} \to \ldots \to T_0 \to 0$ be an exact sequence of finitely generated torsion $A$-modules. Then $\sum_{i=0}^n (-1)^i \chi(T_i) = 0$.

In view of Chapter II, § 2, no. 4, Theorem 1, this follows again from the analogous property of the $l_p$ (Algebra, Chapter II, § 1, no. 10, Corollary 3 to Proposition 16).

Recall (Chapter II, § 5, no. 4) that we may speak of the set $F(A)$ of classes of finitely generated A-modules with respect to the relation of isomorphism; for every finitely generated A-module M, let $\mathrm{cl}(M)$ denote the corresponding element of $F(A)$; we shall denote by $T(A)$ the subset of $F(A)$ consisting of the classes of finitely generated torsion A-modules. Clearly $\chi$ defines a mapping of $T(A)$ to $D(A)$, also denoted by $\chi$, such that $\chi(\mathrm{cl}(T)) = \chi(T)$.