Proposition 7

Let $C$ be an $A$-complex and let $a \in \mathbf{Z}$ be such that $H_n(C) = 0$ for $n < a$. a) There exists a free $A$-complex $L$ such that $L_n = 0$ for $n < a$ and a homologism $f : L \to C$.

b) Suppose $A$ left noetherian and the $A$-modules $H_n(C)$, $n\in \mathbf{Z}$, finitely generated. There exists a free $A$-complex $L$ such that $L_n=0$ for $n<a$ and that $L_n$ is a finitely generated $A$-module for every $n$, and a homomorphism $f:L\to C$.

Let $C'$ be the subcomplex of $C$ such that $C'_n=C_n$ for $n>a$, $C'_a=Z_a(C)$, $C'_n=0$ for $n<a$; then the canonical injection of $C'$ into $C$ is a homomorphism. Replacing $C$ by $C'$, one may therefore suppose that $C_n=0$ for $n<a$. The statement then results from repeated application of the following lemma, for $r=a,a+1,\ldots$ :