Exercise 5, § 5

Let $P$ be an $A$-complex projective, zero to the right, let $n$ be an integer $\geqslant 0$. Show that the following conditions are equivalent:

α) There exists a complex $P'$ of projective $A$-modules of finite type, such that $P'_i = 0$ for $i > n$ or $i < 0$, and a morphism $f : P' \to P$ such that $H_i(f)$ is bijective for $i < n$ and surjective for $i = n$.

β) For every filtered inductive system of $A$-modules $(M_\alpha)_{\alpha \in I}$, the canonical homomorphism

is bijective for $i < n$ and injective for $i = n$.

γ) For every family of right $A$-modules $(N_\alpha)_{\alpha \in I}$, the canonical homomorphism

is bijective for $i < n$ and surjective for $i = n$.

(To prove that α) implies β) and γ), apply Exercise 4 to the cone of $f$. To prove that β) or $\gamma$ implies $\alpha$), reason by induction on $n$, applying Exercise 4, Exercise 18, p. 170 and Exercise 10, p. 174.)