Exercise 6, § 4

Let $P$ be a flat complex of right A-modules. For every A-module $M$ and every integer $i$, set

if $u : M \to N$ is an A-homomorphism, denote by $T^P_i(u)$ the k-homomorphism H_i(1_P ⊗ u). a) Let $r$ be an integer. Now prove that the following conditions are equivalent: α) For every injection $u : M' \to M$, the mapping $T^P_r(u)$ is injective. β) For every surjection $v : M \to M''$, the mapping $T^P_{r+1}(v)$ is surjective. γ) The right A-module $P_r/B_r(P)$ is flat. δ) There exists a flat complex $P'$ of right A-modules, whose differential $d_{r+1}$ is zero, and for every A-module $M$ and every integer $i$ a k-isomorphism $\varphi^M_i : T^P_i(M) \to T^{P'}_i(M)$, such that one has for every A-homomorphism $u : M \to N$.

BOURBAKI. — Algebra X

(For proving that $\gamma$) implies $\delta)$, define $P'$ by setting $P'_{r+1}=Z_{r+1}(P)$ and $P'_r=P_r/B_r(P)$.)

b) Suppose in addition that $A$ is noetherian and that the $A$-modules $P_i$ are of finite type. Now prove that the conditions of a) are equivalent to the following condition:

e) There exists an $A$-module $Q$ and for every A-module $M$ a k-isomorphism

such that for every A-homomorphism $u:M\longrightarrow N$.

(For proving that $\delta$) implies $\varepsilon)$, reduce to the case where $d_{r+1}=0$ and take for $Q$ the dual of $Z_r(P)$.)

c) Now prove that the result of b) remains valid under the assumption that $A$ is noetherian, $H_i(P)$ is of finite type for every $i$ and $H_i(P)=0$ for $i<i_0$ (use the prop. 10, p. 56).