Proposition 7

a) If the $A$-module $Q$ is flat, homomorphism (9) is bijective. b) If the $A$-module $Q$ is finitely generated projective, homomorphism (10) is bijective.

c) If the $A$-module $Q$ is flat, the ring $B$ noetherian and the $B$-module $M$ finitely generated, homomorphism (10) is bijective.

a) If $Q$ is flat, $\gamma_1$ is bijective ($X$, p. 66, cor. 2).

b) If $Q$ is finitely generated projective, it is flat, hence $\gamma_2$ is bijective, and moreover $\alpha$ is bijective (II, p. 75, prop. 2, a)).

c) Under the hypotheses of c), $\gamma_2$ is bijective since $Q$ is flat. Moreover ($X$, p. 53, prop. 6), there exists a resolution $L$ of $M$ such that each $L_n$ is finitely generated free; let $u : L(M) \to L$ be a homotopism ($X$, p. 49, cor. to prop. 3); in the commutative diagram,

the vertical arrows deduced from $u$ are homotopisms ($X$, p. 64, prop. 3 and p. 83, prop. 3) and $\bar{\alpha}$ is bijective (II, p. 75, prop. 2 (ii)); hence $H(\alpha)$ is bijective, and homomorphism (10) is bijective.