Corollary 4

Tag 0DXJ, permanent. Cite it as [Bourbaki, Tag 0DXJ].

Algebra, chapter X, § 5. Modules d’extensions

If $C$ is bounded on the right and if $C$ and $H(C)$ are projective, then

\lambda(C, C') : H(\mathrm{Homgr}_A(C, C')) \to \mathrm{Homgr}_A(H(C), H(C'))

is bijective. By the theorem, it suffices to prove that $B(C)$ and $Z(C)$ are projective. Now one has exact sequences

\begin{align*} 0 &\to B_n(C) \to Z_n(C) \to H_n(C) \to 0 \\ 0 &\to Z_n(C) \to C_n \to B_{n-1}(C) \to 0, \end{align*}

therefore $(B_{n-1}(C) \text{ is projective}) \Rightarrow (Z_n(C) \text{ is projective}) \Rightarrow (B_n(C) \text{ is projective}).$ One concludes by remarking that $B_n(C) = 0$ for $n$ sufficiently small.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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