Remark

Tag 0DS5, permanent. Cite it as [Bourbaki, Tag 0DS5].

Algebra, chapter X, § 2. Complexes de A-modules

a) Suppose that the canonical exact sequence of graded A-modules

\tag{II} 0\longrightarrow B(C)\longrightarrow Z(C)\xrightarrow{\pi}H(C)\longrightarrow0

be split (this occurs for example if $H(C)$ is projective, or $B_n(C)$ injective for each $n$ ); let $\sigma:H(C)\to Z(C)$ be a graded A-linear section of $\pi$ , and let $\psi$ be the homomorphism $x\mapsto\sigma(x)$ from $H(C)$ into $C$ . Then $\psi$ is a homologism from $(H(C),0)$ into $C$ , such that

H(\psi)=1_{H(C)}.

b) Suppose that the canonical exact sequence of graded A-modules

\tag{IV} 0\longrightarrow H(C)\xrightarrow{i}C/B(C)\longrightarrow B(C)(-1)\longrightarrow0

is split (this occurs for example if $B(C)$ is projective, or $H_n(C)$ injective for each $n$ ); let $\tau : C/B(C) \to H(C)$ be a graded A-linear retraction of $i$ and let $\varphi$ be the homomorphism from $C$ into $H(C)$ which associates to each element of $C$ the image by $\tau$ of its class modulo $B(C)$ . Then $\varphi$ is a homologism from $C$ into $(H(C), 0)$ such that $H(\varphi) = 1_{H(C)}$ .

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

Cited by

Nothing.

Cites

Nothing.