Remark 3

Tag 0DY5, permanent. Cite it as [Bourbaki, Tag 0DY5].

Algebra, chapter X, § 6. Utilisation de résolutions non canoniques

With the notation of b), suppose either that E is injective, or that R, R', R'' are projective; then on the one hand sequence (4) is exact (X, p. 83, prop. 2) and one may apply prop. 3; on the other hand, $\varphi(R', E)$ is bijective (th. 1) ; therefore

\delta((2), N) = \varphi(R'', E) ◦ \partial((4)) ◦ \varphi(R', E)^{-1}.

Let us now consider a commutative diagram

(5)

\begin{array}{ccccccccc} 0 & \rightarrow & N' & \xrightarrow{r} & N & \xrightarrow{s} & N'' & \rightarrow & 0 \\ & & c' & & & & c'' & & \end{array}

(6)

\begin{array}{ccccccccc} 0 & \rightarrow & E' & \xrightarrow{\tilde{r}} & E & \xrightarrow{\tilde{s}} & E'' & \rightarrow & 0 \end{array}

whose first row (5) is an exact sequence of left A-modules, the second row (6) an exact sequence of complexes of left A-modules, and in which the vertical arrows are right resolutions. Reasoning as in Prop. 3, one proves the following proposition:

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

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