Remark 2

Tag 0DY4, permanent. Cite it as [Bourbaki, Tag 0DY4].

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

With the notation of a), suppose either that S is flat, or that R, R', R'' are flat; then on the one hand sequence (3) is exact (X, p. 72, cor. 2) and one may apply prop. 3; on the other hand $\psi(S, R')$ is bijective (th. 1), therefore

\partial(P, (2)) = \psi(S, R')^{-1} ◦ \partial((3)) ◦ \psi(S, R'').

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

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