Remark 2

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

Let

(\mathscr{E}) \quad 0 \to M \to N \to P \to 0

be an exact sequence of a-modules. Using A, X, p. 104, prop. 3 and p. 106, prop. 4, one proves that the following diagrams are commutative:

\begin{array}{ccc} \mathrm{Tor}_{d-i}^A(P, H_A^d(A)) & \xrightarrow{\tau^i(P)} & H_A^i(P) \\ \downarrow \partial_{d-i}(\mathscr{E}, H_A^d(A)) & & \downarrow \partial^i(\mathscr{E}) \\ \mathrm{Tor}_{d-i-1}^A(M, H_A^d(A)) & \xrightarrow{\tau^{i+1}(M)} & H_A^{i+1}(M) \end{array}

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

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