Remark 2

Tag 0DD3, permanent. Cite it as [Bourbaki, Tag 0DD3].

Let $u : M \to M'$ be a homomorphism of $A$ -modules. By Remark 1 of No. 2 and that of $§ 8$ , No. 7, the following diagrams are commutative:

\begin{array}{ccc} H_A^i(M) & \xrightarrow{\gamma^i(M)} & D(\mathrm{Ext}_A^{d-i}(M, \Omega)) \\ \downarrow H_A^i(u) & & \downarrow D(\mathrm{Ext}(u,1)) \\ H_A^i(M') & \xrightarrow{\gamma^i(M')} & D(\mathrm{Ext}_A^{d-i}(M', \Omega)) \end{array}

\begin{array}{ccc} \mathrm{Ext}_A^{d-i}(M', \widehat{\Omega}) & \xrightarrow{\delta^i(M')} & D(H_A^i(M')) \\ \downarrow \mathrm{Ext}(u,1) & & \downarrow D(H_A^i(u)) \\ \mathrm{Ext}_A^{d-i}(M, \widehat{\Omega}) & \xrightarrow{\delta^i(M)} & D(H_A^i(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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