Remark 2

Tag 0DB4, permanent. Cite it as [Bourbaki, Tag 0DB4].

Commutative Algebra, chapter X, § 8. Dualité des modules de longueur finie

Let

(\mathscr{E})

0 \to M' \xrightarrow{j} M \xrightarrow{q} M'' \to 0

be an exact sequence of $A$ -modules. The homomorphism $L(q) : L(M) \to L(M'')$ induced on the canonical free resolutions is surjective, and the complex $Ker\,L(q)$ defines a projective resolution of $M'$ . Applying Proposition 3 of A, X, p. 104 to the exact sequence $0 \to Ker\,L(q) \to L(M) \to L(M'') \to 0$ , one obtains commutative diagrams

\begin{array}{ccc} \mathrm{Ext}_A^i(M', D(P)) & \xrightarrow{\theta^i(M', P)} & D(\mathrm{Tor}_i^A(M', P)) \\ \downarrow & & \downarrow \\ \delta^i(\mathscr{E}, D(P)) & & (-1)^{i+1}D(\partial_{i+1}(\mathscr{E}, P)) \end{array}

\begin{array}{ccc} \mathrm{Ext}_A^{i+1}(M'', D(P)) & \xrightarrow{\theta^{i+1}(M'', P)} & D(\mathrm{Tor}_{i+1}^A(M'', P)) \\ \downarrow & & \downarrow \\ \mathrm{Tor}_{i+1}^A(M'', D(P)) & \xrightarrow{\rho_{i+1}(M'', P)} & D(\mathrm{Ext}_A^{i+1}(M'', P)) \\ \downarrow & & \downarrow \\ \partial_{i+1}(\mathscr{E}, D(P)) & & (-1)^{i+1}D(\delta^i(\mathscr{E}, P)) \end{array}

\begin{array}{ccc} \mathrm{Tor}_{i+1}^A(M'', D(P)) & \xrightarrow{\rho_{i+1}(M'', P)} & D(\mathrm{Ext}_A^{i+1}(M'', P)) \\ \downarrow & & \downarrow \\ \mathrm{Tor}_i^A(M', D(P)) & \xrightarrow{\rho_i(M', P)} & D(\mathrm{Ext}_A^i(M', P)) \end{array}

Let

(\mathscr{F})

0 \to P' \to P \to P'' \to 0

be an exact sequence of $A$ -modules; since the $A$ -module $J$ is injective, one deduces from it an exact sequence

(\mathscr{D}(\mathscr{F}))

0 \to D(P'') \to D(P) \to D(P') \to 0.

Applying A, X, p. 104, Proposition 3 and p. 106, Proposition 4 to the exact sequences $(\mathscr{F})$ and $(\mathscr{D}(\mathscr{F}))$ , one obtains analogously commutative diagrams

\begin{array}{ccc} \mathrm{Ext}_A^i(M, D(P')) & \xrightarrow{\theta^i(M, P')} & D(\mathrm{Tor}_i^A(M, P')) \\ \downarrow & & \downarrow \\ \delta^i(M, D(\mathscr{F})) & & (-1)^{i+1}D(\partial_{i+1}(M, \mathscr{F})) \end{array}

\begin{array}{ccc} \mathrm{Ext}_A^{i+1}(M, D(P'')) & \xrightarrow{\theta^{i+1}(M, P'')} & D(\mathrm{Tor}_{i+1}^A(M, P'')) \end{array}

\begin{array}{ccc} \mathrm{Tor}_{i+1}^{\mathbf{A}}(M, D(P')) & \xrightarrow{\rho_{i+1}(M, P')} & D(\mathrm{Ext}_{A}^{i+1}(M, P')) \\ \downarrow & & \downarrow \\ \partial_{i+1}(M, D(\mathscr{F})) & & (-1)^i D(\delta^i(M, \mathscr{F})) \end{array}

\begin{array}{ccc} \mathrm{Tor}_{i}^{\mathbf{A}}(M, P'') & \xrightarrow{\rho_{i}(M, P'')} & D(\mathrm{Ext}_{A}^{i}(M, P'')) \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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