Corollary

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

Commutative Algebra, chapter X, § 10. Cohomologie locale, dualité de Grothendieck

The homomorphism $D(\omega) : \widehat{\Omega} \to D(H^d_A(A))$ is an isomorphism.

Let $M$ be an $A$ -module, and $i$ an integer. Consider the canonical homomorphisms (§ 8, No. 7)

\rho_{d-i}(M, \Omega) : \mathrm{Tor}^A_{d-i}(M, D(\Omega)) \longrightarrow D(\mathrm{Ext}_A^{d-i}(M, \Omega))

\theta^{d-i}(M, H^d_A(A)) : \mathrm{Ext}_A^{d-i}(M, D(H^d_A(A))) \longrightarrow D(\mathrm{Tor}^A_{d-i}(M, H^d_A(A))) .

With the aid of the isomorphisms $\omega : H^d_A(A) \to D(\Omega)$ , $D(\omega) : \widehat{\Omega} \to D(H^d_A(A))$ (cor. 1 of prop. 2) and $\tau^i(M) : \mathrm{Tor}^A_{d-i}(M, H^d_A(A)) \to H^i_A(M)$ (No. 2), one deduces from them canonical homomorphisms of $A$ -modules

\gamma^i(M) : H^i_A(M) \longrightarrow D(\mathrm{Ext}_A^{d-i}(M, \Omega))

\delta^i(M) : \mathrm{Ext}_A^{d-i}(M, \widehat{\Omega}) \longrightarrow D(H^i_A(M)) .

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

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