Theorem 1 (Grothendieck Duality)

Tag 0DD0, permanent. Cite it as [Bourbaki, Tag 0DD0].

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

Let $A$ be a local Macaulay ring, of dimension $d$ , and let $\Omega$ be a dualizing $A$ -module.

a) The $A$ -module $H_A^d(\Omega)$ is a Matlis module; for every $A$ -module $P$ , let $D(P)$ denote the Matlis dual $\mathrm{Hom}_A(P, H_A^d(\Omega))$ .

b) For every finitely generated $A$ -module $M$ and every integer $i$ , the canonical homomorphism

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

is an isomorphism of artinian

A

-modules.

c) For every $A$ -module $M$ and every integer $i$ , the canonical homomorphism

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

is an isomorphism of

\widehat{A}

-modules.

This follows from prop. 6 of § 8, No. 7.

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

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Cites

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