Example

Tag 0DD5, permanent. Cite it as [Bourbaki, Tag 0DD5].

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

Let $A$ be an integral noetherian local ring of dimension 1; let $K$ denote its field of fractions. Let $\Omega$ be a dualizing $A$ -module, and let $M$ be a finitely generated $A$ -module. The $A$ -modules $H_A^0(M)$ and $H_A^1(M)$ are canonically identified with $T(M)$ and $(K/A) \otimes_A M$ (No. 1, Example 5). With these identifications, the duality isomorphisms

\gamma^0(M) : T(M) \longrightarrow D(\mathrm{Ext}_A^1(M, \Omega)) \quad , \quad \gamma^1(M) : (K/A) \otimes_A M \longrightarrow D(\mathrm{Hom}_A(M, \Omega))

(Theorem 1) are none other than the isomorphisms defined in § 9, No. 6.

Exercises

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

Cited by

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Cites

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