Proposition 2

Let A be a local Macaulay ring, of dimension d, and let $\Omega$ be a dualizing A-module. The homomorphism $\omega : H_A^d(A) \to D(\Omega)$ is bijective.

The homomorphism $\omega$ is the limit of the inductive system of mappings $(\omega_\alpha)_{\alpha \in \mathscr{D}_{cs}}$, where

associates to an element $u$ of $\mathrm{Ext}_A^d(A/\alpha, A)$ the mapping $x \mapsto f_x \circ u$ (A, X, p. 114). It is therefore enough to prove that each of the mappings $\omega_\alpha$ is bijective.

Let $\alpha$ be an ideal of $\mathscr{D}_{cs}$, generated by a completely secant sequence $x = (x_1, \ldots, x_d)$ for $A$. The Koszul complex $K^\bullet(x, A)$ furnishes a projective resolution of $A/\alpha$; for every $A$-module $M$, the $A$-module $H^d(\mathrm{Homgr}_A(K^\bullet(x, A), M))$ identifies canonically with $M/\alpha M$ (A, X, p. 155). From this one deduces an isomorphism (A, X, p. 100)

Let $x \in \Omega$. Taking account of loc. cit., p. 103, prop. 2, one has a commutative diagram

where $f_x$ is the homomorphism deduced from $f_x$ by passing to quotients. It follows that if for every $A$-module $M$ one identifies $\mathrm{Ext}_A^d(A/\alpha, M)$ with $M/\alpha M$ by means of $\varphi_M$, the homomorphism $\omega_\alpha$ is identified with the $A$-linear mapping of $A/\alpha$ into $\mathrm{Hom}_A(\Omega, \Omega/\alpha \Omega)$ which sends 1 to the canonical surjection, that is again with the canonical mapping $A/\alpha \longrightarrow \mathrm{End}_{A/\alpha}(\Omega/\alpha \Omega)$. But since the $A/\alpha$-module $\Omega/\alpha \Omega$ is dualizing (§ 9, No. 2, prop. 4), this mapping is bijective (§ 9, No. 4, prop. 6), which proves the proposition.

Let us identify the Matlis bidual $D(D(\Omega))$ with $\widehat{\Omega}$ by the isomorphism $\widehat{\alpha}_\Omega$ (§ 8, No. 3, th. 2, b)).