Corollary 2

Let A be a local Macaulay ring, \Omega a finitely generated A-module, and x a maximal secant sequence of elements of m_A, generating an ideal J. The following conditions are equivalent:

(i) the A-module \Omega is dualizing ; (ii) the A-module \Omega is a Macaulay module of dimension equal to \dim(A), and \Omega/J\Omega is an indecomposable injective module over the artinian local ring A/J ; (iii) the sequence x is \Omega-regular and \Omega/J\Omega is an indecomposable injective module over the artinian local ring A/J ; (iv) the sequence x is \Omega-regular, one has \operatorname{long}_A(\Omega/J\Omega) = \operatorname{long}_A(A/J) and the \kappa_A-vector space \operatorname{Hom}_A(\kappa_A, \Omega/J\Omega) is of dimension 1.

(i) \Rightarrow (ii) : if \Omega is dualizing, it is Macaulay and of dimension \dim(A) (n° 1, prop. 1). The sequence x is A-regular since A is a Macaulay ring; by prop. 4, the A/J-module \Omega/J\Omega is dualizing, and hence is a Matlis A/J-module (n° 1, example 1).

(ii) \Rightarrow (iii) : under assumption (ii), one has \dim(\Omega) = \dim(A) and \dim(\Omega/J\Omega) = \dim(A/J) = 0, so that the sequence x is secant for \Omega, hence \Omega-regular (§ 2, n° 3, th. 1).

(iii) \Rightarrow (i) : under the assumptions of (iii), the A/J-module \Omega/J\Omega is a Matlis A-module, and hence is dualizing (n° 1, example 1) ; by prop. 4 the A-module \Omega is dualizing.

(iii) \Leftrightarrow (iv) : this follows from the remark in § 8, n° 3.