Exercise 7, § 9

Let $A$ be a Macaulay local ring, and let $M$ be a Macaulay finitely generated $A$-module. We denote by $r(M)$ (resp. $t(M)$) the dimension of the $\kappa_A$-vector space $\kappa_A \otimes_A M$ (resp. $\mathrm{Ext}_A^p(\kappa_A, M)$), with $p = \mathrm{prof}(M)$), cf. exerc. 16 of § 1.

a) Suppose that $A$ possesses a dualizing module $\Omega$; put $M' = \mathrm{Ext}_A^c(M, \Omega)$, with $c = \dim(A) - \dim_A(M)$. Prove the equalities $r(M') = t(M)$ and $t(M') = r(M)$ (reduce, with the help of the Prop. 7 of § 3, to the case $c = 0$, then, on passing to the quotient by a maximal regular sequence, to the case where $A$ is artinian, and use the Prop. 6 of § 8).

b) Deduce that one has $t(M_p) \leq t(M)$ for every $p \in \mathrm{Spec}(A)$ (reduce, with the help of exerc. 2 of § 2 and 16 of § 1, to the case where $A$ is complete, and apply $a$).

c) For the module $M$ to be dualizing, it is necessary and sufficient that it be faithful and that one have $t(M) = 1$ (reduce to the case where $A$ is complete; apply $a$) and the cor. of th. 1).