Example 6

Let us take up again the hypotheses and notations of Example 3. One obtains that $I = \varinjlim \mathrm{Ext}_A^d(A/\mathfrak{m}_A^n, \Omega)$ is a Matlis $A$-module. For every $A$-module of finite length $M$, there is a canonical $A$-isomorphism

moreover this isomorphism is $\mathrm{End}_A(M)$-linear (Theorem 3, c)).

In particular, if the ring $A$ is Gorenstein of dimension $d$, the $A$-module $\varinjlim \mathrm{Ext}_A^d(A/\mathfrak{m}_A^n, A)$ is a Matlis module.