Exercise 5, § 9

Let $A$ be a noetherian local ring, admitting a dualizing module $\Omega$. Let $T$ be a finitely generated $A$-module, of finite injective dimension.

a) Prove that $\mathrm{Ext}_A^i(\Omega, T)$ is zero for $i > 0$ (use Exercise 7 of § 3).

b) Prove that the canonical homomorphism $\Omega \otimes_A \mathrm{Hom}_A(\Omega, T) \to T$ is bijective (apply a) and Exercise 10 of § 8).

c) Prove with the aid of Exercise 8, b) of § 8 the equalities $\mathrm{dp}_A(\mathrm{Hom}_A(\Omega, T)) = \dim(A) - \mathrm{prof}(T)$ and $\mathrm{prof}(\mathrm{Hom}_A(\Omega, T)) = \mathrm{prof}(T)$. Deduce that if $\mathrm{prof}(T) = \dim(A)$, the $A$-module $T$ is a direct sum of modules isomorphic to $\Omega$.

d) Put $c = \dim(A) - \mathrm{prof}(T)$. Prove that there exist integers $n_0, \ldots, n_c$ and an exact sequence

(reason by induction on $c$, constructing with the aid of b) a surjective homomorphism $\Omega^{n_0} \to T$).