Exercise 6, § 9

Let $A$ be a noetherian local ring, admitting a dualizing module $\Omega$; let $M$ be a finitely generated $A$-module, of finite projective dimension.

a) Proving that one has $\mathrm{Tor}_i^A(\Omega, M) = 0$ for $i > 0$. (Reason by induction on $\dim(A)$, by considering an exact sequence $0 \to N \to L \to M \to 0$ where $L$ is a finitely generated free module. If $x$ is a cancellable element of $m_A$, deduce from the induction hypothesis that $\mathrm{Tor}_i^A(\Omega, N)$ is zero for $i > 0$, which entails $\mathrm{Tor}_i^A(\Omega, M) = 0$ for $i > 1$, and that the homothety of ratio $x$ is injective in $\Omega \otimes_A N$, which entails that every prime ideal $p$ associated with $\mathrm{Tor}_1^A(\Omega, M)$ is a minimal ideal of $\mathrm{Spec}(A)$; observe that at such a $p$ the $A_p$-module $M_p$ is free).

b) Proving that the $A$-module $\Omega \otimes_A M$ has finite injective dimension.

c) Proving that the mapping $\theta : M \to \mathrm{Hom}_A(\Omega, \Omega \otimes_A M)$ defined by $\theta(m)(w) = w \otimes m$ for $m \in M, w \in \Omega$ is an isomorphism (reduce, with the aid of a projective resolution, to the case where $M$ is free).

d) The mappings $[T] \mapsto [\mathrm{Hom}_A(\Omega, T)]$ and $[M] \mapsto [\Omega \otimes_A M]$ define reciprocal bijections of one another between the set of isomorphism classes of finitely generated $A$-modules and of finite injective dimension and the set of isomorphism classes of finitely generated $A$-module and of finite projective dimension, \* and quasi-inverse equivalences of categories between the corresponding categories *.

EXERCISES

AC X.177