Lemma 47.16.4. Let $(A, \mathfrak m, \kappa )$ be a Noetherian local ring with normalized dualizing complex $\omega _ A^\bullet $. Let $E$ be an injective hull of $\kappa $. Then there exists a functorial isomorphism

\[ R\mathop{\mathrm{Hom}}\nolimits _ A(N, \omega _ A^\bullet ) = \mathop{\mathrm{Hom}}\nolimits _ A(N, E)[0] \]

for $N$ running through the finite length $A$-modules.

**Proof.**
By induction on the length of $N$ we see that $R\mathop{\mathrm{Hom}}\nolimits _ A(N, \omega _ A^\bullet )$ is a module of finite length sitting in degree $0$. Thus $R\mathop{\mathrm{Hom}}\nolimits _ A(-, \omega _ A^\bullet )$ induces an anti-equivalence on the category of finite length modules. Since the same is true for $\mathop{\mathrm{Hom}}\nolimits _ A(-, E)$ by Proposition 47.7.8 we see that

\[ N \longmapsto \mathop{\mathrm{Hom}}\nolimits _ A(R\mathop{\mathrm{Hom}}\nolimits _ A(N, \omega _ A^\bullet ), E) \]

is an equivalence as in Lemma 47.16.3. Hence it is isomorphic to the identity functor. Since $\mathop{\mathrm{Hom}}\nolimits _ A(-, E)$ applied twice is the identity (Proposition 47.7.8) we obtain the statement of the lemma.
$\square$

## Comments (2)

Comment #3231 by Niels on

Comment #3245 by Johan on