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