Theorem 47.18.3 (0A84)—The Stacks project

Theorem 47.18.3. Let $(A, \mathfrak m, \kappa )$ be a Noetherian local ring. Let $\omega _ A^\bullet $ be a normalized dualizing complex. Let $E$ be an injective hull of the residue field. Let $Z = V(\mathfrak m) \subset \mathop{\mathrm{Spec}}(A)$. Denote ${}^\wedge $ derived completion with respect to $\mathfrak m$. Then

\[ R\mathop{\mathrm{Hom}}\nolimits _ A(K, \omega _ A^\bullet )^\wedge \cong R\mathop{\mathrm{Hom}}\nolimits _ A(R\Gamma _ Z(K), E[0]) \]

for $K$ in $D(A)$.

Proof. Observe that $E[0] \cong R\Gamma _ Z(\omega _ A^\bullet )$ by Lemma 47.18.1. By More on Algebra, Lemma 15.91.13 completion on the left hand side goes inside. Thus we have to prove

\[ R\mathop{\mathrm{Hom}}\nolimits _ A(K^\wedge , (\omega _ A^\bullet )^\wedge ) = R\mathop{\mathrm{Hom}}\nolimits _ A(R\Gamma _ Z(K), R\Gamma _ Z(\omega _ A^\bullet )) \]

This follows from the equivalence between $D_{comp}(A, \mathfrak m)$ and $D_{\mathfrak m^\infty \text{-torsion}}(A)$ given in Proposition 47.12.2. More precisely, it is a special case of Lemma 47.12.3. $\square$

The Stacks project

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