The Stacks project

Lemma 47.18.1. Let $(A, \mathfrak m, \kappa )$ be a Noetherian local ring. Let $\omega _ A^\bullet $ be a normalized dualizing complex. Let $Z = V(\mathfrak m) \subset \mathop{\mathrm{Spec}}(A)$. Then $E = R^0\Gamma _ Z(\omega _ A^\bullet )$ is an injective hull of $\kappa $ and $R\Gamma _ Z(\omega _ A^\bullet ) = E[0]$.

Proof. By Lemma 47.10.1 we have $R\Gamma _{\mathfrak m} = R\Gamma _ Z$. Thus

\[ R\Gamma _ Z(\omega _ A^\bullet ) = R\Gamma _{\mathfrak m}(\omega _ A^\bullet ) = \text{hocolim}\ R\mathop{\mathrm{Hom}}\nolimits _ A(A/\mathfrak m^ n, \omega _ A^\bullet ) \]

by Lemma 47.8.2. Let $E'$ be an injective hull of the residue field. By Lemma 47.16.4 we can find isomorphisms

\[ R\mathop{\mathrm{Hom}}\nolimits _ A(A/\mathfrak m^ n, \omega _ A^\bullet ) \cong \mathop{\mathrm{Hom}}\nolimits _ A(A/\mathfrak m^ n, E')[0] \]

compatible with transition maps. Since $E' = \bigcup E'[\mathfrak m^ n] = \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Hom}}\nolimits _ A(A/\mathfrak m^ n, E')$ by Lemma 47.7.3 we conclude that $E \cong E'$ and that all other cohomology groups of the complex $R\Gamma _ Z(\omega _ A^\bullet )$ are zero. $\square$

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