The Stacks project

Remark 47.18.2. Let $(A, \mathfrak m, \kappa )$ be a Noetherian local ring with a normalized dualizing complex $\omega _ A^\bullet $. By Lemma 47.18.1 above we see that $R\Gamma _ Z(\omega _ A^\bullet )$ is an injective hull of the residue field placed in degree $0$. In fact, this gives a “construction” or “realization” of the injective hull which is slightly more canonical than just picking any old injective hull. Namely, a normalized dualizing complex is unique up to isomorphism, with group of automorphisms the group of units of $A$, whereas an injective hull of $\kappa $ is unique up to isomorphism, with group of automorphisms the group of units of the completion $A^\wedge $ of $A$ with respect to $\mathfrak m$.