The Stacks project

Lemma 47.22.3. Let $A$ be a Noetherian ring and let $I \subset A$ be an ideal. Let $\omega _ A^\bullet $ be a dualizing complex.

  1. $\omega _ A^\bullet \otimes _ A A^ h$ is a dualizing complex on the henselization $(A^ h, I^ h)$ of the pair $(A, I)$,

  2. $\omega _ A^\bullet \otimes _ A A^\wedge $ is a dualizing complex on the $I$-adic completion $A^\wedge $, and

  3. if $A$ is local, then $\omega _ A^\bullet \otimes _ A A^ h$, resp. $\omega _ A^\bullet \otimes _ A A^{sh}$ is a dualzing complex on the henselization, resp. strict henselization of $A$.

Proof. Immediate from Lemmas 47.22.1 and 47.22.2. See More on Algebra, Sections 15.11, 15.43, and 15.45 and Algebra, Sections 10.96 and 10.97 for information on completions and henselizations. $\square$

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