Lemma 47.22.4. The following types of rings have a dualizing complex:
fields,
Noetherian complete local rings,
$\mathbf{Z}$,
Dedekind domains,
any ring which is obtained from one of the rings above by taking an algebra essentially of finite type, or by taking an ideal-adic completion, or by taking a henselization, or by taking a strict henselization.
Proof. Part (5) follows from Proposition 47.15.11 and Lemma 47.22.3. By Lemma 47.21.3 a regular local ring has a dualizing complex. A complete Noetherian local ring is the quotient of a regular local ring by the Cohen structure theorem (Algebra, Theorem 10.160.8). Let $A$ be a Dedekind domain. Then every ideal $I$ is a finite projective $A$-module (follows from Algebra, Lemma 10.78.2 and the fact that the local rings of $A$ are discrete valuation ring and hence PIDs). Thus every $A$-module has finite injective dimension at most $1$ by More on Algebra, Lemma 15.69.2. It follows easily that $A[0]$ is a dualizing complex. $\square$
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