Definition 47.15.1. Let $A$ be a Noetherian ring. A dualizing complex is a complex of $A$-modules $\omega _ A^\bullet$ such that

1. $\omega _ A^\bullet$ has finite injective dimension,

2. $H^ i(\omega _ A^\bullet )$ is a finite $A$-module for all $i$, and

3. $A \to R\mathop{\mathrm{Hom}}\nolimits _ A(\omega _ A^\bullet , \omega _ A^\bullet )$ is a quasi-isomorphism.

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