Lemma 47.21.5. Let $(A, \mathfrak m, \kappa )$ be a Noetherian local ring. Then $A$ is Gorenstein if and only if $\mathop{\mathrm{Ext}}\nolimits ^ i_ A(\kappa , A)$ is zero for $i \gg 0$.

**Proof.**
Observe that $A[0]$ is a dualizing complex for $A$ if and only if $A$ has finite injective dimension as an $A$-module (follows immediately from Definition 47.15.1). Thus the lemma follows from More on Algebra, Lemma 15.66.7.
$\square$

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