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 15.69.7. \square
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