Lemma 15.69.7. Let $(R, \mathfrak m, \kappa )$ be a local Noetherian ring. Let $K \in D^+(R)$ have finite cohomology modules. Then the following are equivalent

1. $K$ has finite injective dimension, and

2. $\mathop{\mathrm{Ext}}\nolimits ^ i_ R(\kappa , K) = 0$ for $i \gg 0$.

Proof. This is a special case of Lemma 15.69.6. $\square$

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