Lemma 15.69.7 (0AVJ)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 15: More on Algebra
Section 15.69: Injective dimension
Lemma 15.69.7 (cite)

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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

$K$ has finite injective dimension, and
$\mathop{\mathrm{Ext}}\nolimits ^ i_ R(\kappa , K) = 0$ for $i \gg 0$ .

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

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