Theorem 59.18.8 (03OS)—The Stacks project

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Part 3: Topics in Scheme Theory
Chapter 59: Étale Cohomology
Section 59.18: Čech cohomology
Theorem 59.18.8 (cite)

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Theorem 59.18.8. Notation and assumptions as in Definition 59.18.1. On $\textit{PAb}(\mathcal{C})$ the functors $\check{H}^ p(\mathcal{U}, -)$ are the right derived functors of $\check{H}^0(\mathcal{U}, -)$ .

Proof. By the Lemma 59.18.7, the functors $\check H^ p(\mathcal{U}, -)$ are universal $\delta$ -functors since they are effaceable. So are the right derived functors of $\check H^0(\mathcal{U}, -)$ . Since they agree in degree $0$ , they agree by the universal property of universal $\delta$ -functors. For more details see Cohomology on Sites, Lemma 21.9.6. $\square$

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