The Stacks project

Lemma 52.11.1. Let $U$ be the punctured spectrum of a Noetherian local ring $A$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ U$-module. Let $I \subset A$ be an ideal. Then

\[ H^ i(R\Gamma (U, \mathcal{F})^\wedge ) = \mathop{\mathrm{lim}}\nolimits H^ i(U, \mathcal{F}/I^ n\mathcal{F}) \]

for all $i$ where $R\Gamma (U, \mathcal{F})^\wedge $ denotes the derived $I$-adic completion.

Proof. By Lemmas 52.6.20 and 52.7.2 we have

\[ R\Gamma (U, \mathcal{F})^\wedge = R\Gamma (U, \mathcal{F}^\wedge ) = R\Gamma (U, R\mathop{\mathrm{lim}}\nolimits \mathcal{F}/I^ n\mathcal{F}) \]

Thus we obtain short exact sequences

\[ 0 \to R^1\mathop{\mathrm{lim}}\nolimits H^{i - 1}(U, \mathcal{F}/I^ n\mathcal{F}) \to H^ i(R\Gamma (U, \mathcal{F})^\wedge ) \to \mathop{\mathrm{lim}}\nolimits H^ i(U, \mathcal{F}/I^ n\mathcal{F}) \to 0 \]

by Cohomology, Lemma 20.37.1. The $R^1\mathop{\mathrm{lim}}\nolimits $ terms vanish because the inverse systems of groups $H^ i(U, \mathcal{F}/I^ n\mathcal{F})$ satisfy the Mittag-Leffler condition by Lemma 52.5.2. $\square$

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