Lemma 52.5.4. Let $(A, \mathfrak m)$ be a Noetherian local ring. Let $I \subset A$ be an ideal. Let $M$ be a finite $A$-module. Then

for all $i$ where $R\Gamma _\mathfrak m(M)^\wedge $ denotes the derived $I$-adic completion.

Lemma 52.5.4. Let $(A, \mathfrak m)$ be a Noetherian local ring. Let $I \subset A$ be an ideal. Let $M$ be a finite $A$-module. Then

\[ H^ i(R\Gamma _\mathfrak m(M)^\wedge ) = \mathop{\mathrm{lim}}\nolimits H^ i_\mathfrak m(M/I^ nM) \]

for all $i$ where $R\Gamma _\mathfrak m(M)^\wedge $ denotes the derived $I$-adic completion.

**Proof.**
Apply Dualizing Complexes, Lemma 47.12.4 and Lemma 52.5.2 to see the vanishing of the $R^1\mathop{\mathrm{lim}}\nolimits $ terms.
$\square$

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