Lemma 15.94.3. Let $I$ be an ideal of a Noetherian ring $A$. Let $M$ be an $A$-module with derived completion $M^\wedge $. Then there are short exact sequences

A similar result holds for $M \in D^-(A)$.

Lemma 15.94.3. Let $I$ be an ideal of a Noetherian ring $A$. Let $M$ be an $A$-module with derived completion $M^\wedge $. Then there are short exact sequences

\[ 0 \to R^1\mathop{\mathrm{lim}}\nolimits \text{Tor}_{i + 1}^ A(M, A/I^ n) \to H^{-i}(M^\wedge ) \to \mathop{\mathrm{lim}}\nolimits \text{Tor}_ i^ A(M, A/I^ n) \to 0 \]

A similar result holds for $M \in D^-(A)$.

**Proof.**
Immediate consequence of Proposition 15.94.2 and Lemma 15.87.4.
$\square$

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