The Stacks project

Lemma 15.93.4. Let $A$ be a Noetherian ring and $I \subset A$ an ideal. Let $K$ be an object of $D(A)$ such that $H^ n(K)$ a finite $A$-module for all $n \in \mathbf{Z}$. Then the cohomology modules $H^ n(K^\wedge )$ of the derived completion are the $I$-adic completions of the cohomology modules $H^ n(K)$.

Proof. The complex $\tau _{\leq m}K$ is pseudo-coherent for all $m$ by Lemma 15.63.17. Thus $\tau _{\leq m}K$ is represented by a bounded above complex $P^\bullet $ of finite free $A$-modules. Then $\tau _{\leq m}K \otimes _ A^\mathbf {L} A/I^ n = P^\bullet /I^ nP^\bullet $. Hence $(\tau _{\leq m}K)^\wedge = R\mathop{\mathrm{lim}}\nolimits P^\bullet /I^ nP^\bullet $ (Proposition 15.93.2) and since the $R\mathop{\mathrm{lim}}\nolimits $ is just given by termwise $\mathop{\mathrm{lim}}\nolimits $ (Lemma 15.86.1) and since $I$-adic completion is an exact functor on finite $A$-modules (Algebra, Lemma 10.97.2) we conclude the result holds for $\tau _{\leq m}K$. Hence the result holds for $K$ as derived completion has finite cohomological dimension, see Lemma 15.90.20. $\square$


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