The Stacks project

Proof. Choose generators $f_1, \ldots , f_ r \in I$ of $I$. The inverse system $I^ n$ is pro-isomorphic to the inverse system $(f_1^ n, \ldots , f_ r^ n)$ in the category of $A$-modules. With notation as in Lemma 15.101.5 we find that it suffices to prove the inverse system of maps

defines an isomorphism of pro-objects of $D(A)$. Say we have $a \leq b$ such that $M^ i = 0$ if $i \not\in [a, b]$. Then source and target of the arrows above have cohomology only in degrees $[-r + a, b]$. Thus it suffices to show that for any $p \in \mathbf{Z}$ the inverse system of maps

defines an isomorphism of pro-objects of $A$-modules, see Derived Categories, Lemma 13.42.5. Using the pro-isomorphism between $I_ n^\bullet \otimes _ A^\mathbf {L} M^\bullet $ and $I^ n \otimes _ A^\mathbf {L} M^\bullet $ and the pro-isomorphism between $(f_1^ n, \ldots , f_ r^ n)M^\bullet $ and $I^ nM^\bullet $ this is equivalent to showing that the inverse system of maps

defines an isomorphism of pro-objects of $A$-modules Choose a bounded above complex of finite free $A$-modules $P^\bullet $ and a quasi-isomorphism $P^\bullet \to M^\bullet $. Then it suffices to show that the inverse system of maps

is a pro-isomorphism. This follows from Lemma 15.101.1 as $H^ p(P^\bullet ) = H^ p(M^\bullet )$. $\square$