The Stacks project

Proof. It is elementary to show that 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. Consider the inverse system of distinguished triangles

where $C_ n^\bullet$ is the cone of the first arrow. By Derived Categories, Lemma 13.42.4 it suffices to show that the inverse system $C_ n^\bullet$ is pro-zero. The complex $I_ n^\bullet$ has nonzero terms only in degrees $i$ with $-r + 1 \leq i \leq 0$ hence $C_ n^\bullet$ is bounded similarly. Thus by Derived Categories, Lemma 13.42.3 it suffices to show that $H^ p(C_ n^\bullet )$ is pro-zero. By the discussion above we have $H^ p(C_ n^\bullet ) = H^ p(K_ n^\bullet )$ for $p \leq -1$ and $H^ p(C_ n^\bullet ) = 0$ for $p \geq 0$. The fact that the inverse systems $H^ p(K_ n^\bullet )$ are pro-zero was shown in the proof of Lemma 15.94.1 (and this is where the assumption that $A$ is Noetherian is used). $\square$