The Stacks project

Lemma 15.100.5. In Situation 15.90.15 assume $A$ is Noetherian. With notation as above, the inverse system $(I^ n)$ is pro-isomorphic in $D(A)$ to the inverse system $(I_ n^\bullet )$.

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

\[ I_ n^\bullet \to (f_1^ n, \ldots , f_ r^ n) \to C_ n^\bullet \to I_ n^\bullet [1] \]

where $C_ n^\bullet $ is the cone of the first arrow. By Derived Categories, Lemma 13.41.4 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 13.41.3 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 15.93.1 (and this is where the assumption that $A$ is Noetherian is used). $\square$


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