The Stacks project

Lemma 15.100.3. Let $A$ be a Noetherian ring. Let $I \subset A$ be an ideal. Let $M^\bullet $ be a bounded complex of finite $A$-modules. The inverse system of maps

\[ M^\bullet \otimes _ A^\mathbf {L} A/I^ n \longrightarrow M^\bullet /I^ nM^\bullet \]

defines an isomorphism of pro-objects of $D(A)$.

Proof. Say $I = (f_1, \ldots , f_ r)$. Let $K_ n \in D(A)$ be the object represented by the Koszul complex on $f_1^ n, \ldots , f_ r^ n$. Recall that we have maps $K_ n \to A/I^ n$ which induce a pro-isomorphism of inverse systems, see Lemma 15.94.1. Hence it suffices to show that

\[ M^\bullet \otimes _ A^\mathbf {L} K_ n \longrightarrow M^\bullet /I^ nM^\bullet \]

defines an isomorphism of pro-objects of $D(A)$. Since $K_ n$ is represented by a complex of finite free $A$-modules sitting in degrees $-r, \ldots , 0$ there exist $a, b \in \mathbf{Z}$ such that the source and target of the displayed arrow have vanishing cohomology in degrees outside $[a, b]$ for all $n$. Thus we may apply Derived Categories, Lemma 13.42.5 and we find that it suffices to show that the maps

\[ H^ i(M^\bullet \otimes _ A^\mathbf {L} A/I^ n) \to H^ i(M^\bullet /I^ nM^\bullet ) \]

define isomorphisms of pro-systems of $A$-modules for any $i \in \mathbf{Z}$. To see this choose a quasi-isomorphism $P^\bullet \to M^\bullet $ where $P^\bullet $ is a bounded above complex of finite free $A$-modules. The arrows above are given by the maps

\[ H^ i(P^\bullet /I^ nP^\bullet ) \to H^ i(M^\bullet /I^ nM^\bullet ) \]

These define an isomorphism of pro-systems by Lemma 15.100.1. Namely, the lemma shows both are isomorphic to the pro-system $H^ i/I^ nH^ i$ with $H^ i = H^ i(M^\bullet ) = H^ i(P^\bullet )$. $\square$

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