The Stacks project

Lemma 15.86.8. Let $(K_ n)$ be an inverse system of objects of $D(\textit{Ab})$. Then there exists an object $M = (M_ n^\bullet )$ of $D(\textit{Ab}(\mathbf{N}))$ and isomorphisms $M_ n^\bullet \to K_ n$ in $D(\textit{Ab})$ such that the diagrams

\[ \xymatrix{ M_{n + 1}^\bullet \ar[d] \ar[r] & M_ n^\bullet \ar[d] \\ K_{n + 1} \ar[r] & K_ n } \]

commute in $D(\textit{Ab})$.

Proof. Namely, let $M_1^\bullet $ be a complex of abelian groups representing $K_1$. Suppose we have constructed $M_ e^\bullet \to M_{e - 1}^\bullet \to \ldots \to M_1^\bullet $ and maps $\psi _ i : M_ i^\bullet \to K_ i$ such that the diagrams in the statement of the lemma commute for all $n < e$. Then we consider the diagram

\[ \xymatrix{ & M_ n^\bullet \ar[d]^{\psi _ n} \\ K_{n + 1} \ar[r] & K_ n } \]

in $D(\textit{Ab})$. By the definition of morphisms in $D(\textit{Ab})$ we can find a complex $M_{n + 1}^\bullet $ of abelian groups, an isomorphism $M_{n + 1}^\bullet \to K_{n + 1}$ in $D(\textit{Ab})$, and a morphism of complexes $M_{n + 1}^\bullet \to M_ n^\bullet $ representing the composition

\[ K_{n + 1} \to K_ n \xrightarrow {\psi _ n^{-1}} M_ n^\bullet \]

in $D(\textit{Ab})$. Thus the lemma holds by induction. $\square$

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