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. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0CQ9. Beware of the difference between the letter 'O' and the digit '0'.