The Stacks project

Lemma 15.87.7. Let $(A_ n)$ be an inverse system of rings. Every $K \in D(\textit{Mod}(\mathbf{N}, (A_ n)))$ can be represented by a system of complexes $(M_ n^\bullet )$ such that all the transition maps $M_{n + 1}^\bullet \to M_ n^\bullet $ are surjective.

Proof. Let $K$ be represented by the system $(K_ n^\bullet )$. Set $M_1^\bullet = K_1^\bullet $. Suppose we have constructed surjective maps of complexes $M_ n^\bullet \to M_{n - 1}^\bullet \to \ldots \to M_1^\bullet $ and homotopy equivalences $\psi _ e : K_ e^\bullet \to M_ e^\bullet $ such that the diagrams

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

commute for all $e < n$. Then we consider the diagram

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

By Derived Categories, Lemma 13.9.8 we can factor the composition $K_{n + 1}^\bullet \to M_ n^\bullet $ as $K_{n + 1}^\bullet \to M_{n + 1}^\bullet \to M_ n^\bullet $ such that the first arrow is a homotopy equivalence and the second a termwise split surjection. The lemma follows from this and induction. $\square$

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