Lemma 15.88.7. 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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