The Stacks project

Lemma 15.87.8. 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 $(K_ n^\bullet )$ such that each $K_ n^\bullet $ is K-flat.

Proof. First use Lemma 15.87.7 to represent $K$ by a system of complexes $(M_ n^\bullet )$ such that all the transition maps $M_{n + 1}^\bullet \to M_ n^\bullet $ are surjective. Next, let $K_1^\bullet \to M_1^\bullet $ be a quasi-isomorphism with $K_1^\bullet $ a K-flat complex of $A_1$-modules (Lemma 15.59.10). Suppose we have constructed $K_ n^\bullet \to K_{n - 1}^\bullet \to \ldots \to K_1^\bullet $ and maps of complexes $\psi _ e : K_ e^\bullet \to M_ e^\bullet $ such that

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

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

\[ \xymatrix{ C^\bullet \ar@{..>}[d] \ar@{..>}[r] & K_ n^\bullet \ar[d]^{\psi _ n} \\ M_{n + 1}^\bullet \ar[r]^{\varphi _ n} & M_ n^\bullet } \]

in $D(A_{n + 1})$. As $M_{n + 1}^\bullet \to M_ n^\bullet $ is termwise surjective, the complex $C^\bullet $ fitting into the left upper corner with terms

\[ C^ p = M_{n + 1}^ p \times _{M_ n^ p} K_ n^ p \]

is quasi-isomorphic to $M_{n + 1}^\bullet $ (details omitted). Choose a quasi-isomorphism $K_{n + 1}^\bullet \to C^\bullet $ with $K_{n +1}^\bullet $ K-flat. Thus the lemma holds by induction. $\square$

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