Lemma 15.87.5. Let (A_ n) be an inverse system of rings. Suppose that we are given
for every n an object K_ n of D(A_ n), and
for every n a map \varphi _ n : K_{n + 1} \to K_ n of D(A_{n + 1}) where we think of K_ n as an object of D(A_{n + 1}) by restriction via A_{n + 1} \to A_ n.
There exists an object M = (M_ n^\bullet ) \in D(\textit{Mod}(\mathbf{N}, (A_ n))) and isomorphisms \psi _ n : M_ n^\bullet \to K_ n in D(A_ n) such that the diagrams
\xymatrix{ M_{n + 1}^\bullet \ar[d]_{\psi _{n + 1}} \ar[r] & M_ n^\bullet \ar[d]^{\psi _ n} \\ K_{n + 1} \ar[r]^{\varphi _ n} & K_ n }
commute in D(A_{n + 1}).
Proof.
We write out the proof in detail. For an A_ n-module T we write T_{A_{n + 1}} for the same module viewd as an A_{n + 1}-module. Suppose that K_ n^\bullet is a complex of A_ n-modules representing K_ n. Then K_{n, A_{n + 1}}^\bullet is the same complex, but viewed as a complex of A_{n + 1}-modules. By the construction of the derived category, the map \psi _ n can be given as
\psi _ n = \tau _ n \circ \sigma _ n^{-1}
where \sigma _ n : L_{n + 1}^\bullet \to K_{n + 1}^\bullet is a quasi-isomorphism of complexes of A_{n + 1}-modules and \tau _ n : L_{n + 1}^\bullet \to K_{n, A_{n + 1}}^\bullet is a map of complexes of A_{n + 1}-modules.
Now we construct the complexes M_ n^\bullet by induction. As base case we let M_1^\bullet = K_1^\bullet . Suppose we have already constructed M_ e^\bullet \to M_{e - 1}^\bullet \to \ldots \to M_1^\bullet and maps of complexes \psi _ i : M_ i^\bullet \to K_ i^\bullet such that the diagrams
\xymatrix{ M_{n + 1}^\bullet \ar[d]_{\psi _{n + 1}} \ar[rr] & & M_{n, A_{n + 1}}^\bullet \ar[d]^{\psi _{n, A_{n + 1}}} \\ K_{n + 1}^\bullet & L_{n + 1}^\bullet \ar[l]_{\sigma _ n} \ar[r]^{\tau _ n} & K_{n, A_{n + 1}}^\bullet }
above commute in D(A_{n + 1}) for all n < e. Then we consider the diagram
\xymatrix{ & & M_{e, A_{e + 1}}^\bullet \ar[d]^{\psi _{e, A_{e + 1}}} \\ K_{e + 1}^\bullet & L_{e + 1}^\bullet \ar[r]^{\tau _ e} \ar[l]_{\sigma _ e} & K_{e, A_{e + 1}}^\bullet }
in D(A_{e + 1}). Because \psi _ e is a quasi-isomorphism, we see that \psi _{e, A_{e + 1}} is a quasi-isomorphism too. By the definition of morphisms in D(A_{e + 1}) we can find a quasi-isomorphism \psi _{e + 1} : M_{e + 1}^\bullet \to K_{e + 1}^\bullet of complexes of A_{e + 1}-modules such that there exists a morphism of complexes M_{e + 1}^\bullet \to M_{e, A_{e + 1}}^\bullet of A_{e + 1}-modules representing the composition \psi _{e, A_{e + 1}}^{-1} \circ \tau _ e \circ \sigma _ e^{-1} in D(A_{e + 1}). Thus the lemma holds by induction.
\square
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