The Stacks project

Lemma 15.87.5. Let $(A_ n)$ be an inverse system of rings. Suppose that we are given

  1. for every $n$ an object $K_ n$ of $D(A_ n)$, and

  2. 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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