The Stacks project

Lemma 24.35.1. In the situation above, suppose that $\mathcal{A} = (A_ n)$ and $\mathcal{B} = (B_ n)$ are inverse systems of differential graded $R$-algebras. If $\varphi : (A_ n) \to (B_ n)$ is an isomorphism of pro-objects, then the functor $\mathit{QC}(\mathcal{A}, \text{d}) \to \mathit{QC}(\mathcal{B}, \text{d})$ constructed above is an equivalence.

Proof. Let $\psi : (B_ n) \to (A_ n)$ be a morphism of pro-objects which is inverse to $\varphi $. According to the discussion in Categories, Example 4.22.6 we may assume that $\varphi $ is given by a system of maps as above and $\psi $ is given $n(1) < n(2) < \ldots $ and a commutative diagram

\[ \xymatrix{ \ldots \ar[r] & B_{n(3)} \ar[d]^{\psi _3} \ar[r] & B_{n(2)} \ar[d]^{\psi _2} \ar[r] & B_{n(1)} \ar[d]^{\psi _1} \\ \ldots \ar[r] & A_3 \ar[r] & A_2 \ar[r] & A_1 } \]

of differential graded $R$-algebras. Since $\varphi \circ \psi = \text{id}$ we may, after possibly increasing the values of the functions $n(\cdot )$ and $m(\cdot )$ assume that $B_{n(m(i))} \to A_{m(i)} \to B_ i$ is the identity. It follows that the composition of the functors

\[ \mathit{QC}(\mathcal{B}, \text{d}) \to \mathit{QC}(\mathcal{A}, \text{d}) \to \mathit{QC}(\mathcal{B}, \text{d}) \]

sends a good sheaf of differential graded $\mathcal{B}$-modules $\mathcal{N} = (N_ n)$ to the inverse system $\mathcal{N}' = (N'_ i)$ with values

\[ N'_ i = N_{n(m(i))} \otimes _{B_{n(m(i))}} B_ i \]

which is canonically quasi-isomorphic to $\mathcal{N}$ exactly because $\mathcal{N}$ is an object of $\mathit{QC}(\mathcal{B}, \text{d})$ and because $N_ j$ is a K-flat differential graded module for all $j$. Since the same is true for the composition the other way around we conclude. $\square$

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