Proposition 22.37.6 (09SA)—The Stacks project

Proposition 22.37.6. Let $R$ be a ring. Let $(A, \text{d})$ and $(B, \text{d})$ be differential graded $R$ -algebras. Let $F : D(A, \text{d}) \to D(B, \text{d})$ be an $R$ -linear equivalence of triangulated categories. Assume that

$A = H^0(A)$ , and
$B$ is K-flat as a complex of $R$ -modules.

Then there exists an $(A, B)$ -bimodule $N$ as in Lemma 22.37.2.

Proof. As in Remark 22.37.5 above, we set $N = F(A)$ in $D(B, \text{d})$ . We may assume that $N$ is a differential graded $B$ -module with property (P). Set

$(E, \text{d}) = \mathop{\mathrm{Hom}}\nolimits _{\text{Mod}^{dg}_{(B, \text{d})}}(N, N)$

Then $H^0(E) = A$ and $H^ k(E) = 0$ for $k \not= 0$ by Lemma 22.22.3. Moreover, by the discussion in Remark 22.37.5 and by Lemma 22.37.2 we see that $N$ as a $(E, B)$ -bimodule induces an equivalence $- \otimes _ E^\mathbf {L} N : D(E, \text{d}) \to D(B, \text{d})$ . Let $E' \subset E$ be the differential graded $R$ -subalgebra with

$(E')^ i = \left\{ \begin{matrix} E^ i & \text{if }i < 0 \\ \mathop{\mathrm{Ker}}(E^0 \to E^1) & \text{if }i = 0 \\ 0 & \text{if }i > 0 \end{matrix} \right.$

Then there are quasi-isomorphisms of differential graded algebras $(A, \text{d}) \leftarrow (E', \text{d}) \rightarrow (E, \text{d})$ . Thus we obtain equivalences

$D(A, \text{d}) \leftarrow D(E', \text{d}) \rightarrow D(E, \text{d}) \rightarrow D(B, \text{d})$

by Lemma 22.37.1. Note that the quasi-inverse $D(A, \text{d}) \to D(E', \text{d})$ of the left vertical arrow is given by $M \mapsto M \otimes _ A^\mathbf {L} A$ where $A$ is viewed as a $(A, E')$ -bimodule, see Example 22.33.6. On the other hand the functor $D(E', \text{d}) \to D(B, \text{d})$ is given by $M \mapsto M \otimes _{E'}^\mathbf {L} N$ where $N$ is as above. We conclude by Lemma 22.34.3. $\square$

The Stacks project

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