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
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
Then there are quasi-isomorphisms of differential graded algebras $(A, \text{d}) \leftarrow (E', \text{d}) \rightarrow (E, \text{d})$. Thus we obtain equivalences
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. $\square$
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)