The Stacks project

Remark 22.37.5. Let $R$ be a ring. Let $(A, \text{d})$ and $(B, \text{d})$ be differential graded $R$-algebras. Suppose given an $R$-linear equivalence

\[ F : D(A, \text{d}) \longrightarrow D(B, \text{d}) \]

of triangulated categories. Set $N = F(A)$. Then $N$ is a differential graded $B$-module. Since $F$ is an equivalence and $A$ is a compact object of $D(A, \text{d})$, we conclude that $N$ is a compact object of $D(B, \text{d})$. Since $A$ generates $D(A, \text{d})$ and $F$ is an equivalence, we see that $N$ generates $D(B, \text{d})$. Finally, $H^ k(A) = \mathop{\mathrm{Hom}}\nolimits _{D(A, \text{d})}(A, A[k])$ and as $F$ an equivalence we see that $F$ induces an isomorphism $H^ k(A) = \mathop{\mathrm{Hom}}\nolimits _{D(B, \text{d})}(N, N[k])$ for all $k$. In order to conclude that there is an equivalence $D(A, \text{d}) \longrightarrow D(B, \text{d})$ which arises from the construction in Lemma 22.37.2 all we need is a left $A$-module structure on $N$ compatible with derivation and commuting with the given right $B$-module structure. In fact, it suffices to do this after replacing $N$ by a quasi-isomorphic differential graded $B$-module. The module structure can be constructed in certain cases. For example, if we assume that $F$ can be lifted to a differential graded functor

\[ F^{dg} : \text{Mod}^{dg}_{(A, \text{d})} \longrightarrow \text{Mod}^{dg}_{(B, \text{d})} \]

(for notation see Example 22.26.8) between the associated differential graded categories, then this holds. Another case is discussed in the proposition below.

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