The Stacks project

Lemma 22.37.4. Let $R$ be a ring. Let $(A, \text{d})$ and $(B, \text{d})$ be differential graded $R$-algebras. Assume that $A = H^0(A)$. The following are equivalent

  1. $D(A, \text{d})$ and $D(B, \text{d})$ are equivalent as $R$-linear triangulated categories, and

  2. there exists an object $P$ of $D(B, \text{d})$ such that

    1. $P$ is a compact object of $D(B, \text{d})$,

    2. if $N \in D(B, \text{d})$ with $\mathop{\mathrm{Hom}}\nolimits _{D(B, \text{d})}(P, N[i]) = 0$ for $i \in \mathbf{Z}$, then $N = 0$,

    3. $\mathop{\mathrm{Hom}}\nolimits _{D(B, \text{d})}(P, P[i]) = 0$ for $i \not= 0$ and equal to $A$ for $i = 0$.

The equivalence $D(A, \text{d}) \to D(B, \text{d})$ constructed in (2) sends $A$ to $P$.

Proof. Let $F : D(A, \text{d}) \to D(B, \text{d})$ be an equivalence. Then $F$ maps compact objects to compact objects. Hence $P = F(A)$ is compact, i.e., (2)(a) holds. Conditions (2)(b) and (2)(c) are immediate from the fact that $F$ is an equivalence.

Let $P$ be an object as in (2). Represent $P$ by a differential graded module with property (P). Set

\[ (E, \text{d}) = \mathop{\mathrm{Hom}}\nolimits _{\text{Mod}^{dg}_{(B, \text{d})}}(P, P) \]

Then $H^0(E) = A$ and $H^ k(E) = 0$ for $k \not= 0$ by Lemma 22.22.3 and assumption (2)(c). Viewing $P$ as a $(E, B)$-bimodule and using Lemma 22.37.2 and assumption (2)(b) we obtain an equivalence

\[ D(E, \text{d}) \to D(B, \text{d}) \]

sending $E$ to $P$. 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. $\square$

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