## 22.28 Equivalences of derived categories

Let $R$ be a ring. Let $(A, \text{d})$ and $(B, \text{d})$ be differential graded $R$-algebras. A natural question that arises in nature is what it means that $D(A, \text{d})$ is equivalent to $D(B, \text{d})$ as an $R$-linear triangulated category. This is a rather subtle question and it will turn out it isn't always the correct question to ask. Nonetheless, in this section we collection some conditions that guarantee this is the case.

We strongly urge the reader to take a look at the groundbreaking paper [Rickard] on this topic.

Lemma 22.28.1. Let $R$ be a ring. Let $(A, \text{d}) \to (B, \text{d})$ be a homomorphism of differential graded algebras over $R$, which induces an isomorphism on cohomology algebras. Then

\[ - \otimes _ A^\mathbf {L} B : D(A, \text{d}) \to D(B, \text{d}) \]

gives an $R$-linear equivalence of triangulated categories with quasi-inverse the restriction functor $N \mapsto N_ A$.

**Proof.**
By Lemma 22.24.6 the functor $M \longmapsto M \otimes _ A^\mathbf {L} B$ is fully faithful. By Lemma 22.24.4 the functor $N \longmapsto R\mathop{\mathrm{Hom}}\nolimits (B, N) = N_ A$ is a right adjoint, see Example 22.24.5. It is clear that the kernel of $R\mathop{\mathrm{Hom}}\nolimits (B, -)$ is zero. Hence the result follows from Derived Categories, Lemma 13.7.2.
$\square$

When we analyze the proof above we see that we obtain the following generalization for free.

Lemma 22.28.2. Let $R$ be a ring. Let $(A, \text{d})$ and $(B, \text{d})$ be differential graded algebras over $R$. Let $N$ be an $(A, B)$-bimodule which comes with a grading and a differential such that it is a differential graded module for both $A$ and $B$. Assume that

$N$ defines a compact object of $D(B, \text{d})$,

if $N' \in D(B, \text{d})$ and $\mathop{\mathrm{Hom}}\nolimits _{D(B, \text{d})}(N, N'[n]) = 0$ for $n \in \mathbf{Z}$, then $N' = 0$, and

the map $H^ k(A) \to \mathop{\mathrm{Hom}}\nolimits _{D(B, \text{d})}(N, N[k])$ is an isomorphism for all $k \in \mathbf{Z}$.

Then

\[ - \otimes _ A^\mathbf {L} N : D(A, \text{d}) \to D(B, \text{d}) \]

gives an $R$-linear equivalence of triangulated categories.

**Proof.**
By Lemma 22.24.6 the functor $M \longmapsto M \otimes _ A^\mathbf {L} N$ is fully faithful. By Lemma 22.24.4 the functor $N' \longmapsto R\mathop{\mathrm{Hom}}\nolimits (N, N')$ is a right adjoint. By assumption (3) the kernel of $R\mathop{\mathrm{Hom}}\nolimits (N, -)$ is zero. Hence the result follows from Derived Categories, Lemma 13.7.2.
$\square$

Sometimes the $B$-module $P$ in the lemma below is called an “$(A, B)$-tilting complex”.

Lemma 22.28.4. 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

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

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

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

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 $N = 0$,

$\mathop{\mathrm{Hom}}\nolimits _{D(B, \text{d})}(P, P[i]) = 0$ 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

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

Then $H^0(E) = A$ and $H^ k(E) = 0$ for $k \not= 0$ by Lemma 22.15.3 and assumption (2)(c). Viewing $P$ as a $(E, B)$-bimodule and using Lemma 22.28.2 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 22.28.1.
$\square$

Proposition 22.28.6. 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 22.28.2.

**Proof.**
As in Remark 22.28.5 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 22.15.3. Moreover, by the discussion in Remark 22.28.5 and by Lemma 22.28.2 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 22.28.1. 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 $A^{opp} \otimes _ R E'$-module, see Example 22.24.5. 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 22.25.3.
$\square$

Lemma 22.28.8. Let $R$ be a ring. Let $A$ and $B$ be $R$-algebras. The following are equivalent

there is an $R$-linear equivalence $D(A) \to D(B)$ of triangulated categories,

there exists an object $P$ of $D(B)$ such that

$P$ can be represented by a finite complex of finite projective $B$-modules,

if $K \in D(B)$ with $\mathop{\mathrm{Ext}}\nolimits ^ i_ B(P, K) = 0$ for $i \in \mathbf{Z}$, then $K = 0$, and

$\mathop{\mathrm{Ext}}\nolimits ^ i_ B(P, P) = 0$ for $i \not= 0$ and equal to $A$ for $i= 0$.

Moreover, if $B$ is flat as an $R$-module, then this is also equivalent to

there exists an $(A, B)$-bimodule $N$ such that $- \otimes _ A^\mathbf {L} N : D(A) \to D(B)$ is an equivalence.

**Proof.**
The equivalence of (1) and (2) is a special case of Lemma 22.28.4 combined with the result of Lemma 22.27.6 characterizing compact objects of $D(B)$ (small detail omitted). The equivalence with (3) if $B$ is $R$-flat follows from Proposition 22.28.6.
$\square$

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