Lemma 22.37.8. 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.

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