The Stacks project

Lemma 22.37.8. Let $R$ be a ring. Let $A$ and $B$ be $R$-algebras. The following are equivalent

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

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

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

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

    3. $\mathop{\mathrm{Ext}}\nolimits ^ i_ B(P, P) = 0$ 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

  1. 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 22.37.4 combined with the result of Lemma 22.36.6 characterizing compact objects of $D(B)$ (small detail omitted). The equivalence with (3) if $B$ is $R$-flat follows from Proposition 22.37.6. $\square$

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