Remark 22.37.9. Let $R$ be a ring. Let $A$ and $B$ be $R$-algebras. If $D(A)$ and $D(B)$ are equivalent as $R$-linear triangulated categories, then the centers of $A$ and $B$ are isomorphic as $R$-algebras. In particular, if $A$ and $B$ are commutative, then $A \cong B$. The rather tricky proof can be found in [Proposition 9.2, Rickard] or [Proposition 6.3.2, KZ]. Another approach might be to use Hochschild cohomology (see remark below).

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