An associative algebra $A$ is said to be Morita equivalent to another one $B$ if there is an equivalence $$\mathsf{Mod}\_A\simeq \mathsf{Mod}\_B$$ between its corresponding abelian categories of modules. Moreover, whenever $A$ and $B$ are commutative, they are Morita equivalent iff they are isomorphic. On the other hand, we say that $A$ is derived Morita equivalent to $B$ if there is an equivalence $$D(A)\simeq D(B)$$ between its triangulated derived categories. My question is:


Q. If $A$ is commutative and derived Morita equivelent to $B$, is $A$ Morita equivalent to $B$? 



Derived Morita equivalence of associative algebras
This seems to be true by <https://arxiv.org/pdf/math/9810134.pdf> , theorem 2.7.


