Let $(X,\mathcal{O}_X)$ and $(Y,\mathcal{O}_Y)$ be schemes such that their derived categories of modules are equivalent via $F:D(\mathcal{O}_X)\to D(\mathcal{O}_Y)$. Is it true that $F(\mathcal{O}_X)=\mathcal{O}_Y$, as complexes centered in degree zero?
My attempt is to use some intrinsic characterization of the structure sheaf, such as the unique satisfying $\mathrm{Hom}(\mathcal{O}_X,E^{\bullet})\cong E^0(X)$, in order to say that this is isomorphic to $\mathrm{Hom}(F\mathcal{O}_X,F(E^{\bullet}))$, but I really don't know whether $E^0(X)$ is actually isomorphic to $(FE^{\bullet})^0(Y)$ in order to deduce $F\mathcal{O}_X$ to be $\mathcal{O}_Y$.
Any help is appreciated.
No, it is not true. For instance the derived equivalence of an abelian variety and its dual takes the structure sheaf to a skyscraper sheaf.