Let X and Y be to varieties and $F\colon D\mathrm{QCoh}(X) \to D\mathrm{QCoh}(Y)$ a continuous functor between the corresponding unbounded derived categories of quasi-coherent sheaves (given by a kernel on X×Y). Assume that $F(D^b\mathrm{Coh}(X)) \subseteq D^b\mathrm{Coh}(Y)$ and that F is conservative.


Are there any conditions that ensure that $F(\mathcal{G}) \in D^b\mathrm{Coh}(Y)$ implies $\mathcal{G} \in D^b\mathrm{Coh}(X)$?


The corresponding statement for the abelian categories is easy (if $\mathcal G$ is not coherent, then there is an infinite increasing sequence of coherent subsheaves giving rise to such a sequence of subsheaves of $F(\mathcal G)$, which has to stabilize), but I don't seem to be able to transport this proof to the derived setting.



When is a sheaf coherent if its image under a Fourier-Mukai transform is coherent?
Your proof works in the derived case as well.


That is, assume smoothness so that $D^bCoh$ is identified with the full subcategory of compact objects (in general the argument will apply to the subcategory of perfect complexes). Then every object $\mathcal{G}$ of $DQCoh$ can be written as a filtered homotopy colimit $colim \mathcal{G}\_\alpha$ of objects of $D^bCoh$, as there is an equivalence of infinity-categories $DQCoh(X) = Ind(D^bCoh(X))$. If $F(G) = colim F(\mathcal{G}\_\alpha)$ is compact, then you get $F(G) = F(G\_\beta)$ for some $\beta$, and use conservativity of $F$ to conclude.


