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.

  • $\begingroup$ well, closed immersions detects compact objects. $\endgroup$ – Elden Elmanto Sep 14 '16 at 21:46

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.

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  • $\begingroup$ Thanks. For non-smooth schemes the same proof works for $D^bCoh$ and IndCoh. $\endgroup$ – Clemens Koppensteiner Sep 14 '16 at 23:09

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