6
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ well, closed immersions detects compact objects. $\endgroup$ – Elden Elmanto Sep 14 '16 at 21:46
3
$\begingroup$

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.

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.