Is $\mathcal{D} \bigl( \mathrm{QCoh}(\mathfrak{X}) \bigr)$ compactly generated?

Question

An object $E$ in a triangulated category $\mathcal{T}$ with (small) coproducts is called compact if the functor $\mathrm{Hom}_{\mathcal{T}}(E,-)$ commutes with arbitrary coproducts or, equivalently, if any morphism from $E$ to some coproduct factors through a finite subcoproduct.

Neeman (The Grothendieck Duality Theorem..., Prop 2.5) showed that $\mathcal{D} \bigl( \mathrm{QCoh}(X) \bigr)$ is compactly generated if $X$ is a quasi-compact and separated scheme and the proof uses only the semi-separatedness of $X$. Note that compact = perfect. Bondal-Van den Bergh (Generators and representability of functors, Cor 3.1.2) prove this statement for $X$ quasi-compact and quasi-separated.

Whenever it comes to semi-separated and quasi-compact schemes I look for connections to algebraic stacks $\mathfrak{X}$ in the sense of Goerss, Naumann, ..., i.e., a stack in the flat topology with affine diagonal and some faithfully flat atlas from some affine scheme. Those schemes which are algebraic stacks are precisely the semi-separated and quasi-compact ones. Has anyone ever tried to extend the proof of Neeman to show that $\mathcal{D} \bigl(\mathrm{QCoh}(\mathfrak{X})\bigr)$ is compactly generated?

The 2-category of algebraic stacks with some fixed atlas is equivalent to the 2-category of flat Hopf algebroids (Naumann, The stack of formal groups..., §3.3). Under this equivalence quasi-coherent sheaves correspond to comodules over the associated Hopf algebroid. So a reformulation is: Is $\mathcal{D} \bigl( \mathrm{Comod}(\Gamma) \bigr)$ compactly generated for every flat Hopf algebroid $(A,\Gamma)$?

Answer 1

I don't know the answer to your question in general (on what I would call "fpqc-stacks with affine diagonal") but the answer is at least true for many algebraic stacks. In particular, in Perfect complexes on algebraic stacks arXiv:1405.1887 it is shown that:

(1) $\mathcal{D}_{qc}(X)$ is compactly generated by a single object for every quasi-compact algebraic stack with quasi-finite and separated diagonal (e.g., $X$ Deligne–Mumford).

(2) $\mathcal{D}_{qc}(X)$ is compactly generated for every algebraic stack that étale-locally is of the form $[U/GL_n]$ with $U$ quasi-affine of characteristic zero.

Some earlier results (see references) for stacks are due to Toën and Lurie. All current proofs of compact generation of derived categories for schemes, algebraic spaces and stacks involve gluing compact generators (over open coverings in the scheme case and over étale coverings in the case above) via Thomason's localization theorem. Proving compact generation in the generality you ask for would thus probably require completely new methods.

When $X$ has affine diagonal (as for your stacks), then sometimes $\mathcal{D}_{qc}(X)=\mathcal{D}(\mathrm{QCoh}(X))$ but this is closely related to compact generation (see arXiv:1405.1888).

Answer 2

Neeman found a counterexample in a 2014 paper, $B\mathbb{G}_a,$ the classifying stack of the additive group scheme in positive characteristic. The original reference is rather inexplicit, but this followup paper by Hall, Neeman, and Rydh proves an even stronger fact, that $D_{qc}(B\mathbb{G}_a)$ in positive characteristic has no nonzero compact objects whatsoever.