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

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 topoplogy 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 $\mathrm{QCoh}(\mathfrak{X})$ 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 $(A,\Gamma)$?

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(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.
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).