Are the derived categories of modular representations of algebraic groups compactly generated? (e.g. consider SL\_2 in characteristic 2). Note modular reps of finite groups are compactly generated (by the regular representation) - that's an example of compact generation of modules for an algebra. But here we're asking about comodules for a coalgebra that's not dualizable so it's not immediately clear (to me).


This makes more specific my other question for any "nice" examples of non-compactly generated categories.



Compact generation for modular representations
It appears this question is resolved in a definitive fashion in today's preprint
[Algebraic Groups and compact generation of
their derived categories of representations](http://arxiv.org/pdf/1405.1890v1.pdf) by Hall and Rydh. Their first theorem asserts that the quasicoherent derived category of the stack $BG$ for $G$ a group scheme of finite type over a field $k$ is compactly generated if and only if either the characteristic of $k$ is zero, or the component group of the reductive part of $G$ (after
base change to $\overline{k}$) is SEMI-ABELIAN - or equivalently, contains no additive groups. In particular semisimple groups in (any) positive characteristic are out. 
They then deduce (using another paper of theirs from today, with Neeman) that even for $G$ affine in these ``poor" cases the quasicoherent derived category differs from the derived category of representations 
(ie quasicoherent sheaves on $BG$), and other striking results.


