Compact generation for modular representations 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.
 A: 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 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.
A: This is a good question the answer to which I unfortunately do not know, so let me give answers to three different questions instead.


*

*The derived category of comodules over any coalgebra (over a field) is well generated in the sense of Neeman and Krause.  The same applies to the derived category of DG-comodules over a DG-coalgebra.  One can prove this using the results of Krause's paper on localization theory for triangulated categories together with the next assertion #2.  Well-generated triangulated categories are technically more complicated than compactly generated ones, but for some applications they are just as good.  The contravariant (= most usual) Brown representability holds in well-generated triangulated categories, while the covariant version may not.

*I have a philosophy that one is not supposed to consider the derived categories of comodules.  Derived categories are good for modules or DG-modules, perhaps sometimes for sheaves, but not for comodules.  For comodules, one is supposed to consider the coderived category.  This is the quotient category of the category of complexes of comodules by an equivalence relation more delicate than the conventional quasi-isomorphism.  The simplest, if not always the best, definition is that the coderived category of comodules is the homotopy category of arbitrary complexes of injective comodules.  The point is that the coderived category of comodules (DG-comodules, CDG-comodules) is always compactly generated, the compact generators being the totally finite-dimensional complexes.  The subcategory of compactly generated objects is simply the bounded derived category of finite-dimensional comodules (this is true for comodules, not for DG-comodules).

*When the category of comodules over a coalgebra C has a finite homological dimension, its derived and coderived categories coincide.  So the derived category of comodules over a coalgebra of finite homological dimension is compactly generated.  It may follow that the derived category of representations of a (finite-dimensional) algebraic group in characteristic 0 is always compactly generated, but this does not apply to modular representations in general.
I am interested in any examples that may tend to argue pro or contra my philosophy as stated in #2, so if anyone knows of a situation when the unbounded derived category of comodules, as distinguished from the homotopy category of complexes of injective comodules, turns out to be good or bad for whatever purpose, please let me know.
A: You might want to try tilting modules.  Those at least provide a generating set with trivial self-extensions.
If I recall correctly, each tilting is left and right orthogonal to all but finitely many simples, and vice-versa, and every finite-dimensional module has a finite-length tilting resolution.  
That is, the derived category of finite-dimensional modular representations is derived equivalent to bounded, finite rank perfect complexes over the endomorphism ring of the tiltings. 
