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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.

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For the education of the rest of us, would you mind posting a link to the definition of a compactly generated category? (It's not in n-Lab). Thanks! –  David Speyer Nov 11 '09 at 15:13
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Suppose that T is a triangulated category with all small coproducts. An object c of T is compact if Hom(c,-) commutes with coproducts, i.e. every map from c to a coproduct factors through a finite sub-coproduct. T is compactly generated if there exists a small suspension closed set G of compact objects such that Hom(g,X) = 0 for all g in G iff X=0 for any object X of T. Comments are not really the ideal place for this - but it works for the moment I guess. –  Greg Stevenson Nov 12 '09 at 6:17
    
Thanks, Greg! That's very helpful. –  David Speyer Nov 28 '09 at 17:49

3 Answers 3

This is a good question the answer to which I unfortunately do not know, so let me give answers to three different questions instead.

  1. 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.

  2. 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).

  3. 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.

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Not an example per se, but in Mark Hovey's book on model categories he constructs a model category of chain complexes of comodules over a coalgebra whose homotopy category seems to be what you call the coderived category. –  Reid Barton Nov 28 '09 at 17:35
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Leonid - thanks, very interesting! Perhaps I'm confused, but isn't descent a good reason to have the derived category of comodules? i.e. if X->Y is a morphism of stacks say we get a coalgebra on X (coming from the adjunction comonad), and I think the comonadic version of sheaves on Y is given by the unbounded derived category of comodules? (more precisely, this should be the homotopy category of the descent dg- or oo-category)? if the right answer in descent turns out to be the coderived category, I'll be completely sold. –  David Ben-Zvi Nov 28 '09 at 19:49
    
This is a more complicated situation, since sheaves on Y are described as comodules over a coring over a ring (assuming X is affine), rather than simply over a coalgebra over a field. Comodules give you the abelian category of sheaves on Y, and then it is up to you how do you want to derive it. When X is smooth and finite-dimensional, you can just do the coderived category of comodules and I am not aware of any reason why it might be not good enough or how the derived category could be better. –  Leonid Positselski Nov 28 '09 at 20:14
    
When the homological dimension of X is infinite, I am not sure what the "right" derived category is or how to even pose the question. I tried to think about this in terms of the Kontsevich-Rosenberg noncommutative stacks and attempted to construct a double-sided derived functor of the functor of tensor product of a right sheaf with a left sheaf, producing a vector space (the cotensor product of comodules). I found a definition of such a derived functor, but it wasn't pretty and involved replacing vector spaces with provector spaces everywhere. –  Leonid Positselski Nov 28 '09 at 20:26
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Thanks! that settles the question: the object that appears in descent is then the derived, not coderived, category of comodules over the corresponding coalgebra. e.g. for G a finite group scheme, Rep G= sheaves on BG is the derived category of the group coalgebra C, which is the derived category of the group algebra C* (dual to functions on the group), not the coderived category. Likewise for example in cyclic homology theory the category that naturally arises is the derived category of Omega, not its coderived category - so not exactly D-modules (but rather a completion of D-modules). –  David Ben-Zvi Nov 28 '09 at 22:00
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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.

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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.

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