Timeline for Compact generation for modular representations
Current License: CC BY-SA 2.5
18 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 28, 2009 at 22:10 | comment | added | Leonid Positselski | Well, derived categories of comodules may be hard to work with. For example the derived category of DG-comodules is not an invariant of the quasi-isomorphism class of the DG-coalgebra (Kaledin). The coderived category isn't either, but it isn't supposed to be. On the other hand, there exist Spaltenstein-style unbounded injective resolutions for the derived category of DG-comodules all right. | |
| Nov 28, 2009 at 22:00 | comment | added | David Ben-Zvi | 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). | |
| Nov 28, 2009 at 21:52 | comment | added | Leonid Positselski | I am not sure what a dualizable coalgebra is. If your mean finite-dimensional, then, no, for a finite-dimensional C the coderived category of C-comodules is the coderived category of C*-modules and is not the derived category of C*-modules. If you mean bar-duality, then the coderived category of C is equivalent to the derived category of Cobar(C) provided that C is conilpotent. | |
| Nov 28, 2009 at 21:45 | comment | added | Leonid Positselski | Yes, the coderived category of comodules over the exterior coalgebra is equivalent to the derived category of modules over the symmetric algebra, and the coderived category of Omega-(DG)-modules is equivalent to the derived category of D-modules. That's quite right. | |
| Nov 28, 2009 at 21:43 | comment | added | David Ben-Zvi | Here's a concrete question: if your coalgebra is dualizable, is the coderived category the same as the derived category of modules for the dual algebra? | |
| Nov 28, 2009 at 21:41 | comment | added | David Ben-Zvi | Unfortunately I don't understand model categories. In any case dg categories form a symmetric monoidal oo-category, and given a symmetric monoidal dg category we can speak about associative coalgebras in it, and pass to a dg category of module objects. I don't know how to say this terribly concretely. The Barr-Beck theorem here is just what tells you that (given suitable hypotheses) these comodule categories on X recover sheaves on Y. But I'm now convinced (based on usual Koszul duality) that the homotopy category of this dg category is the derived, not coderived, category. | |
| Nov 28, 2009 at 21:36 | comment | added | David Ben-Zvi | I think I'm beginning to understand: in the case of usual exterior/symmetric duality, you would consider the coderived category of the exterior coalgebra, which is defined in such a way as to get an equivalence - like how one defines D-equivalence of Omega-modules to get an equivalence between Omega and D-modules. In that case, I am quite certain that what I want here is NOT the coderived category -- ie the limit of dg categories (which for me is the canonical answer) will give the derived rather than coderived category (and hence fail to satisfy Koszul duality without size restrictions). | |
| Nov 28, 2009 at 21:35 | comment | added | Leonid Positselski | Could you spell it more precisely -- what is "description as dg comodules for the group coalgebra" and "its homotopy category"? Is it some model structure on the category of complexes of comodules, and if it is, how am I supposed to know anything about this model structure? I am not familiar with Lurie's Barr-Beck theorem, unfortunately; I located it now, but it wasn't immediately clear what to do with it. | |
| Nov 28, 2009 at 21:19 | comment | added | David Ben-Zvi | My favorite example (related to the usual exterior/symmetric Koszul duality) is X=pt, Y=BS^1 (but we work with rational coefficients). We obtain the dg category of S^1 equivariant sheaves on a point as comodules for the coalgebra of cochains on S^1 (with respect to convolution), which we can rewrite as modules for chains on S^1, which is the usual exterior algebra. | |
| Nov 28, 2009 at 21:16 | comment | added | David Ben-Zvi | Of course we could replace pt to BG by any X to Y satisfying descent, and we can take X to be Spec of a field (eg Spec of an algebraic closure mapping to Spec k.) In any case there is an a priori well defined notion of what we mean by derived category of Y, and I'm asking, does it agree with the derived or coderived category of the corresponding coalgebra on X? | |
| Nov 28, 2009 at 21:11 | comment | added | David Ben-Zvi | Right -- the usual bar construction realizes BG as a colimit of affine schemes (for G affine), and its dg category of sheaves is then realized as the limit (in the oo-category of dg categories) of sheaves on the simplices (if you'd like, we're totalizing the resulting cosimplicial dg category). By the Barr-Beck theorem (Lurie's version), we can equivalently write sheaves on BG in terms of a comonad on k-modules, which gives a description as dg comodules for the group coalgebra. Now the question: is its homotopy category what you'd call "derived" or "coderived"? (I would guess the former..?) | |
| Nov 28, 2009 at 20:53 | comment | added | Leonid Positselski | Now I'm not sure that I understand you, either -- you present BG as a (co)limit of affine schemes? And then assign DG-versions of the derived categories of modules to these affine schemes? And take the limit of these DG-categories? And you believe that you obtain the derived category rather than the coderived category of comodules in this way? | |
| Nov 28, 2009 at 20:35 | comment | added | David Ben-Zvi | I'm not sure I understand your comment about it being "up to you" - there is a well defined unbounded derived category of Y, and the question is how to recover it. The motivating example (in the question above) is X=pt, Y=BG, so indeed we have comodules over a coalgebra over a field. In this case descent holds and we get the derived category of the stack BG. This seems to me like convincing motivation for derived categories of comodules: we know what we want the derived category of a stack to be (the unique limit preserving extension of the assignment rings ---> modules to Stacks^{op}).. | |
| Nov 28, 2009 at 20:26 | comment | added | Leonid Positselski | 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. | |
| Nov 28, 2009 at 20:14 | comment | added | Leonid Positselski | 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. | |
| Nov 28, 2009 at 19:49 | comment | added | David Ben-Zvi | 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. | |
| Nov 28, 2009 at 17:35 | comment | added | Reid Barton | 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. | |
| Nov 28, 2009 at 17:16 | history | answered | Leonid Positselski | CC BY-SA 2.5 |