Другие фрагменты из той же переписки
Mar. 14th, 2020 10:16 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
posicIt all depends on what you expect of it. If you want to have a full version of "Two kinds of derived categories..." over a ring, with Quillen equivalences of algebra/coalgebra and module/comodule model categories, then it will require a lot of extra care. In particular, you would certainly have to replace an algebra or coalgebra with a weakly equivalent one before applying the bar/cobar construction (as the tensor products over a ring are not exact).
Nonflat coalgebras may be particularly problematic, or even outright impossible. I would resign from the outset to the compromise of considering suitably flat coalgebras only (which means that, in the contemporary terminology, you may have a model structure, but not a model category, since limits and colimits will not exist; not even finite limits and colimits).
On the other hand, if you have a fixed Koszul dual algebra-coalgebra pair, which are suitably flat over the base ring, and you want to construct a Quillen equivalence of module/comodule categories -- it should be much easier (because all the tensor products involved are exact). This is the setting of my Appendix B (which is also a special case in many other ways, so some care is needed when generalizing it).
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Concerning Koszul duality on schemes, this is generally pretty straightforward and unproblematic. Ivan Mirkovic with Simon Riche and maybe other collaborators may have written several papers on a particular version of it under the banner of "linear Koszul duality". I myself wrote about it, in a special (but rather advanced) particular case of D-Omega duality, in Appendix B to the "Two kinds of derived categories..." memoir.
The difference between base ring and base scheme is not that large. One thing which exists over a base ring, but is lacking over a base scheme, is projective modules. Over a scheme, you may have to use flat sheaves as a replacement. There are enough flat sheaves over any quasi-compact semi-separated scheme (but not over quasi-separated Noetherian schemes!) At worst, when working with coderived categories, my "very flat sheaves" may be needed. After the very flat conjecture has been proved, they should be reasonably convenient to use. Once again, there are enough very flat quasi-coherent sheaves over any quasi-compact semi-separated scheme.
The only caveat is that nonhomogeneous Koszul duality, as developed in my memoir, happens on the coderived and the contraderived sides. Now if you want to do it with quasi-coherent sheaves, then on the coderived side it is fine. But on the contraderived side, the definition of the contraderived category involves infinite products. And infinite products of quasi-coherent sheaves are not well-behaved. That is where contraherent cosheaves enter the picture. Infinite products of contraherent cosheaves behave well (they are local in the scheme and exact w.r.t. the Quillen exact structure).
You are supposed to use quasi-coherent sheaves on the coderived/comodule side and contraherent cosheaves on the contraderived/contramodule side. Then you will have "Koszul triality"-type results and other mixtures of Koszul duality with co-contra correspondence over schemes. This was in fact my main motivation for developing the theory of contraherent cosheaves. There is a discussion of it in the introduction to my preprint about them.
Nonflat coalgebras may be particularly problematic, or even outright impossible. I would resign from the outset to the compromise of considering suitably flat coalgebras only (which means that, in the contemporary terminology, you may have a model structure, but not a model category, since limits and colimits will not exist; not even finite limits and colimits).
On the other hand, if you have a fixed Koszul dual algebra-coalgebra pair, which are suitably flat over the base ring, and you want to construct a Quillen equivalence of module/comodule categories -- it should be much easier (because all the tensor products involved are exact). This is the setting of my Appendix B (which is also a special case in many other ways, so some care is needed when generalizing it).
***
Concerning Koszul duality on schemes, this is generally pretty straightforward and unproblematic. Ivan Mirkovic with Simon Riche and maybe other collaborators may have written several papers on a particular version of it under the banner of "linear Koszul duality". I myself wrote about it, in a special (but rather advanced) particular case of D-Omega duality, in Appendix B to the "Two kinds of derived categories..." memoir.
The difference between base ring and base scheme is not that large. One thing which exists over a base ring, but is lacking over a base scheme, is projective modules. Over a scheme, you may have to use flat sheaves as a replacement. There are enough flat sheaves over any quasi-compact semi-separated scheme (but not over quasi-separated Noetherian schemes!) At worst, when working with coderived categories, my "very flat sheaves" may be needed. After the very flat conjecture has been proved, they should be reasonably convenient to use. Once again, there are enough very flat quasi-coherent sheaves over any quasi-compact semi-separated scheme.
The only caveat is that nonhomogeneous Koszul duality, as developed in my memoir, happens on the coderived and the contraderived sides. Now if you want to do it with quasi-coherent sheaves, then on the coderived side it is fine. But on the contraderived side, the definition of the contraderived category involves infinite products. And infinite products of quasi-coherent sheaves are not well-behaved. That is where contraherent cosheaves enter the picture. Infinite products of contraherent cosheaves behave well (they are local in the scheme and exact w.r.t. the Quillen exact structure).
You are supposed to use quasi-coherent sheaves on the coderived/comodule side and contraherent cosheaves on the contraderived/contramodule side. Then you will have "Koszul triality"-type results and other mixtures of Koszul duality with co-contra correspondence over schemes. This was in fact my main motivation for developing the theory of contraherent cosheaves. There is a discussion of it in the introduction to my preprint about them.
