Re: homotopy flat DG modules
Dear Justin,
I may have had some thoughts about this question since it was asked, but I cannot now remember
what these were (if any). No, I certainly have not found anything like a satisfactory answer. I also
no longer remember what was my own original motivation for asking this question back then (though
I am pretty sure that I had a rather specific motivation).
Thinking of it now, perhaps the best approach would be to construct something that might be called
"the homotopy flat and cotorsion model structure" on the category of DG-modules over a DG-ring.
Say, a DG-module C over a DG-ring A is called homotopy cotorsion if the complex Hom_A(F,C)
is acyclic for any acyclic graded flat and homotopy flat DG-module F over A.
The homotopy flat and cotorsion model structure (on the category of DG-modules over A and closed
morphisms between them) would have quasi-isomorphisms as weak equivalences, injective
morphisms with graded flat and homotopy flat cokernels as cofibrations, and surjective morphisms
with graded cotorsion and homotopy cotorsion kernels as fibrations. One you've constructed such
a model structure, you would probably have a variety of alternative descriptions of the classes
of DG-modules involved coming out from this construction.
This is a rather complicated approach based on modern advanced techniques (you may want to
look up Enochs' books and papers on cotorsion modules, Hovey's papers on "abelian model
structures", Hanno Becker's most recent arXiv preprint, and Chapter 9 of my book on semi-infinite
homological algebra for further details).
Compared to that, the question about h-projective DG-modules is easy. Firstly, any triangulated
category with countable direct sums or countable products is idempotent closed by a theorem
of Boekstedt and Neeman, so any triangulated subcategory in the homotopy category of
DG-modules that is closed either with respect to infinite direct sums or with respect to infinite
products is also closed with respect to direct summands. So allowing direct summands in
the construction is certainly unnecessary.
To prove the assertion in my "alternative descrption 1", you just need to look into your favorite proof
of the claim that any DG-module is quasi-isomorphic to a homotopy projective one. I would guess
that any such proof involves a more or less explict construction of this homotopy projective
DG-module, and when you look into this construction, you will find that it uses essentially no other
operations than shifts, cones, and sums, applied to the DG-module A over A. A kind of countable
filtered inductive limit may be also used, but then you can apply the telescope construction to
express this in terms of countable sums and a cone.
The specific references are given right there in my MO question.
With best wishes,
Leonid
04.03.2014, 00:00, Justin Campbell wrote:
> Dear Professor Positselski,
> I came across your MO question Homotopy flat DG modules today, and found it quite interesting. It was asked over three years ago: did you ever find a satisfactory answer?
>
> I also have a question about one of the assertions made in the question. Say I have a DG module which is h-projective in the sense that the complex of maps into any acyclic module is acyclic. Then how do we obtain the "alternative description" 1 from your question? Probably we should allow direct summands as well as shifts, sums, and cones, right?
>
> Thanks,
> Justin Campbell
I may have had some thoughts about this question since it was asked, but I cannot now remember
what these were (if any). No, I certainly have not found anything like a satisfactory answer. I also
no longer remember what was my own original motivation for asking this question back then (though
I am pretty sure that I had a rather specific motivation).
Thinking of it now, perhaps the best approach would be to construct something that might be called
"the homotopy flat and cotorsion model structure" on the category of DG-modules over a DG-ring.
Say, a DG-module C over a DG-ring A is called homotopy cotorsion if the complex Hom_A(F,C)
is acyclic for any acyclic graded flat and homotopy flat DG-module F over A.
The homotopy flat and cotorsion model structure (on the category of DG-modules over A and closed
morphisms between them) would have quasi-isomorphisms as weak equivalences, injective
morphisms with graded flat and homotopy flat cokernels as cofibrations, and surjective morphisms
with graded cotorsion and homotopy cotorsion kernels as fibrations. One you've constructed such
a model structure, you would probably have a variety of alternative descriptions of the classes
of DG-modules involved coming out from this construction.
This is a rather complicated approach based on modern advanced techniques (you may want to
look up Enochs' books and papers on cotorsion modules, Hovey's papers on "abelian model
structures", Hanno Becker's most recent arXiv preprint, and Chapter 9 of my book on semi-infinite
homological algebra for further details).
Compared to that, the question about h-projective DG-modules is easy. Firstly, any triangulated
category with countable direct sums or countable products is idempotent closed by a theorem
of Boekstedt and Neeman, so any triangulated subcategory in the homotopy category of
DG-modules that is closed either with respect to infinite direct sums or with respect to infinite
products is also closed with respect to direct summands. So allowing direct summands in
the construction is certainly unnecessary.
To prove the assertion in my "alternative descrption 1", you just need to look into your favorite proof
of the claim that any DG-module is quasi-isomorphic to a homotopy projective one. I would guess
that any such proof involves a more or less explict construction of this homotopy projective
DG-module, and when you look into this construction, you will find that it uses essentially no other
operations than shifts, cones, and sums, applied to the DG-module A over A. A kind of countable
filtered inductive limit may be also used, but then you can apply the telescope construction to
express this in terms of countable sums and a cone.
The specific references are given right there in my MO question.
With best wishes,
Leonid
04.03.2014, 00:00, Justin Campbell wrote:
> Dear Professor Positselski,
> I came across your MO question Homotopy flat DG modules today, and found it quite interesting. It was asked over three years ago: did you ever find a satisfactory answer?
>
> I also have a question about one of the assertions made in the question. Say I have a DG module which is h-projective in the sense that the complex of maps into any acyclic module is acyclic. Then how do we obtain the "alternative description" 1 from your question? Probably we should allow direct summands as well as shifts, sums, and cones, right?
>
> Thanks,
> Justin Campbell