Three responses to one of yesterday's questions.............DMD
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Subject: Re: three questions
From: Clark Barwick
Date: Wed, 10 May 2006 15:01:09 +0200
This is in response to the question posed by Ronnie Brown and Tim Porter.
I believe that the answer is yes. R. Waldmüller and M. Wiethaup
(Göttingen)
have constructed a cofibrantly generated "projective" model structure on
the
total space of any "right Quillen presheaf" on a cofibrantly generated
model
category satisfying some (rather mild) extra hypotheses. The details
should
appear very soon in a forthcoming preprint.
As I understand it, if one takes the trivial model structure on the
category of rings (equivalences are isomorphisms), and the right
Quillen presheaf assigning to any ring R the category of chain
complexes of R-modules (with the usual model structure, where the
fibrations are the surjections), then this will satisfy their
hypotheses, and thus their result will give the desired model structure.
Best wishes,
Clark Barwick
On 10 May 2006, at 14:13, Don Davis wrote:
> Subject: Chain complexes over variable rings?
> From: Tim Porter
> Date: Tue, 09 May 2006 15:33:15 +0100
>
> Dear All,
>
> Does anyone know a reference (or even if it is true) as to
> whether the category of all chain complexes form a Quillen model
> category or similar? To be more precise, we are thinking of the
> category of pair (C,R) where C is a chain complex of modules over
> the ring R, and the morphsims are similarly pairs in the obvious way.
>
> Thanks
>
> Ronnie Brown and Tim Porter.
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Subject: Re: three questions
From: Justin Smith
Date: Wed, 10 May 2006 12:58:24 -0400
Yes, they do form a model category as follows:
1. weak equivalences are chain-homotopy equivalences, i.e. quadruples
(f,g,phi,theta):C \to D
where f:C \to D, g:D\to C are chain-maps and
f\circ g-1 = d\circ phi + phi\circ d
g\circ f - 1 = d\circ theta + theta\circ d
2. cofibrations are injections of chain-complexes that are split as
morphisms of graded modules
3. fibrations are surjections of chain-complexes that are split as
morphisms of graded modules.
A reference for this:
Quillen model structures for relative homological algebra by * J.
Daniel Christensen
, Mark
Hovey in
*Math. Proc.
Cambridge Philos. Soc (I'm not sure of the volume)
http://arxiv.org/abs/math.KT/0011216
They also prove that this model structure is not cofibrantly generated
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Subject: Ronnie and Tim question
From: Marek Golasinski
Date: Wed, 10 May 2006 23:56:55 +0200 (CEST)
I think the following might be helpful:
(1) There is paper by me and a co-author,
The homotopy category of chain complexes is a homotopy category,
Colloq. Math. 47 (1982) no. 2, 173-178
on the Quillen model category structure on the category of
chain complexes.
(2) Now, form the category given by pairs $(C,R)$, where $C$
is a chain complex complex of modules over the ring $R$.
Morphisms $(C,R)\to (D,S)$ are given by pairs $(f,g)$,
where $f : R\to S$ is a ring homomorphism and $g : C\to D$ is
a chain map over $R$.
(3) Following (1) and the paper by Roig A., Model
category structures in bifibred categories, J. Pure
Appl. Algebra 95 (1994), 203-223
a Quillen model category structure on the
category defined in (2) might be derived.
Marek (Golasinski)