Holmstrom Brown representability

For the classical theorem in topology, see Classical Brown representability. For an abstraction of this, see Abstract Brown representability

LNM0012: Halbexakte Homotopiefunktoren, studies systematically the functors satisfying the conditions for Brown rep. Unfortunately in German…

http://mathoverflow.net/questions/71812/on-brown-representability-theorem

http://mathoverflow.net/questions/11458/applications-of-the-brown-representability-theorem

Jardine preprint June 2009.

See Motivic homotopy theory for a summary of Voevodsky’s ICM talk.

Joyal and Tierney: Classifying spaces for sheaves of simplicial groupoids. http://www.ams.org/mathscinet-getitem?mr=1239557 Has a rep result for H1 in the sense of iso classes of G-torsors, in a cat of simplicial sheaves.

One can express sheaf cohomology as Hom in the bounded derived category of abelian sheaves on $X$, from the constant sheaf Z to the sheaf $F[i]$ which is $F$ placed in degree $-i$ I think.

A survey of well generated triangulated categories. (English summary) Representations of algebras and related topics, 307–329, Fields Inst. Commun., 45. Summarizes Neeman's book on triang cats. I don't have this book I think

http://ncatlab.org/nlab/show/Brown+representability+theorem

http://mathoverflow.net/questions/32287/representing-cohomology-of-a-sheaf-a-la-eilenberg-maclane

http://ncatlab.org/nlab/show/representable+functor

http://www.ncatlab.org/nlab/show/Eilenberg-MacLane+object

Naumann in talk July 2009: SH(Z) is Brown, meaning that there is a Brown rep thm I think.

arXiv:0909.1943 Brown representability in $\A^1$-homotopy theory from arXiv Front: math.AG by N. Naumann, M. Spitzweck We prove the following result of V. Voevodsky. If $S$ is a finite dimensional noetherian scheme such that $S=\cup_\alpha\Spec(R_\alpha)$ for {\em countable} rings $R_\alpha$, then the stable motivic homotopy category over $S$ satisfies Brown representability.

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Email from Oriol Raventos Feb 2010:

Hi Andreas,

I’m actually writing my PhD on this. It’s not finished yet, but I’m also writing a small preprint, since some other people asked me about it. I’ll send it to you as soon as I have it.

In the meanwhile, you can look at the slides of the talk of Fernando Muro in a recent conference in Prague:

http://personal.us.es/fmuro/talks.htm

I can provide you any details if you need them.

We haven’t looked carefully at the case of Motivic Homotopy Theory, although we thought the paper in the Arxiv by Nauman and Spitzweck about it could be very interesting.

I hope it helped. If you have some more concrete question I’ll may try to answer it. Anyway I’ll send you the first draft when I have it.

Best wishes,

Oriol Raventós

nLab page on Brown representability