God this room is so great...

@OmarAntolín-Camarena No, I meant Bredon cohomology not Borel.

@SaalHardali Yes, this is true for a field

Probably some flatness issues are involved.

@SeanTilson it's got two polynomial generators, a_\sigma in dimension \sigma and u_{2\sigma} in dimension 2-2\sigma. The first has order 2. Then there's some anderson dual crap: the torsion gets reflected thru the origin then shifted by 2\sigma-3, and the torsion-free bit is reflected and shifted by 2\sigma-2.

And all the dual stuff has stupid multiplication and is a module over the polynomial bit

I may have gotten the degrees wrong but to be fair it's 6am

but it's supposed to look like you wrote down the standard chain complex for H^*(C_2) and for H_*(C_2) and then listed the homologies of all the brutal truncations, except there's some fuss near degree zero because it's the one place in the cell structure of S^n\sigma where there's a trivial sphere instead of a free one

And I've just now seen that Aaron gave you a reference already. Oh well...

Anyone have a digital copy of Friedlander's paper on the "infinite loop Adams conjecture?"

It should be possible to commute the homology of ko (connective real k theory) by smashing its postnikov tower with HZ/2 and working through the spectral sequence. Have the details been written somewhere?

@TomBachmann I think Rognes has some notes somewhere, I don't know if he does it that way.

Rognes has a lot of notes ^^. I'll see what I can find. Thanks!

@TomBachmann I don't know that I entirely understand what spectral sequence you want to work through.

You could just run the AHSS with coefficients in ko_* and KO_* and compare them to try and determine sifferentials. It would start with the cohomology or homology of EM spaces. This works out nicely for ku and KU.

iirc.

well I look at the tower ... -> KO_{\ge 2} -> KO_{\ge 1} -> KO _{\ge 0}. The cofibres are all known. Now I smash the tower with HZ/2, the cofibres are still known. This gives me a spectral sequence converging to the homology of connective KO

I guess smashing commutes with inverse limits.

I always forget these things. I think that this might be identical to what I describe.

I suppose it is not entirely obvious that this commutes ^^.

(As in, I have no clue.)

smashing doesn't commute with inverse limits in general, but it does when you have some connectivity bounds - for instance, smashing a connective thing with a Postnikov tower does, because each homotopy group eventually stabilizes

right

Yeah, that is just literally the AHss. When you compare it with the spectral sequence for the periodic theory it helps because you know that everything has to vanish there.

I believe.

@TomBachmann that's known, and it "degenerates"

the E_1-term is: H_* HZ, H_* HZ/2, H_* HZ/2, 0, H_* HZ, 0, 0, 0, and then repeating with period 8

the "first nontrivial differentials" are H_* HZ --d_1--> H_* HZ/2 --d_1--> H_* HZ/2 --d_2--> H_* HZ --d_4--> H_* HZ --d_1--> ...

those differentials are known (they can be identified with "Sq^2, Sq^2, Sq^3, and Sq^2 Sq^3" respectively)

and they actually give an exact sequence 0 -> H_* ko -> H_* HZ -> H_* HZ/2 -> ...

@TylerLawson Is it easy to deduce these differentials?

so in particular the "Postnikov" spectral sequence degenerates completely at E_5

@Sean it's not too bad, esp. if you do cohomology instead of homology. the cohomology version is A/Sq^1 <- A <- A <- A/Sq^1 <- ... and everything is tensored up from a chain complex over A(1)

over A(1) the differentials are kind of forced, in the sense that there's nothing else in the corresponding degree to map to

right, and then you would compare it to the periodic theory where you know everything has to vanish?

(except 0 -- so to give a proof of this you need some argument)

Yeah, comparing with the periodic theory would definitely do that

As I recall, that is how it went for the complex case.

But then I have to know that there are no maps from KO to HF_2.

The Sq^2's can be deduced because Sq^2's job is to detect \eta

Oh, you can use ... -> H_* Sigma KO -> H_* KO -> H_* KU -> ... to do that, because the first map is multiplication-by-eta and induces zero on homology

@TylerLawson thanks! Is this written somewhere?

I guess one has to be careful about the rules of the game

@TomBachmann Not sure, but I know it as a verbally given exercise

@TylerLawson Thanks!

In the proof of Proposition 8.7 of arxiv.org/pdf/math/0703204v3.pdf (an early version of Lurie's DAGIII) coCartesian lifts of $N_{\Delta}(M^o)^{\otimes}\rightarrow N(Fin_*)$ are given by "componentwise acyclic cofibrations". I have been told that one can omit the cofibrancy condition, such that coCartesian morphisms are just "componentwise equivalences" in $N_{\Delta}(M)$. Does anyone have a reference for such a characterization?

What's the standard reference of the Quillen equivalence between chain complex of abelian groups and HZ-modules? (if possible) any reference which is oo-category free?

@MingcongZeng I think this is in a paper by Schwede

Let me have a try... he has quite some papers

Apparently the connective case is in

Stable homotopical algebra and Γ-spaces, Math. Proc. Cambridge Philos. Soc. 126 (1999), 329-356

And the general case is in S. Schwede and B. Shipley, Stable model categories are categories of modules, Topology 42 (2003)

@EspenNielsen He's not saying these are precisely the coCartesian morphisms, but that choosing that data gives one coCartesian morphism. The same argument shows that any weak equivalence to a fibrant-cofibrant object gives a coCartesian morphism

@RuneHaugseng Cool! Thank you!

What's the current state of the art regarding grothendieck duality in the derived setting? Where can I read about this?

@SeanTilson Yeah, I decided that was must be what you meant a little too late.

@SaalHardali Depends on exactly what you want, but, e.g., arxiv.org/pdf/1501.01999.pdf

@RuneHaugseng Thanks. I was confused about why he specified that the maps should be cofibrations if this was not needed in the argument.