What kind of limits does suspension spectrum preserve? Finite?

@JonathanBeardsley it doesn’t even preserve products!

Ah yeah fair point.

sad trombone

Haha

Is any of you attending CT2019 that is happening in Edinburgh??

@DenisNardin Sorry, I am confused. In the simplest case, if we consider two homotopy fixed point spectral sequence associated to a chain complex with (linear) G-action, they are in fact two spectral sequences associated to a bicomplex. I don't think that two spectral sequence are equivalent (though they are induced by the same total differential)?

@Yai0Phah The SS induced by the Postnikov filtration on the spectrum doesn't correspond to either of those two (although it coincides with the SS coming from the skeletal filtration of $BG$ starting at the $E_2$-page). There's a difference between the Postnikov truncation and the truncation bête of a chain complex

To give you an example, if you take the complex $[\mathbb{Z}\xrightarrow{2}\mathbb{Z}]$ representing $\mathbb{Z}/2$, its Postnikov filtration is concentrated in degree 0, while the filtration bête has one filtration piece given by $\mathbb{Z}$ ("including" into $\mathbb{Z}/2$ via the usual projection)

I know that there are differences between brute ones and "good" ones.

So, when you consider your bicomplex, one of the two spectral sequences is associated to the skeletal filtration of $BG$ and the other to the filtration bête of the chain complex you're taking the homotopy fixed points of

But you were asking about comparing the first of those two SS to another one, the one arising from the Postnikov filtration of the chain complex, that is neither of them

In particular, as far as I understand, the Postnikov-based SS doesn't arise from a double complex in any way I can see

You can see that the second SS is not what you want if you look at its E_2-page: its signature is $H^p(G; E_q)⇒H_{q-p}(E^{hG})$, which is different from $H^p(G;H_qE)$ (which arises from the Postnikov filtration)

I perceived this difference, but I did not believe that this should have been the only reason that they are different.

Well, it's a pretty significant difference

What is E_q?

The q-th group in your complex

I'm seeing the complex as $[...→E_q→E_{q-1}→E_{q-2}→...]$

It is difficult to trace back the indexing, but it seems to me that what you wrote is the E_1-page for brute truncations

Ugh I'm terrible at indexing the pages, but isn't the E_1-page $C^\ast(G,E_\ast)$?

I think this is the page numbering that gives you the "expected" degrees of the differential

Ofc that's obviously just a convention and not really important

The problem is that if you choose different r's, then it is natural that two spectral sequences don't coincide.

But do you agree that the Postnikov SS does not come from any of the SS of a double complex?

I don't know when r is big.

I can only tell that they do not coincide when r=1

No, I mean. They come from different filtrations

Yeah, they will coincide later with one of the two (it's proven in the reference I gave you, at least for the case of a trivial action), but the double complex is a red herring here

Maybe I'm not explaining myself well

I just wanted to say that your counterexample is not a counterexample

I was also not clear. I did not claim for counterexamples. I only remember that there are arguments that take advantage of both vertical and horizontal filtrations. It seems very weird to my ears that the two are essentially "same" (this is why I don't believe that brute vs. good is very essential. I believe that this difference could be somewhat overcome) when r is large.

But they are not essentially the same

What I'm saying is that the Postnikov filtration is neither the vertical nor the horizontal

In fact it's very different from either filtration, and its SS coincides with the one coming from the skeletal filtration of BG only when r is at least 2 (and you have to do a bit of reindexing for this to make sense)

OK, let's omit the brute truncation

What is the most general thing that could be deduced from that paper? I mean, whether we can generalize it to bi-filtered spectra so that one filtration induces Postnikov filtration in the associated grated to the other filtration?

That paper proves that the two different exact couples that can be used to describe the AHSS are isomorphic

Or it really exploits some properties of Map(X,Y).

The key property it uses is that if $X$ is an $n$-dimensional spectrum and $Y$ is a n-truncated spectrum, then $[X,Y]=\ast$ (warning: off by one errors here)

Ah, so it does not generalize easily to arbitrary bi-filtered structures?

Definitely not

Sorry, I've got to go. Hopefully someone else will chime in

Possibly someone who understands spectral sequences more than me :)

Thanks. At first, I thought that this kind of result should be an interaction between Beilinson t-structure and another filtration structure, therefore I came up with bi-filtered structures.

@DenisNardin By the way, literally, it seems to me that Lemma 4.3.1 in that paper means that the exact couples induces spectral sequences are isomorphic, but not exact couples themselves, as he did not claim the isomorphism between A's (the difference is killed modulo complete Hausdorff).

Hi. Does anyone know if there's any general result proving that the derived category of mixed motives DM(S, \Lambda) is equivalent to the category of HZ-algebras (after choosing suitable topologies)? I know that the result was proved for S = Spec (k) (I don't remember who proved it, though) and for \Lambda = Q and S excellent by Cisinski-Déglise, however I was hoping for some result regarding integral coefficients and some reasonable base S.

Also does the difficult only lie is constructing transfers or its more subtle? If so, is the result trivially true for D_{A^1}?

@user40276 (assuming you mean HZ-modules) It's been proven by essentially the argument that Röndigs-Østvær over fields that admit resolution of singularities. I'm pretty sure you can use a localization-type argument to extend the result for schemes for which all residue fields have resolution of singularities (although I've never seen it written down, so maybe I should be more suspicious). From there the ball is on the algebraic geometer's side :)

You can also get it for all fields if the exponential characteristic of the field is invertible in $\Lambda$, by an argument in Hoyois-Kelly-Østvær

There is a natural adjunction between $H\mathbb{Z}$-mod and $DM$, the difficulty is in proving that the unit is an equivalence. It is fairly straightforward to prove it when $SH$ is compact-rigidly generated, but that is known only for fields with resolution of singularites

@DenisNardin Whoops! Yes, HZ-modules. Thanks for the reply. Do you know if the problem in proving this equivalence is only when including transfers? I mean if one uses instead D (Sh_Nis (Sm/S, \Lambda)) the equivalence would follows trivially or no? If that's the case, it looks like that the result for the case with transfers would follow using the qfh topology (or maybe even the h topology).

@JonathanBeardsley Thanks!