@PeterNelson yeah, i'll ask him. thanks

@DenisNardin, thanks for your answer. Is there anyway to turn my question into something that makes sense?

when is the map R -> THH(R) an equivalence?

@skd Such a map is called THH-étale, In particular étale maps (of connective rings) are THH-étale.

@jmc Well, what are you trying to do?

Ok, so I have a local system L on X. Let x be a point of X. Then I want to understand to what extent H^1(X,L) is the same as the group cohomology H^1( \pi_1(X,x), L_x)

So I thought, Ok, lets put G = \pi_1(X,x), and make maps X -> BG -> * and write down the Leray spectral sequence.

But I don't know if this can be made precise.

Ah, the Leray SS is a good idea. Oh I understand now, you're trying to do it with sheaves, but you don't need it

Although I'd use the Leray-Serre SS for the map F→X→BG

(where F is the homotopy fiber, that is P_2X)

Ok, so actually, I want to do this with a scheme (smooth, quasi-proj) and a lisse l-adic sheaf... and the (pro?)etale \pi_1

But I thought I would first try to understand the picture in topology.

Well, if L is a (finite) local system I'm pretty sure H^*(X;L) is just H^*(ht(X);L) where ht(X) is the étale homotopy type of X, so the picture in topology is more or less all you need

Yes, I hoped that the pictures would match up.

So how does H^*(ht(X);L) relate to group cohomology?

In topology you get a SS H^p(BG;H^q(\tilde X;L))=> H^{p+q}(X;L) and the map you're trying to understand is the inclusion of the 0-line

Well, I'm interested in p+q = 1

So I need to understand what H^q(P_2X;L) is, for q = 0,1

And now I'm sort of lost...

Well, in general it can be a lot of things. You'll need some other input

Ok, so it's not enough to know that G is the fundamental group of X, and L is a local system?

I hoped that would already have some implications...

No, I mean, take L the constant local system at Z. You're basically asking to compute the cohomology of X from knowing the cohomology of BG

Yes, you have a 5-term exact sequence

0→H^1(BG;H^0(\tilde X;L))→H^1(X;L)→H^1(\tilde X;L)^G→H^2(BG;H^0(\tilde X;L))→H^2(X;L)

Right, I understand that in high degree you can't say anything. But I can say a lot about H^i(X,Z) for i=0,1 if I know G

Right, so let's see H^0(\tilde X;L) should just be the stalk of L at the basepoint of X

Exactly, that is what I hoped.

But in fact, I don't know much homotopy theory. So I don't know how to work with P_2X

and H^1(\tilde X;L)=0 unless I'm mistaken, 'cause \tilde X is simply connected

right, so P_2X is just the universal cover of X ?

Uh sorry, I made a stupid mistake

The thing I was calling P_2X is not P_2X, but the first Whitehead cover of X, also known as the universal cover of X

Ok. right. You have a cartesian square...

X~ ---> X, and * --> BG

and then some vertical maps

Ugh, I was thinking the right thing, but writing the wrong one, sorry

Right, so this means that H^1(X,L) is just group cohomology of pi_1 with coefficients in the fibre of L?

It seems so. I'm kind of confused because it's the kind of thing I'm supposed to know if it's true, but the proof seems to work out ok

Right... I'm also a bit scared...

It feels like we are applying proper base change to that cartesian square

But the map X -> BG is not really proper

Nah, we're applying some kind of basechange to a homotopy cartesian square. Different things, with different rules

Rules that I don't know

Well, in this case it's just the Serre SS, so nothing to be scared about

And also, this works only in low degrees. In higher degrees the cohomology of \tilde X comes and screws up your pretty picture (as it should do!)

Exactly

I did realise that

So now I need to convince myself that a similar thing is true in the l-adic setting

But it should work

@dhy Can you send me an email with your email address? I don't remember your real world name, sorry.

@jmc The Serre SS works in pro spaces, I think I can dig out some reference in Friedlander or Artin-Mazur

Ok cool

That would be very helpful

So, Friedlander is not exactly explicit in his construction, but you can extract it from here: numdam.org/article/PMIHES_1973__42__5_0.pdf

(essentially he's taking the Serre SS for all levels in the pro-system and then taking the limit)

Ok, I will try to understand that article

Thanks a lot for your help!

Does anyone know if the left Kan extension along an exact functor of stable ∞-cats of an n-excisive functor is n-excisive?

Unless I'm mistaken it should follow from the equivalence F_!(T_nG)→T_n(F_!G) coming since F_! commutes with finite limits, but I'd love to have a reference for it

@GeoffroyHorel thanks. i was going to ask when THH-etale implies ordinary etale, but I think this happens if R is finitely presented over S (and likewise for a map of connective rings A->B)

@DenisNardin when I first asked my question, you mentioned that the map $X \to BG$ is only defined up to homotopy. Why is that now no longer a problem? Because we work in some sort of homotopy category?

So there are two things: the Leray SS (the derived functor SS for Rf o Rg) and the Serre SS. I know how to prove they are the same when f is a fiber bundle. At best I can rig things so that f is a fibration, and I don't know if it is enough (maybe yes, but I don't see a proof). Either way I'm going to use the Serre SS who doesn't care if things are only defined up to homotopy

Ok, I see.

I always thought that Serre SS = "Leray SS for fibre bundles"

I really need to learn more homotopy theory. (-;

I mean, the point here is that Rf_*L is not well defined in general, but we are choosing a very special f for which Rf_*L is something we can control homotopically. Your original question was "how to compute Rf_*L"

Ok, I meant: "how to compute Rf_*L for this very special f"

Well, but you didn't specify f :) You just gave us the homotopy class and that's not enough

Hmmm, ok... I'm very confused now

So how did you specify f? Because I missed where you did that.

To give a stupid example: let f_1,f_2:X→Y be constant maps at different points. Then Rf_1_* and Rf_2_* are not the same functor, but if Y is connected f_1 and f_2 are homotopic

I chose an f who is a fibration

(actually, I used a different definition for the SS, but I'm almost positive it is equivalent to choosing an f who is a fibration)

So, are you saying that even though cohomology is homotopy-invariant, the derived functors are not?

I didn't realisate that

I would expect a homotopy to induce a natural isomorphism between derived functors

Well, sheaf cohomology is not homotopy invariant

Aah, yes, of course...

Brainfart

Again, look at the example of the constant maps: if f is a constant map at y∊Y, R^nf_*Z is the skyscraper sheaf H^n(X;Z) at y. This is clearly not independent of y

Yes, it's indeed obvious

So I want to use the fact that I have a local system... and remark that its sheaf cohomology is the same as some other kind of cohomology that is homotopy invariant. Does that make sense?

More or less, yeah. The category of local system depends only on the homotopy type of your space (or for less nice spaces, of the shape of your space, but we need not concern ourselves with them)

Do you know a reference where this is made precise?

There is a notion of pushforward of local systems, that recovers the cohomology of the homotopy fiber, but it's not the same pushforward

What, the fact that the category of local systems depends only on the homotopy type of X?

No, that fact is pretty clear I think, but how you continue

to deduce that you can work in the homotopy category, get the Serre SS and use it to compute H^1(X,L) as group cohomology

Well, it depends on how much technology you want to throw in. If you just want to prove your theorem you can just construct the Serre SS (e.g., like in Hatcher or in Boardman's spectral sequences papers)

Oh you mean the fact that the cohomology with coefficients in a local system is the same as the sheaf cohomology?

What do you mean with "cohomology with coefficients in a local system"?

For me that = sheaf cohomology

What's you definition of local system?

Locally constant sheaf

Ok, we can give another definition which is equivalent for locally nice spaces

A local system is the datum for each point x∊X of an abelian group (of whatever) A_x and for each homotopy class of maps \gamma:x→x' of a map of abelian groups A_x→A_{x'} s.t the obvious compatibilities are satisfied

In other words, a local system is a functor from the fundamental groupoid of X to Ab

Right, that makes sense

Then classical covering space theory tells you that for locally nice spaces these two notions of local system coincide

Yes, I know that these two are equivalent

(Although I just use reps of pi_1(X,x)...)

In particular they are both the datum of an abelian group object in covering spaces A→X

But of course using the groupoid is more canonical

For pro-spaces in general you don't have a basepoint, so I'm trying to be as independent as possible :)

Yes, that's good

And now you want to use this abelian group object to define cohomology with coefficients in A

By using some fibration sequence, I guess

No, I'm going to write down some explicit complex

By the way, this is in appedix 3.H of Hatcher's book

Although I don't like his presentation

Ok, thanks for the reference

So an n-cochain is a function phi sending every n-simplex Δ^n→X to a lift to A

ok

There's magic that tells you that this complex is in fact computing the sheaf cohomology

But I guess that the magic is the same as for constant coefficients?

More or less, yeah

Anyway the construction of the Serre SS eats this gadget

Ok

And this gadget is homotopy invariant, with the same proof as ordinary cohomology

Right, that seems reasonable

And Friedlander builds the étale analogue of this machinery?

Well, you don't really need the étale analogue of this machinery, you need the pro analogue of this machinery

What do you mean with pro ?

In the end, my goal is that I want to show that H^1(X,L) = H^1( pi^et_1(X,x), L_x)

where X is a reasonable scheme, and L is a lisse l-adic sheaf on X

The étale homotopy type is a functor from varieties (in fact topoi) to pro-spaces (as in pro-systems of spaces). All the stuff you wrote in your equation depends only on the étale homotopy type, so if you can run the proof above for pro(Spaces) you're good

I don't know if there's a way of doing the proof directly at the level of schemes

because the map X→Bπ_1X does not exists at the level of schemes (the rhs is not even a scheme...)

true, so I wanted to use some stacky category of

but your approach might be better

I don't know anything about pro-spaces

Frielander does not do exactly what you need, but it's close

So you say I replace X with a prospace, pi^et_1(X,x) with the fundamental groupoid

and L with an abelian group object covering X

is that right?

More or less, except the correct notion of local system is a bit subtle, but you don't need to worry about that. You only care that its cohomology is the cohomology you're interested in (this is in Friedlander) and that it satisfies a Serre SS

Ok, so is all of this in Friedlander? Or is there some other reference on pro-spaces that is also very useful?

Btw, thanks for you help. I really appreciate it!

I just wish I had better references to give. But maybe someone else will pop up and help me out on this People certainly use the Serre SS for pro-spaces, but I don't remember a reference where it is set up properly

Ok, I'll start digging around the interwebs a bit

So with spaces, in this very general context, do you mean something like simplicial sets, or spectra?

Uh, take simplicial sets for simplicity

@jmc The comment and answer to this question may be useful: mathoverflow.net/questions/266970/…

@MarcHoyois Ah of course using the finite étale site was the right idea... I went into a weird rabbit hole there

@MarcHoyois Thanks! That is indeed helpful!

Is there a good description of d_3 in the Serre spectral sequence? (In the setting I'm thinking about, $\pi_1$ of the base acts trivially on the fiber, and I have $\mathbb Q$ coefficients, so feel free to make those simplifying assumptions.)

I.e. taking the limit of Z/l^nZ and tensoring with Q_l