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.
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
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
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"
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)
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
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)
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?
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
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...)
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
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
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.)