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Results tagged with etale-cohomology
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user 6074
for questions about etale cohomology of schemes, including foundational material and applications.
17
votes
Is there a "universal" cohomology theory for varieties over p-adic fields?
I'm seeing this old question just now, and simply wanted to remark that the situation may be slightly better.
Namely, enlarging the category of $\mathbb Q$-vector spaces into the larger semisimple $\m …
- 21.3k
13
votes
Accepted
Homology of the étale homotopy type
I'm sure there are easier and better ways to think about this, but here's how I like to think about it.
Work on the big pro-etale site on all schemes, which maps to the pro-etale site of a point, $\pi …
- 21.3k
12
votes
Accepted
Does $0\to I\to\mathrm{Gal}_K\to\mathrm{Gal}_k\to 0$ always split?
Good question! Let me try to guess what Gabber had in mind there. (Note that he only says "known" (to him), not "well-known"...)
The claim is that the extension splits. Note that to prove this, we are …
- 21.3k
30
votes
Accepted
When (or why) is a six-functor formalism enough?
When defining a homotopy-coherent structure, you have to strike the correct balance between supplying enough data (so that all isomorphisms (between isomorphisms, ...) that you need later are actually …
- 21.3k
5
votes
Accepted
Fpqc-locally constant if and only if étale-locally constant?
The answer is Yes, but it fails for some slight variants, so let me first discuss an analogue for sheaves of sets. In that case, this is closely related to the discussion of pro-etale fundamental grou …
- 21.3k
13
votes
Accepted
On the definition of the etale site of an adic space
Great question!
The short answer is that Huber simply wanted to be in a setting where everything is (stably) sheafy, and so put some assumptions ensuring this. Note that Huber's work remained somewhat …
- 21.3k
18
votes
Accepted
Is there a ring stacky approach to $\ell$-adic or rigid cohomology?
This is an interesting question.
First, I think the [PS] reference does not give the "correct" Betti stack. In my notes on 6 functors, I define a different stack $X_B$ such that $D_{\mathrm{qc}}(X_B)$ …
- 21.3k