For questions on the étale cohomology groups of an algebraic variety or scheme, algebraic analogues of the usual cohomology groups with finite coefficients of a topological space.
Questions tagged [etale-cohomology]
314 questions
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Semisimplicity of Frobenius action on H^1(X,Q_p)
Let $X$ be a smooth projective curve over a finite field of characteristic $p$, with Hasse-Witt invariant $\lambda>1$.
For $\ell\neq p$, the Frobenius action on the etale cohomology group $H^1(X,\mathbb Q_\ell)$ is semisimple. Is the statement still…
Brett Frankel
- 9,817
2
votes
1 answer
Confusion about Milne -- locally constant sheaves on étale site?
In Lectures on Étale Cohomology by J. S. Milne, Example 8.5(b) on Page 60, it is claimed that locally constant sheaves on [the étale site of] $U := \Bbb A^1_k \setminus \{0\}$, where $k$ is an algebraically closed field, correspond to modules over…
Kenny Lau
- 25,049
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Why define étale cohomological dimension as it is defined?
I am learning étale cohomology, and we defined the étale cohomological dimension of a scheme $X$ as the minimum of $n$ where $H_{ét} ^n (X,F)$ vanishes for all the torsion sheaves $F$, yet I don't get why we only restrict our view to torsion…
2
votes
2 answers
To use Etale methods in proofs or not
My question concerns a statement made by Fong and James in "Geometry of the Simple groups..."(1998); the statement says that they offer a proof "independent of etale methods." Am I to presume that "etale methods" are not as desirable as other…
user406419
- 21
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Why do we have $H_2(\mathbb P^1) \cong H_1(\mathbb A^1\setminus \{0\})$?
This question comes from Matt E's answer in
What is the intuition behind the concept of Tate twists?
He remarks that $H_2(\mathbb P^1) \cong H_1(\mathbb A^1\setminus \{0\})$, but I don't see where this comes from. Could someone give more detail?
Bun
- 301
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vote
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What is alias etale map?
I think that etal map, is a local homeomorphism(or a local diffeomorphism for manifolds). I want a reference that explain alias etale map.
mehrnaz
- 21