Questions tagged [etale-cohomology]
for questions about etale cohomology of schemes, including foundational material and applications.
724
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When inverse image presheaf is already a sheaf
Following proof from Milne's Étale Cohomology (page 94) contains an equality I not understand.
Setting: assume $X$ is a variety (=absolutely reduced, irreducible scheme of finite type over base field ...
- 5,780
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1
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Some questions about splitting of sequence $0\to I\to\mathrm{Gal}_K\to\mathrm{Gal}_k\to 0$ for Henselian val field $K$
I have a couple of questions about following proof by Peter Scholze on splitting of the ses (...does it have a name?...)
$$0\to I\to\mathrm{Gal}_K\to\mathrm{Gal}_k\to 0$$
for $K$ henselian valuation ...
- 5,780
2
votes
0
answers
207
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Using the Dold-Thom Theorem to define \'etale cohomology
For reasonable spaces $X$, the Dold-Thom Theorem states that $\pi_i(SP(X)) \cong \tilde{H}_i(X)$ where $SP(X) = \bigsqcup_i \mathrm{Sym}^i(X)$. There is a purely algebro-geometric realization of this ...
- 7,716
5
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228
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Locally acyclic morphism which is not flat
Let $k$ be a closed field of characteristic $p \geqq 0$ and $\Lambda = \mathbf{Z}/\ell$, $\ell \neq p$. Recall that a morphism $f \colon X \to S$ of $k$-varieties is said to be locally acyclic if for ...
- 253
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A stalk criterion for unit map to be an isomorphism on étale site
Let $f: X \to Y$ be a morphism of schemes and $\mathcal{F}$ sheaf of sets/Abelian groups on the small étale site $Y_{ét}$. Assume we manage somehow to show thatat every geometric point $\overline{y} \...
- 5,780
3
votes
1
answer
457
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Characterization of étale locally constant sheaves over a normal scheme
I have a question about the verification of remark 1.2 in James Milne's book Étale Cohomology stated on page 156:
Assume $X$ be a normal & connected scheme with generic
point $g: \eta \to X$.
Then ...
- 5,780
1
vote
0
answers
251
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A hard-Lefschetz theorem with torsion coefficients?
Let $X$ be a smooth projective surface over $\overline{\mathbb{F}_{q}}$. Let $\ell$ be a prime distinct from the characteristic.
Assume we have a Lefschetz pencil of hyperplane sections on $X$. Let $...
- 11
4
votes
1
answer
359
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Unit map on étale site under $(f^*,f_*)$ adjunction
Let $f: X \to Y$ be a morphism between two irreducible schemes and $\mathcal{F}$ sheaf on the small étale site $Y_{ét}$. My question is more or less "dual" to this one:
Question: Under which ...
- 5,780
3
votes
1
answer
340
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Diagonal morphism of henselization is an open immersion?
Let $(R,\mathfrak{m})$ be a local ring, denote by $R \rightarrow R^h$ its henselization. Write $S = \operatorname{Spec} R$ and $S^h = \operatorname{Spec} R^h$. Is it true that the diagonal morphism $\...
- 53
1
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1
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126
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An etale cover of a semiperfect ring
Assume that $R$ is a semiperfect ring in characteristic $p$, i.e the frobenius is surjective on $R$. I think one can prove that an etale cover of $R$ should again be semiperfect by considering the ...
- 65
5
votes
1
answer
404
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Cohomology of Shimura varieties before and after completion at some prime
Let $(G,X)$ be a Shimura datum with reflex field $E\subset \mathbb C$. For any neat open compact subgroup $K \subset G(\mathbb A_f)$, let $\mathrm{Sh}_K$ denote the associated Shimura variety. It is a ...
- 717
1
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0
answers
141
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Multiplicity and the perfect projective line
Let $\mathbf{F}_p$ be the field with $p$ elements, and $X = (\mathbf{P}^1_{\overline{\mathbf{F}}_p})^\text{perf}$ the inverse perfection of the projective line over $\mathbf{F}_p$.
Let $\Gamma$ be the ...
- 85
1
vote
1
answer
332
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Self-intersection of the diagonal on a surface
Let $X$ be a smooth projective curve over the complex numbers, and take $\Delta$ the diagonal divisor on $X\times X$. Using the adjunction formula, one computes $\Delta\cdot\Delta =2-2g$ for $g$ the ...
- 85
3
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0
answers
187
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Beilinson-Lichtenbaum conjecture for algebraic extensions of $\mathbb{Z}/m$
Let $X$ be smooth over some field $k$ and $m\in\mathbb{Z}$ so that $m$ maps to a unit in $k^{\times}$. By Beilinson-Lichtenbaum one has an isomorphism of cohomology groups
\begin{equation*}
\...
- 1,745
3
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0
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169
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Relations between some categories of étale sheaves
I asked this question on math.stackexchange but nobody answers, so I try here even if I'm not sure my question is a research level one..
Let $X$ be a scheme over a number field $k$. Feel free to add ...
- 1,125