Questions tagged [etale-cohomology]
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.
300
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Etale cohomology of $Spec(𝔽ₚ^{\text{sep}}((t)))$
I am thinking about how norms $ν : L ⭢ ℤ$ on higher local fields could induce long exact sequences in different cohomologies.
$𝔽ₚ^{\text{sep}}((t)),ℚₚ^{\text{sep}}$, and $ℂ$ are a local fields. What ...
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Understanding a lemma in Milne's etale cohomology
I am reading chapter VI of J. Milne's book, Etale Cohomology. This chapter begins with cohomological dimensions: if $X$ is a finitely generated scheme over a separably closed field $k$, then (étale) ...
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Constructibility of étale sheaves in cohomological dimension
In the appendix of the book of R. Kiehl and R. Weissauer, Weil Conjectures, Perverses Sheaves and l'adic Fourier transform, they quoted the following theorem.
Let $X$ be a finitely generated scheme ...
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Do we ever use etale coherent sheaves?
In my experience with algebraic geometry we are interested in either
Coherent sheaves with the Zariski topology (for geometry)
Abelian group or sets or groupoid sheaves with the etale topology (...
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Compact generators of $D^b_c(X)$
Let $X$ be a scheme and $n$ be an integer invertible on $X$, consider the category $D^b(X,\mathbb{Z}/n)$ (resp. $D^b_c(X,\mathbb{Z}/n)$) of bounded chain complexes (up to quasi-isomorphisms) of ...
- 1,386
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Frobenius action on the Picard group
I have been concerned for the better part of today with the following problem: let $X/\mathbb{F}_q$ be a geometrically connected smooth projective curve. The trinity of Frobenii on $X_{\overline{\...
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Fixed part of the induced representation. ntation.
Suppose the topological groups $h, G$ satisfy that $H < G$ and that $[G \colon H] < \infty$. Let $V$ be a $K$-vector space on which $H$ acts continuously. Then we consider the induced $G$-...
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Direct Image and the induced representation.
Suppose $f \colon \mathrm{Spec}\, L \to \mathrm{Spec}\, K$ be a finite covering. Given a smooth sheaf ${\cal F}_{\rho}$ on $\mathrm{Spec}\,L$ corresponding to the representation $\rho \colon {\pi}_1(\...
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Composition of Gysin and restriction maps on $\ell$-adic cohomology
I follow the notations of Milne's lectures notes on etale cohomology, most specifically the section titled "The Gysin map" in chapter 24, p. 145.
Let $k$ be an algebraically closed field, ...
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Etale cohomology of a variety over a number field VS of its $p$-adic completion
Let $X$ be a quasi-projective variety over a number field $E$. Let $v$ be a place in $E$ and let $E_v$ denote the $v$-adic completion. Let $p$ be the prime number determined by $v$, and let $\Lambda = ...
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Reference request: ramification divisor of simple perverse sheaf
I'm reading Beilinson's paper "Constructible sheaves are holonomic"
In 4.6 of this paper, he used the following fact:
Consider sheaf of $\Lambda$-modules, where $\Lambda=\mathbb{Z}/l^N\...
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Nearby cycle complex of a scheme over a DVR with finite number of singularities
Let $R$ be a DVR and $S = \mathrm{Spec}(R)$ with closed point $s$ and generic point $\eta$. Let $X\to S$ be a scheme over $S$, and denote by $X_s$ and $X_{\eta}$ the special and generic fibers. We ...
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Reference request: Étale cohomology of quotient by finite group
Let $X$ be a variety over $\mathbb{F}_q$. Let $G$ be a finite group of automorphisms of $X$ such that the map $f:X \to X/G$ is Galois. It is then the case that $H^i(X/G)=H^i(X)^G$ (where $H^i$ denotes ...
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Why is there one and only one $f\in X_2(\gamma)$ such that $g_2f=\gamma$ and the image of $f$ is $x_2$?
I'm reading Lei Fu's "Etale Cohomology Theory".
Proposition 3.2.7. Let $(S,\gamma)$ be a pointed connected noetherian scheme, $X_1$ and $X_2$ two etale covering spaces of $S$, $u: X_1\to ...
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The number $2$ in cohomology theories
I've started feeling this rather curious mystique coming from an unaddressed - at least in my experience - excessive presence of the number $2$ in a few different areas of maths. My curiosity really ...
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direct image of lisse sheaves
I am reading the book Weil Conjectures, perverse sheaves and l'adic Fourier transform by Kiehl and Weissauer. In I.6, this book claims that for a smooth affine curve $U_0$ over finite field $\mathbf{F}...
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Compare two definition of $Rf_!$ (derived pushforward with proper support)
I'm talking about etale sheaves.
For a morphism $f:X\to Y$ of schemes, there are two definition of $Rf_!:D(X)\to D(Y)$.
The usual derived functor:
$\forall\mathcal{F}\in Sh(X)$, let $f_!\mathcal{F}$ ...
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Can we infer Betti numbers of a family degenerating to a variety from the variety?
I'm noticing I know positive results to the following question in a number of special cases, but I don't know the general situation.
Let $S$ be a complete trait, to make sure I'm not misspeaking ...
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Is the (strict) henselization map etale?
Let $R$ be a local ring. I wonder if the henselization map $R \to R^h$ or the strict henselization map $R \to R^{sh}$ etale? I manage to prove that it is flat and unramified. But I cannot prove it is ...
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Locally free sheaves on flat site versus Zariski site
Suppose $\mathscr{M}$ and $\mathscr{N}$ are locally free sheaves of rank $n$ on the Zariski site of some variety $X_{zar}$. I would like to show $\mathscr{M}\cong \mathscr{N}$ if and only if $\mathscr{...
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Uniqueness of finite etale covering
Let $X1$,$X2$ be two finite etale covering over $X$ with the same number of the fibers over $x_0$,which is a geometric point in $X$.And I assume that $X$ is a smooth projective variety over an ...
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Is a closed subscheme of a curve just a disjoint union of closed points?
Let $C$ be a smooth proper geometrically integral curve over a number field $k$, and consider the reduced effective divisor $Z = p_1+...+p_n$, where $p_i$ are distinct closed points of $C$. Here $Z$ ...
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Book containing the following line
I came across the image below on social network and I feel quite excited about the line. I wonder if anyone knows what is the book (maybe a paper or an introductory note)? To the best of my knowledge, ...
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Open subgroups of the arithmetic fundamental groups of algebraic curves over finite fields
Let $q$ be a power of a fixed prime number $p$, and let $\mathbb{F}_q$ be the finite field of $q$ elements. For a geometrically connected algebraic curve $X$ defined over $\mathbb{F}_q$, we will ...
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Etale homotopy fibre and etale classifying space
I am looking for answers/references for the following constructions/statements.
Let $X$ be a scheme over a field $K$, with geometric etale fundamental group $\pi_1(\overline{X},\overline{b})$.
There ...
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Why is the etale fundamental group of $\mathbb{P}^n$ trivial?
For $n=1$, one just uses the Riemann-Hurwtiz formula. I am curious about how to do it for $n\geq 2$. I spoke to a colleague and their proof did not click with me and I've since forgotten how it goes.
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Etale cohomology of pushforward on affine scheme
Let $X = Spec(\mathcal O_K)$ be the spectrum of the ring of integers of a number field $K$. Let $\mathfrak p$ be a non-zero prime ideal of $\mathcal O_K$ and let $i: Spec(\mathcal O_K/\mathfrak p) \to ...
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Structure of the stalk of a pushforward etale sheaf
Let $X$ be an algebraic scheme over an algebraically closed field of characteristic $p$, let $U$ be an
open dense subset and let $\mathcal F$ be a lisse $\overline{\mathbb Q_l}$ sheaf on $U$.
It is ...
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Pullback of Etale Cover is Trivial
In Milne's notes on 'Etale Cohomology, Proposition 6.16 on page 49 is the following:
Proposition 6.16 Assume $X$ is connected, and let $\overline{x}$ be a geometric point of $X$. The map $\mathscr{F}\...
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Why $\overline{\mathbb{Q}_\ell}$?
In étale cohomology usually the only well-behaving coefficient sheaves are the finite ones. But one needs a characteristic zero coefficient sheaf for the fixed-point formula to work, thus the $\mathbb{...
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Triangulated category of bounded constructible complexes of $\mathbb{Q}_l$-sheaves
Let $k$ be a field, $l$ be a prime number invertible in $k$, $X/k$ be an algebraic variety, then the triangulated category $D_c^b(X)$ of bounded constructible complexes of étale $\mathbb{Q}_l$-sheaves ...
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"sheaf of germs of functions" VS "sheaf of functions"?
I encountered phrases such as "sheaf of germs of continuous functions" (e.g., page 164 of the book "Foundations of Differentiable Manifolds and Lie Groups" by Warner 1983) and &...
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Henselisation of Normal local rings (in Milne's Etale Cohomology)
The usual way to define the Henselisation $A^h$ of a local ring $(A, \mathfrak{m})$ is to take direct limit $\varinjlim (B, q)$ over all etale neighborhoods of $A$
(i.e. pairs $(B,q)$ where $B$ is an ...
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Interpretation of etale cohomology
We know that
$$\operatorname{FiniteEtale}(X) \cong \operatorname{LocalSystem}_{X}(\text {FiniteSet})$$
So how does one interpret the etale cohomology of a finite local system with the corresponding ...
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Characterizing vanishing étale cohomology in higher degrees
Let $X$ be a scheme and suppose that the étale cohomology $H^i(X,\mathcal{F})$ vanishes for all quasi coherent sheaves $\mathcal{F}$ and $i > 0$. Clearly if $X = \text{Spec}(k)$ for an ...
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Local monodromy of Kummer Sheaves
Let $\mathbb{F}_q$ be a finite field, $\chi$ a complex (or $\overline{\mathbb{Q}_\ell}$) character of $\mathbb{F}_q^\times$, and $\mathcal{L}_\chi$ the Kummer sheaf on the multiplicative group $\...
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Why a homomorphism of schemes induces a homomorphism of automorphism groups? [duplicate]
In Milne's Lectures of Etale Cohomology, page 27,the first line, he says(all schemes are supposed to be integral.):
Suppose $X_i$ and $X_j$ are finite etale over scheme X, then a homomorphism $\phi\...
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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 ...
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How do we explicitly compute the Galois action on etale cohomology?
The general theorems about etale cohomology are usually enough to let us compute a given $\mathrm{H}^i(X,\mathbf{Q}_\ell)$ as a $\mathbf{Q}_\ell$-vector space without too much difficulty.
I would like ...
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Algebraic group extensions of $\mathbb{Z}/p\mathbb{Z}$ (Milne's AG, Exercise 2.6)
This is exercise 2.6 in Milne's "Algebraic Groups". It asks to show that the commutative algebraic group extensions over a field $k$ of characteristic $p$
$$0\to\mu_p\to G\to \mathbb{Z}/p\...
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Question about description of $H^1_{et}(X, \mu_n)$
On this page of Stacks project the pair $(\mathcal{L},\alpha)$ mentioned inside is described as follows:
Fix an integer $n$ and a scheme $X$. We form a pair $(\mathcal{L},\alpha)$ where $\mathcal{L}$ ...
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Independence of Etale Fundamental Group on Base Point
I'm reading Milne's notes on Etale Cohomology, where on page 27, there Remark 3.3 says that
(a) If $\overline{\overline{x}}$ is a second geometric point of $X$, then there is an isomorphism, $\pi_1(X,...
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Is $\mu_n$ and $\mathbb{Z}/n\mathbb{Z}$ the same thing?
As a $\mathbb{Z}/n\mathbb{Z}$-module (not a sheaf), I know that the roots of unity $\mu_n$ is noncanonically isomorphic to the $\mathbb{Z}/n\mathbb{Z}$. Also the nLab says that it is also so in the $\...
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Does an exact sequence of commutative group schemes define an exact sequence of sheaves?
Suppose we are given an exact sequence
$$ 0\to A\to B\to C\to 0$$
of commutative group scheme over a field $k$. Then does the sequence define an exact sequence of étale sheaves on a $k$-scheme $X$?
I ...
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Is $H^1_{et}( \overline{C},\mathbb Q_l)$ irreducible as a Galois representation?
Let $C$ be a smooth, projective, geometrically connected curve over $\mathbb Q$. We know that for any prime $\ell$ outside a finite set of primes, we have a $G_{\mathbb Q}$-representation $H^1_{et}( \...
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142
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Etale cohomology groups as Galois representations
I am trying to understand how etale cohomology groups are Galois representations. Let $X$ be a scheme over a perfect field $K$ and write $\overline{X}=X \times_{{Spec }K} \textrm{Spec } \overline{K}$ ...
- 429
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59
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Ring of total fractions of the strict henselization of a non-normal local ring
Let $A$ be an integral Noetherian local ring of dimension 1 which is not normal and with residue field a finite field $\mathbb{F}_q$ (so typically some local ring of a singular curve on a finite field)...
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100
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When the presheaf inverse image of a sheaf is already a sheaf
In Milne's book Etale cohomology, p93, he defines $0$-th cohomology with compact support of a separated variety $X$
(in this book a variety is a geometrically irreducible, geometrically reduce scheme ...
- 519
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129
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How the Frobenius behaves under the change of base field
Let $k'/k$ be an extension of finite fields, $X$ be a scheme over $k'$ and thus over $k$. Then $X\otimes_{k} \bar{k}$ is $n = [k':k]$ disjoint union of $X\otimes_{k'} \bar{k}$. How the Frobenius of ...
- 519
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A question about Flasque Sheaves on a site..
We define a Flasque Sheaf on a site as one whose first Čech Cohomology vanishes for every covering of every object of the site. I know this definition is non-standard, but Lei Fu's 'Étale Cohomology ...
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