Skip to main content

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.

Filter by
Sorted by
Tagged with
0 votes
0 answers
87 views

Formally etale and etale

Let $f:X \to Y$ be a morphism of schemes over an algebraically closed field $k$ which is formally etale and an injection on $k$-points. Is $f$ etale? If not, what are some additional properties that ...
E. KOW's user avatar
  • 359
4 votes
1 answer
106 views

Compactly supported étale cohomology and affine space fibration

Suppose $f: X \to Y$ is an $n$-dimensional affine space fibration of schemes: by this I mean that there exists an étale cover $\{U_i \to Y\}_{i \in I}$ such that the map $X \times_Y U_i \to U_i$ is ...
Ashwin Iyengar's user avatar
0 votes
0 answers
63 views

Do étale coordinates give rise to a regular sequence of diagonal elements?

Fix an algebraic extension $k\subseteq K$ of fields of characteristic zero and consider a map of commutative rings $\phi\colon K\left[T_{1}^{\pm},\dots,T_{n}^{\pm}\right]\to A$ which is étale. Now ...
user141099's user avatar
1 vote
1 answer
46 views

étale fundamental group of a connected algebraic group over an algebraically closed field of characteristic p

I am looking for an explanation of the fact that the étale fundamental group of a connected unipotent algebraic group $G$ over an algebraically closed field $k$, where $char(k)=p$ has no non-trivial ...
Pambra iskra's user avatar
0 votes
0 answers
58 views

Etale cohomology of locally constant sheaf on affine curve

In the proof of Proposition 14.12 in Milne's Lectures on 'Etale Cohomology, Next, let $\mathcal{F}$ be a locally constant sheaf on $U$ with finite fibres. There exists a finite Galois covering $\pi:U'...
dwg's user avatar
  • 76
2 votes
0 answers
74 views

References for Grothendieck-Verdier Duality in the original spirit

Does anyone know of good expository notes going through the theory of Verdier duality? I have been trying to read the original paper by Verdier entitled "A Duality theorem in the étale cohomology ...
Pambra iskra's user avatar
1 vote
1 answer
75 views

Cycle class of a smooth complete intersection $X \hookrightarrow \mathbb P^n$

Let $f:X \hookrightarrow \mathbb P^n$ be a $(d_1,\ldots ,d_r)$ smooth complete intersection over an algebraically closed field $k$. Let $\ell$ be a prime number different from the characteristic of $k$...
Suzet's user avatar
  • 5,571
1 vote
0 answers
73 views

Using étale fundamental group to show unramifiedness of Tate module

Let $E$ be an elliptic curve over $\mathbb{Q}_p$ with good reduction at $p$, i.e., there exists an elliptic curve $\mathcal{E}$ over $\mathbb{Z}_p$ whose generic fiber is isomorphic to $E$. I have ...
WLOG's user avatar
  • 1,336
1 vote
0 answers
100 views

Pullback of Constant Sheaf: A "Very Plain Proof"

Let $Y$ be any scheme, $S$ a set and $\underline{S}_Y$ the constant sheaf on some reasonble site over $X$, say to have something concrete in mind the small etale site $X_{\text{et}}$. Let $f: X \to Y$ ...
user267839's user avatar
  • 7,601
0 votes
1 answer
51 views

Long exact sequence of $\ell$-adic cohomology with compact support for $X = X_1 \cup X_2$ union of two irreducible components

Let $X$ be a separated scheme of finite type over an algebraically closed field $k$ of characteristic $p>0$, possibly not equidimensional. Assume that $X$ has only two irreducible components $X_1$ ...
Suzet's user avatar
  • 5,571
2 votes
1 answer
54 views

Why is the bilinear form on $H^d(X,\mathbb Q_{\ell})$ afforded by Poincaré duality alternating when $d = \dim(X)$ is odd?

Let $X$ be a smooth projective irreducible variety of pure dimension $d$ over an algebraically closed field of positive characteristic $p$. Let $\ell \not = p$ be a prime number. There is an ...
Suzet's user avatar
  • 5,571
1 vote
0 answers
39 views

Fix points of Galois action on etale homology

Let $k = \mathbb{F}_q$ be a finite field of characteristic $p$ and $C$ a projective, smooth curve over $k$. Denote by $\bar C$ the base change of $C$ to a separable closure $\bar k$ of $k$. Let $\ell$ ...
Erich's user avatar
  • 245
3 votes
1 answer
154 views

étale $\ell$-adic cohomology is a Weil cohomology theory

I was reading https://mathoverflow.net/questions/85078/ell-adic-weil-cohomology-theory and in the first paragraph it is said that $\ell$-adic cohomology is a Weil cohomology theory over separably ...
FreeFunctor's user avatar
0 votes
1 answer
228 views

Does the boundary map $H^k(Z,\Lambda) \to H^{k+1}_c(U,\Lambda)$ factor through the closure of $U$?

Edit: Fixed the triangles following a discussion in the comment section. Let $X$ be a reducible algebraic variety, and let $j:U\hookrightarrow X$ be an open subvariety which is not dense in $X$. Let $...
Suzet's user avatar
  • 5,571
2 votes
1 answer
58 views

Correspondence, cycle class map and Bloch's decomposition of the diagonal

I'm studying Weil cohomology theories, in particular étale $\ell$-adic cohomology, and I have found some problems related to the cycle class map. Let $X$ be a scheme of dimension $d$, my ingredients ...
FreeFunctor's user avatar
2 votes
1 answer
45 views

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 ...
ALi1373's user avatar
  • 31
0 votes
0 answers
45 views

Corollary 5.3.4 in Szamuely, Galois group and fundamental groups

There is a specific point in the proof of Corollary 5.3.4 in Szamuely, Galois group and fundamental groups that I can't seem to fully understand . Specifically the statement is Corollary 5.3.4. If $𝜙:...
Mouthfullofearth's user avatar
5 votes
1 answer
62 views

Cohomology $H(X^{(p)}, \mathbb Q_{\ell})$ of the Frobenius twist of a variety over a finite field

Let $X$ be a quasi-projective variety over a finite field $\mathbb F_q$ where $q$ is a power of a prime $p$. Let $\ell$ be a prime number different from $p$. We have the relative Frobenius morphism $$\...
Suzet's user avatar
  • 5,571
2 votes
0 answers
116 views

A localization sequence for étale cohomology of $\mathcal O_K$, where $K$ is a local field

Given a non-archimedean local field $K$, let $\mathcal O_K$ be the associated valuation ring and $k$ its residue field. According to this MO answer, we have short exact sequence $$0 \to H^2(\mathcal ...
Lukas Heger's user avatar
  • 21.9k
1 vote
0 answers
120 views

Is it possible to construct geometrically the ($\phi$, $\Gamma$)-module corresponding to a $p$-adic representation coming from geometry?

The $p$-adic étale cohomology of algebraic varieties over $p$-adic fields is a fundamental subject in the study of $p$-adic representations. Moreover, thanks to the comparison theorems in $p$-adic ...
Hiroyuki Sunata's user avatar
0 votes
0 answers
24 views

What is the ramification group of a curve at a point

I'm reading the book "Weil conjectures, perverse sheaves and l-adic fourier transform" I can't understand the following lemma:  where $X_0$ is a smooth curve over $\kappa=\mathbb{F}_q$, $X=...
Xiong Jiangnan's user avatar
0 votes
0 answers
29 views

Relations between (the category of étale sheaves over $X$) and (the category of étale sheaves over a model of $X$)

Let $X$ be a quasi-projective scheme over $\mathbb{C}$ and $\frak{X}$ be a (quasi-projective) model over a number field. Are there relations between the derived categories $$D(\mathbb{Sh}(\text{Et}/X))...
Marsault Chabat's user avatar
1 vote
1 answer
99 views

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 ...
user avatar
2 votes
0 answers
107 views

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) ...
Alexey Do's user avatar
  • 2,169
2 votes
0 answers
50 views

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 ...
Alexey Do's user avatar
  • 2,169
0 votes
1 answer
120 views

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 (...
user135743's user avatar
1 vote
1 answer
102 views

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 ...
Alexey Do's user avatar
  • 2,169
3 votes
1 answer
153 views

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{\...
Basil's user avatar
  • 73
0 votes
0 answers
35 views

Fixed part of the induced representation

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 $G$-...
Pierre MATSUMI's user avatar
1 vote
0 answers
29 views

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(\...
Pierre MATSUMI's user avatar
2 votes
0 answers
162 views

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, ...
Suzet's user avatar
  • 5,571
0 votes
1 answer
74 views

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 = ...
Suzet's user avatar
  • 5,571
0 votes
0 answers
89 views

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 ...
Suzet's user avatar
  • 5,571
3 votes
0 answers
91 views

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 ...
Fat Lip's user avatar
  • 111
4 votes
0 answers
121 views

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 ...
Thomas Manopulo's user avatar
3 votes
1 answer
133 views

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}...
Runner's user avatar
  • 145
2 votes
0 answers
161 views

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 $f_!\mathcal{F}$ ...
Xiong Jiangnan's user avatar
2 votes
0 answers
74 views

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 ...
Curious's user avatar
  • 593
2 votes
1 answer
121 views

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 ...
usr9999's user avatar
  • 55
1 vote
1 answer
76 views

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{...
Shrugs's user avatar
  • 1,609
1 vote
1 answer
82 views

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 ...
wsh's user avatar
  • 127
0 votes
1 answer
179 views

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, ...
Alexey Do's user avatar
  • 2,169
3 votes
1 answer
80 views

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 ...
stupid boy's user avatar
2 votes
0 answers
63 views

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 ...
kindasorta's user avatar
  • 1,260
1 vote
1 answer
275 views

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. ...
Shrugs's user avatar
  • 1,609
0 votes
0 answers
147 views

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 ...
bsbb4's user avatar
  • 3,676
3 votes
1 answer
484 views

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 ...
Liu Y's user avatar
  • 315
1 vote
1 answer
162 views

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}\...
user940160's user avatar
2 votes
0 answers
72 views

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{...
Hypatia du Bois-Marie's user avatar
2 votes
0 answers
35 views

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 ...
Doug's user avatar
  • 1,308

1
2 3 4 5
7