# Questions tagged [etale-cohomology]

for questions about etale cohomology of schemes, including foundational material and applications.

516
questions

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### Discriminant ideal in a member of Barsotti-Tate Group

Let $S = \operatorname{Spec} R$ an affine scheme (in our case latter a complete dvr) and $p$ a prime. Then Barsotti-Tate group or $p$-divisible group $G$ of height $h$ over $S$
is an inductive system
...

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87 views

### A lisse mixed sheaf as an extension of pure lisse sheaves

I am trying to understand Corollary 1.8.11 in Deligne's Weil II paper. The statement is that for a normal scheme $X_0$ that is of finite type over $\mathbb{F}_q$, every lisse $\ell$-adic $\iota$-mixed ...

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104 views

### Decomposition of cohomology via algebraic cycles (or correspondences)

In the paper " On the calculation of local terms in the Lefschetz-Verdier trace formula and its application to a conjecture of Deligne" by Richard Pink, he wrote in the first page that
To obtain ...

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123 views

### Counting the image of a map of varieties using the trace formula

Suppose $f: X\to Y$ is a finite map of varieties over a finite field $\mathbb F_q$. Is there an etale constructible $\mathbb Q_\ell$ sheaf $\mathscr F$ on $Y$ which counts the number of rational ...

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95 views

### Čech for $\ell$-adic sheaves

On the étale site of a scheme $X$, there is a spectral sequence associated to the data of an étale hypercover $K$ of $X$ and an abelian étale sheaf $\mathcal F$ on $X$:
$$E_2^{p,q}=\check H^p(K,\...

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**1**answer

61 views

### Does a morphism of etale sheaves restricting to a closed subscheme $Z$ induce a morphism of their subsheaves of sections supported on $Z$?

Let $X$ be a locally Noetherian scheme and $i:Z\to X$ be an immersion of closed subschemes.
Let $\mathcal{F},\mathcal{G}$ be two etale abelian sheaves over $X_{et}$.
We can define the subsheaf $\...

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37 views

### local acyclicity when restricting to an hypersurface

Let $X$ be a smooth scheme over $\mathbb{C}$ and a constructible sheaf $K$ of complex vector spaces on $X\times\mathbb{A}^1$ and a function $g:X\rightarrow \mathbb{A}^1$.
Suppose that $K$ is locally ...

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108 views

### Where general mixed Galois representations are defined?

I am interested in etale cohomology of varieties, and respectively, in mixed $\mathbb Q_{\ell}$-adic Galois representations over finitely generated fields. What is the canonical reference for this ...

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191 views

### Help with $\mathbf{Q}_{\ell}$ sheaves

Let $X\to S$ be a morphism of smooth connected varieties over an algebraically closed field $k$; let $j:\eta\to S$ be the inclusion of the generic point into $S$ (not a geometric generic point) and ...

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162 views

### Which $p$-adic valuations of Weil numbers (that is, eigenvalues of Frobenius) are possible?

Let $C$ be a smooth projective curve over a finite field $\mathbb F_q$, $q$ is a power of the characteristic $p$. It is well-known that if $\alpha$ is an eigenvalue of Frobenius acting on $H^1_{et}(C,\...

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170 views

### Henselization and completions of local rings & schemes

That's the second part of my coarse becoming acquainted with Henselizations of fields and local rings. (in this question we focus on local rings as it is more algebro geometric motivated). So let $(R,...

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129 views

### Do Neron-Severi groups of smooth projective unirational varieties contain $\ell$-torsion?

Let $X$ be a smooth projective unirational variety over an algebraically closed field of characteristic $p>0$, and $\ell\neq p$ a prime. My question: can the Neron-Severi group of $X$ contain (non-...

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158 views

### Dimension of $\ell$-adic Eilenberg-Maclane space

I'm currently studying the $\ell$-adic cohomology functor, i.e. the functor $$F:X \rightarrow H^i_{ét}(X,\mathbb{Q}_{\ell}).$$
In some sense, this is a representable functor, i.e. there exists an $\...

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144 views

### What is the etale homotopy type of the Witt group of braided fusion categories?

The Witt group $\mathcal{W}$ of braided fusion categories (see also the sequel paper) can be defined over any field; I am happy to restrict to characteristic $0$ if it matters.
Is $\mathbb k \...

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219 views

### What is known about lower etale cohomology of unirational varieties?

Which information is currently known about $H^1_{et}(X,\mathbb{Z}_l)$ and $H^2_{et}(X,\mathbb{Z}_l)$, where $X$ is a smooth unirational variety over an algebraically closed field of finite ...

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321 views

### Modern context for hypercohomology spectra

In Thomason's paper Algebraic K-theory and étale cohomology, (Ann. ENS 1985, Numdam link) Thomason develops an elaborate theory of hypercohomology spectra, $\mathbb{H}(X,\mathcal{F})$ for presheafs of ...

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95 views

### Are there any explicit (prime-to-l) alterations for interesting varieties (or schemes)?

I have read that it is easier to find regular alterations of varieties than their resolutions of singularities (moreover, I believed in this sentence when I read it). My question is: do there exist ...

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199 views

### What is the etale cohomology of G_m on Spec Z?

I apologize if this is straightforward, but I can't seem to find a reference: What is $H^1_{et}(\mathrm{Spec}(\mathbb{Z}), \mathbf{G}_m)$? It shouldn't be as simple as noting that the etale ...

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744 views

### Is there a version of algebraic de Rham cohomology that can be used to calculate torsion classes?

Much work has gone into the construction of cohomology theories which are defined on algebraic varieties (étale, crystalline, etc.) and comparison isomorphisms between them.
Say $X$ is an algebraic ...

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157 views

### Completed cohomology and variants

I am interested in the following set up:
I have an ind-sequence of curves $\dots X_2\to X_1$ defined over a finite field of characteristic $p$ such that $X_n/X_{n-1}$ is a Galois degree $\ell$ cover ...

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821 views

### Usage of étale cohomology in algebraic geometry

I'm a student interested in arithmetic geometry, and this implies I use étale cohomology a lot. Regarding its definition, étale cohomology is a purely algebro-geometric object. However, almost every ...

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189 views

### Computing weights of $\bar{\mathbb{Q}}_l(1)$ from the definition

This seems to be a trivial question, but I am genuinely confused about it.
The notion of weights as in Deligne's Weil II are defined in terms of eigenvalues of automorphisms that Frobenius morphisms ...

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102 views

### Canonical étale path between a point and its ''nearby'' point

Consider the punctored line $X=\Bbb{A}^1_k\setminus \{s_1,\ldots,s_n\}$ over some field $k$. A(n étale) path in $X$ between two geometric points $x$ and $y$ is, by definition, an isomorphism between ...

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642 views

### What is known about the cohomological dimension of algebraic number fields?

What is the cohomological dimension of algebraic number fields like $\Bbb{Q}$, $\Bbb{Q}[i]$, $\Bbb{Q}[\sqrt{3}]$ or similar? I'm interested in computing the cohomological dimension of $\Bbb{A}^1_k$ ...

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260 views

### Is there a Poincare residue in characteristic $p$?

The Poincare residue I mean is there one here:
https://en.wikipedia.org/wiki/Poincar%C3%A9_residue
Basically, I would like a nice way to use a meromorphic $n$-form on $\mathbf{P}^n_{\mathbf{F}_p}$ ...

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266 views

### Restrictions on the Galois representations coming from singular varieties

Fix a prime number $p$. Choose an algebraic closure $\mathbb{Q}_p\to \overline{\mathbb{Q}_p}$. Given a proper geometrically irreducible scheme $X$ over $\mathbb{Q}_p$ and a non-negative integer $i$, ...

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117 views

### Conjugation action of $Gal(\bar{s}/s)$ on the tame ramification group

There is a statement in SGA 7-1 Exposé 1 (P. Deligne, Résumé des premiers exposés de A. Grothendieck, pdf of SGA7-1), (0.3.1):
$S$ is a Henselian trait (i.e. the spectrum of a henselian discrete ...

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380 views

### Poincaré duality and Galois action

Poincaré duality says that under nice situation we have a canonical perfect pairing
$$ P : H_c^r(X, \mathscr{F}) \times H^{2d - r}(X, \mathscr{F}^\vee(d)) \to \mathbb{Z}/n.$$
I want to show that ...

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192 views

### A question about strict henselian local rings

Let $f: X\to Y$ be a finite morphism of schemes, let $y\in Y, Y(\bar{y}):={\rm Spec}(\mathscr{O}_{Y, \bar{y}}), X_{\bar{y}}:=X\times_Y{\rm Spec}(\kappa_y^s)=X_y\otimes_{\kappa_y}\kappa_y^s, X({\bar{y}}...

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200 views

### Big etale topos vs small etale topos

Are they equivalent?
That is, given a sheaf of sets $\mathscr{F}$ defined on the small etale site on $X$, is there an essentially unique way to extend it to a sheaf on the big etale site on $X$? If ...

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231 views

### Étale fundamental group of multiplicative group over an algebraically/separably closed field

This is a repost of my question here.
Do we know the structure of the étale fundamental group $\pi^\text{et}_1(\mathbb{G}_{m,K^\text{sep}})$ of the multiplicative group, for a given field $K$? For ...

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107 views

### Torsion in crystalline cohomology as a function of the generic fiber

There is an example of a smooth projective surface $X$ over $\mathbb{Z}_2$ such that $H^*_{et}(X_{\overline{\mathbb{Q}_2}}, \mathbb{Z}_2)$ is torsion-free but $H^*_{crys}(X_{\mathbb{F}_2}/\mathbb{Z}_2)...

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126 views

### Endomorphism rings of flat group schemes

Let $R$ be a commutative ring and $X$ be a flat $R$-group scheme.
We call $\text{End}_R(X)$ the ring of endomorphisms of the $R$-group scheme $X$, defined over $R$.
Let $R\to S$ be a ring map ...

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138 views

### Why are traces an analogue to integrals?

In Poincare duality for singular cohomology, one integrates cohomology classes against a fundamental class to get a number $\int_{[M]} \omega$.
In the formulation of Poincare duality in etale ...

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189 views

### Analytic and algebraic torsor of abelian scheme

Let $M$ be an affine complex manifold, let $A$ be an abelian scheme over $M$. Let $\mathcal{A}$ be the sheaf of local sections of $A$ over $M$. We can equip $M$ with etale topology $M_{et}$ or complex ...

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156 views

### Reference request: number of irreducible components and top dimension etale cohomology

Let $k$ be an algebraically closed field, $\ell$ is a prime different with characateristic of $k$, and consider the $\ell$-adic etale cohomology. We know the number of connected components of a scheme ...

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504 views

### Derived base change in étale cohomology

Given a commutative square of ringed topoi
$$\begin{array}{ccc}X'\!\! & \overset{f'}\to & Y'\!\! \\ \!\!\!\!\!{\small g'}\downarrow & & \downarrow{\small g}\!\!\!\! \\ X & \...

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188 views

### Purity of vanishing cycle for proper scheme over DVR with smooth generic fiber

Let $X$ be a scheme proper and flat over a complete discrete valuation ring $O$ with finite residue field $k$, and choose a prime $l$ not equal to characteristic of $k$. Consider the Galois ...

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390 views

### Why care about Grothendieck topology? [closed]

Noah Schweber said here the following:
Why would you want a notion of sheaf theory for objects more general
than topological spaces? Well, the original motivation (to my
understanding) was to ...

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**1**answer

723 views

### Generalized Behrend version for Grothendieck-Lefschetz trace formula

[MOVED HERE FROM MSE.]
The statement of the Grothendieck-Lefschetz fixed point theorem is well-known. For a proper algebraic variety $X$ over $\mathbb F_q$,
$$\#X(\mathbb F_q) =\sum_i (−1)^i Tr(Fr_X, ...

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418 views

### Structure of the variety of $n$-tuples of $m \times m$ matrices with zero product

Consider the functor sending a commutative ring $R$ to $\{(A_1,\dots,A_n) \in ( M_m(R) )^n | A_1 \dots A_n =0 \}$ which defines a scheme over $\mathbb Z$, let $X$ be its base change to $\mathbb C$.
...

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425 views

### Isomorphism of the $\ell$-adic Tate module of an elliptic curve with CM

Let $E$ be an elliptic curve over $K$ (totally real number field) with complex multiplication by the field $L$. Let $\psi$ be the Grössencharacter associated to $E$, assume that $\psi$ of type $(-r,0)$...

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275 views

### How to construct cup-product in a general site?

Let $\mathcal{C}$ be a site on the category of schemes for some Grothendieck topology, $X$ a scheme, and let $F,G$ be two sheaves of abelian groups on $\mathcal{C}$. Do we have cup-product as follows?
...

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155 views

### Is the induced map between etale cohomology groups independent of the compactification?

Let $k$ be an algebraically closed field, and $X$ be a smooth variety. For any compactification $i: X \hookrightarrow Y$ (so $X$ is a dense open subset of $Y$), consider the induced map
$i_!: H^i_{et,...

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318 views

### Comparison of complex and p-adic Hodge structure

Let $X$ be a smooth projective variety over a p-adic local field $K$, and let $\bar{K}$ be the algebraic closure of $K$. Fix an isomorphism $\sigma:\bar{K}\to\mathbb{C}$. Do $\sigma$ induces an (iso)...

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121 views

### A canonical complex computing etale cohomology

Crystalline cohomology can be computed as the hypercohomology of the de Rham-Witt complex.
If we want to compute the etale cohomology of the constant sheaves $\mathbb{Z}_l$ or $\mathbb{Q}_l$ (well, ...

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255 views

### In the definition of big/small étale/fppf/… site, is their covering set really a set?

My definition of a site is a pair $(\mathcal{C},\DeclareMathOperator{\Cov}{Cov}\Cov (\mathcal{C}))$ where $\mathcal{C}$ is a category and $\Cov (\mathcal{C})$ is a set consisting of elements of the ...

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121 views

### Cohomology of modular curves: vanishing and decomposition

Let $\pi:E\to Y$ be a universal elliptic curve over an open modular curve $Y$. Take a prime $\ell$ and take $\mathcal{H}=(R^1\pi_*\mathbb{Q}_\ell)^\vee$ where the dual, $(-)^\vee$, means the internal ...

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275 views

### Serre's examples in rigid analytic geometry

Over $\mathbb{C}$, we have the following phenomenon: there exist algebraic varieties whose etale homotopy types are isomorphic but the homotopy types of their analytifications are not. Such examples ...

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**1**answer

309 views

### Structure theorem for etale algebras over a more general ring than a field

I call etale a finite-type flat $R$-algebra $A$ such that $\Omega_A =0$ (I hope this is the standard definition).
In the case where $R=k$ is a field, any such algebra $A$ decomposes as a finite ...