# Questions tagged [etale-cohomology]

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

659
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### Why do we have $X/G\cong \coprod_jX_j/G_j$?

I'm reading Lei Fu's "Etale Cohomology Theory".
Corollary 3.2.6. Let $S$ be a noetherian scheme,let $X\to S$ be an etale covering space, and let $G$ be a finite group acting on the right of ...

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### Base points and geometric points of schemes

Let $k$ b a field and $X$ a geometrically connected variety over $k.$ Let us fix an algebraic closure $$\ast: \text{Spec } \bar{k} \rightarrow \text{Spec } k$$ and denote by $\bar{X} = X \times_k \bar{...

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### Étale group schemes and specialization

If $A$ is an abelian variety over a finite field $\mathbf{F}_q$, then $A(\mathbf{F}_q)$ (resp. $A(\overline{\mathbf{F}}_q)$) is a finite (resp. infinite torsion) group, but $A(\mathbf{F}_q(t))$ is a ...

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### Flat scheme-theoretic closure

Suppose $R$ is a discrete valuation ring with fraction field $K$. Let $X\subset \mathbf{P}^n_{C_K}$ be a closed subscheme, flat over $C_K$, a smooth projective curve over $K$.
Let $C_R$ be a flat ...

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### Is there a ring stacky approach to $\ell$-adic or rigid cohomology?

Ever since Simpson's paper [Sim], it was observed that many different cohomology theories arise in the following way: we begin with our space $X$, we associate to it a stack $X_\text{stk}$ (which ...

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### $p$-adic étale cohomology group of open smooth varieties

Let $K$ be a finite extension of $\mathbb{Q}_p$ and let $X$ be a smooth variety over $K$.
Dr. Yamashita announced that he had proved the Galois representation of $p$-adic étale cohomology group $H^*_{\...

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### Interpretation of Tate conjecture using motivic homotopy

For a smooth projective variety $X$ over a field $k$ the Tate conjecture says that the cycle class maps
$$CH^i(X)\otimes \mathbb{Q}_l \to H^{2i}(X_{\bar{k}},\mathbb{Q}_l(i))^{G_k}$$
are surjective. To ...

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### Calculating étale fundamental groups from the usual fundamental group

$\newcommand{Spec}{\operatorname{Spec}}$Let $X$ be a connected affine smooth variety over $\mathbb{Q}$, with a point $x\in X(\Spec(\mathbb{Q})$.
For any algebraically closed field $K$ of ...

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### When (or why) is a six-functor formalism enough?

The six functor formalism in a given cohomology theory consists of for each space a derived category of sheaves and six different ways to construct functors between those categories (four involving a ...

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### Cycles contained in ample enough hypersurfaces

Let $X$ be an irreducible smooth projective variety of pure dimension $d$ over the complex numbers and $Z\subset X\times X$ a codimension $d$ irreducible smooth closed subvariety.
Is there a smooth ...

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### Tangential basepoint of a log singular local system

Consider the Legendre family $f: X\longrightarrow Y = \mathbb{P}^1\setminus\{0,1,\infty\}$, defined over $K = \mathbb{Q}/\mathbb{Q}_p$.
having fibre above $t\in Y$, the elliptic curve $E_t := y^2 = x(...

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168
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### Cup products and correspondences

Suppose $X$ is a smooth projective complex variety, connected of dimension $n$.
Let $a$ be an algebraic correspondence in $A^n(X\times X)$, the group of cycles modulo homological equivalence in $H^{2n}...

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### Geometric generic point of a complete linear system

In the following context: Let $S$ be a connected smooth projective surface over $\mathbb{C}$, and let $\Sigma$ be the complete linear system of a very ample divisor $D$ on $S$. Let $d=\dim(\Sigma)$ ...

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284
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### About simple motives

I'm reading through Jannsen's paper Motives, numerical equivalence, and semi-simplicity and I'd like to pose two questions.
Suppose all motives are $F$-linear, for some characteristic zero field $F$, ...

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### $L$-series and Riemann zeta function

I am currently reading SGA 4$\frac{1}{2}$, exposé 2: Rapport sur la formule des traces.
The $L$-series associated to a scheme $X$ of finite type over $\mathbb{F}_{p}$ is defined as
$$L(X,s):=\prod_{x\...

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### K-theory of l-adic sheaves of a curve

I am trying to understand if there is a good notion of a $K$-theory attached to the etale topology on a nice scheme $X$ (say smooth projective goem connected curve over a finite field is enough for me)...

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### Unramified section associated to a rational point

This is a question for those familiar with the section conjecture, so I'll do away with the definition of a ramification map in this case. Here is the definition of a ramification map from an etale ...

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### In which "sense" unramified Milnor-Witt K-groups are unramified

Let $X$ be an integral locally noetherian smooth
scheme over base field $k$. Then for every $x \in X^{(1)}$ point of codimension $1$, the stalk $\mathcal{O}_{X,x}$ is a discrete valuation
ring. ...

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### Intuition for de Rham comparison theorem in $p$-adic Hodge theory

The de Rham comparison theorem from $p$-adic Hodge theory compares the etale cohomology of a variety with the de Rham cohomology of that variety. It says the following:
Let $K/\mathbf{Q}_p$ be a ...

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### Cohomology of a curve and its Jacobian over an algebraic closure of a number field

In this MathOverflow post, the smooth projective curve $C$ was defined over $\mathbb{C}$ and we have an isomorphism of de Rham cohomology groups
$$H^1(C, \mathbb{C}) \cong H^1(J_C, \mathbb{C}),$$
...

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### Why do we care about the eigenvalues of the Frobenius map?

The Riemann hypothesis for finite fields can be stated as follows: take a smooth projective variety X of finite type over the finite field $\mathbb{F}_q$ for some $q=p^n$. Then the eigenvalues $\...

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### What are some concrete applications of Grothendieck's six operations?

In Gallauer's An introduction to six-functor formalisms I read:
Indeed, the language and theory of six-functor formalisms permeates much of modern algebraic geometry and beyond, and has spawned ...

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### Proof of the projection formula (for cohomology of $\mathbf{P}V$)

Let $V\to X$ be a vector bundle (over say a scheme).
Then the cohomology of its projectivisation is
$$\text{H}^*(\mathbf{P}V)\ =\ \text{H}^*(X)[t]/(t^{n+1}+c_1(V)t^n+\cdots+c_n(V))$$
as an algebra, ...

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### Is there the specialisation map of etale K theory?

Take a smooth proper morphism of schemes $X\to S$. Fix a point $t\in S$ and a point $s\in \overline\{s\}$. For a prime $l$ which is invertible in $S$, is there the natural specialization map of etale ...

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### Henselisation of normal rings (Milne's EC)

The usual way to define the Henselisation $A^h$ of a local ring $(A, \mathfrak{m})$ is by taking the direct limit $\varinjlim (B, q)$ over all etale neighborhoods of $A$
(i.e. pairs $(B,q)$ where $B$ ...

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### Purity for proper varieties

Let $X$ be a proper, geometrically connected, geometrically integral variety over $\mathbf{F}_q$. There exists a finite field extension $k/\mathbf{F}_q$ of degree $d$ and an alteration $X'\to X_k$ ...

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### $p$-adic étale cohomology groups are not $\mathbb{C}_p$-admissible

It is stated in Caruso - An introduction to $p$-adic period rings (the remarks following equation (2)) that the $p$-adic étale cohomology groups of an algebraic variety $X$ over a finite extension $K$ ...

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### Is there any theory of "étale cohomology" with algebraic coefficients?

For simplicity, I will restrict attention to untwisted coefficients.
Let $k$ be a finite field of characteristic $p$, and $\ell\ne p$ prime. Can one define a cohomology theory with $\mathbb{Q}_\ell^{\...

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### "Cohomology with compact support isomorphic to homology" theorems

I am collecting theorems throughout different fields which say roughly something of the form "Cohomology with compact support isomorphic to homology".
I'm studying this situation (and its ...

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### Voevodsky's motives and Deligne's systems of realizations

$\newcommand{\gm}{\mathrm{gm}}$Let $\mathbf{DM}_{\gm}(\mathbb{Q},\mathbb{Z})$ be Voevodsky's category of geometric motives over $\mathbb{Q}$ with coefficients in $\mathbb{Z}$ (e.g. as on p.124 of ...

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### Cohomology map induced by inclusion of curves

Let $C$ be a smooth affine geometrically integral curve of genus $\geq 1$ over an algebraically closed field $k$, and let $\iota: C \rightarrow C'$ denote the inclusion into its smooth ...

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### Symmetrical monoidal $2$-category of cohomological correspondences

My question is whether a symmetric monoidal $2$-category of ``cohomological correspondences'' has been been rigorously constructed anywhere in the literature.
Let me be more precise about what I mean.
...

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### Torsors over elliptic curves

Let $G$ be a finite abelian etale group scheme over a number field $k$. Let $E$ be an elliptic curve over $k$ and $C := E\backslash \{O\}$ its affine model of the same equation.
Recall that for a ...

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### A comparison theorem between crystalline cohomology and étale cohomology

Suppose $X/\mathbb F_q$ is a smooth projective variety. Katz-Messing (eudml) shows that the characteristic polynomial of the Frobenius on $H^i_{et}(\overline{X},\mathbb Q_\ell)$ and $H^i_{crys}(X)$ ...

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### Two natural morphisms of sheaves with the same source and target; do they agree?

Suppose we have a diagram
$\require{AMScd}$
\begin{CD}
A @>a>> B\\
@V b V V @VV c V\\
C @>>d> D @>e>> E \\
@VfVV @VVgV @VVhV \\
F @>>i> G @>>j> H
\end{CD}...

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### Strict henselianization of complete intersections

As far as I understand (and tbh for my purposes), one of the main points of strict henselisation of a local ring is that it computes the stalk at a point of a scheme in the étale topology. In the ...

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### Does the Gross-Hopkins period map have a natural interpretation coming from derived algebraic geometry?

The Gross-Hopkins period map is a map on the $W(k)$-points of $LT_n \to P^{n-1}$, where $k = F_{p^n}$. It sends a lift $G$ of the Honda formal group to the 1-$d$ subspace $\omega_G$ of the Dieudonn'e ...

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### Deligne's integrality theorem in the setting of $ \mathbb{F}_{\ell}((t)) $-adic cohomology

Let $ \mathbb{F}_{q} $ be a finite field of characteristic $ p $ and $ \overline{\mathbb{F}_{q}} $ be an algebraic closure of $ \mathbb{F}_{q} $. Let $ X $ be a smooth projective variety over $ \...

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### Every rank 1 local system $L$ satisfying $m^*L=L\boxtimes L$ comes from the Lang torsor? The same holds for D-modules?

Let $G$ be a commutative connected algebraic group over a field $k$, with group operation $m:G\times G\to G$. If $k=\mathbb{F}_q$, we may use a character $\varphi:G(k)\to\overline{\mathbb{Q}}_\ell^\...

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### Computation of cohomology of Eilenberg-Maclane spaces

$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Ext{Ext}\DeclareMathOperator\Spf{Spf}$Background:
If $E$ is a complex-oriented spectrum, then $E^*(K(\mathbb{Z}/p^k,1))$ sits inside a long exact ...

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### When is map of $E_{\infty}$-ring spectra etale iff certain condition is fullfilled

When is it true that a map of $E_{\infty}$-ring spectra $R \to S$ is etale (in Lurie's sense) if and only if, $\operatorname{TAQ}^R(S) = 0$ and $ \pi_*(R)\otimes_{\pi_0(R)} \pi_0(S) = \pi_*(S)$?

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### Groupoid of points, shape and stratified shape of $\operatorname{Sh} (X_\text{pro-ét})$

$\DeclareMathOperator\Sh{Sh}\DeclareMathOperator\Pt{Pt}$Maybe this is well-known or even a stupid misunderstanding of something very basic. It's well-known that the groupoid of points (i.e., groupoid ...

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### Question regarding étale sheaf under finite étale surjective morphism

Let $X$ and $Y$ be smooth projective varieties over $\mathbb{C}$, and suppose we have a surjective finite étale morphism $f:X\rightarrow Y$ (actually $Y=X/G$ for a free action of a finite group $G$), ...

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### Does p-adic etale cohomology know the variety has ordinary reduction or not?

For a smooth proper variety $X$ over discrete valuation ring $\mathcal{O}$ of mixed characteristic $(0,p)$, let $X_K$ be the generic fibre over a generic point $\textbf{Spec} K$ and let $X_k$ be the ...

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### Dimension is an invariant in the Grothendieck ring of algebraic varieties

$\DeclareMathOperator\Var{Var}$Let $k$ be any field and let $K_0(\Var_k)$ be the Grothendieck ring of $k$-algebraic varieties (i.e. algebraic varieties up to cut-and-paste relations). Given an ...

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### Sheafifcation for the étale site

Let $X$ be a scheme and $\mathcal{F}$ a presheaf on $X_{ét}$.
For each $x_{i}\in X$, pick a geometric point $\bar{x}_{i}$ over $x$ and denote by $i_{\bar{x}_{i}}:\text{Spec}(k_{i})_{\text{ét}}\...

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### Degree-2 étale covers of curves in characteristic 2 vs torsion points on the Jacobian

It can be found, at Hartshorne exercise 4.2.7 for example, that in the case where $\operatorname{char}(k) \neq 2$ we have a nice correspondence between etale degree 2 covers of a curve $C$ and 2-...

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### Stratified sites/topoi and constructible sheaves

Is it possible to define (possibly derived) categories of constructible sheaves over sites more general than those of open subsets of topological spaces while still retaining essential features, like ...

2
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429
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### Galois invariants and tensor products

Consider a number field $K$ and a finite Galois field extension $L/K$. Let $E$ be an elliptic curve over $K$ and consider the abelian group
$$E(L)\otimes L^{\times}.$$
Every element $g$ in $\text{Gal}(...

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### Adèlic points and algebraic closure

Consider $\mathcal{X}$ a projective and flat scheme over $\text{Spec}(\mathcal{O}_K)$, with $\mathcal{O}_K$ the ring of integers of a number field $K$.
Let $F/K$ vary over all finite Galois number ...