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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120
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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{...
4
votes
1
answer
314
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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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1
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143
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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 ...
- 65
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1
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937
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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 ...
- 980
1
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0
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178
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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^*_{\...
- 187
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126
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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 ...
- 381
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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 ...
20
votes
1
answer
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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 ...
- 126k
4
votes
1
answer
220
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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 ...
- 537
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108
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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(...
- 685
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votes
1
answer
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}...
user480741
2
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0
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76
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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 $d=\dim(\Sigma)$ ...
- 519
1
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1
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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 $F$, ...
- 537
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0
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144
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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
$$L(X,s):=\prod_{x\...
- 1,587
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111
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K-theory of l-adic sheaves of a curve
I am trying to understand if there is a good notion of a $K$-theory attached to the etale topology on a nice scheme $X$ (say smooth projective goem connected curve over a finite field is enough for me)...
- 11
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79
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Unramified section associated to a rational point
This is a question for those familiar with the section conjecture, so I'll do away with the definition of a ramification map in this case. Here is the definition of a ramification map from an etale ...
- 835
3
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0
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380
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In which "sense" unramified Milnor-Witt K-groups are unramified
Let $X$ be an integral locally noetherian smooth
scheme over base field $k$. Then for every $x \in X^{(1)}$ point of codimension $1$, the stalk $\mathcal{O}_{X,x}$ is a discrete valuation
ring. ...
- 4,994
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answers
340
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Intuition for de Rham comparison theorem in $p$-adic Hodge theory
The de Rham comparison theorem from $p$-adic Hodge theory compares the etale cohomology of a variety with the de Rham cohomology of that variety. It says the following:
Let $K/\mathbf{Q}_p$ be a ...
- 1,611
1
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0
answers
86
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Cohomology of a curve and its Jacobian over an algebraic closure of a number field
In this MathOverflow post, the smooth projective curve $C$ was defined over $\mathbb{C}$ and we have an isomorphism of de Rham cohomology groups
$$H^1(C, \mathbb{C}) \cong H^1(J_C, \mathbb{C}),$$
...
- 835
26
votes
4
answers
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Why do we care about the eigenvalues of the Frobenius map?
The Riemann hypothesis for finite fields can be stated as follows: take a smooth projective variety X of finite type over the finite field $\mathbb{F}_q$ for some $q=p^n$. Then the eigenvalues $\...
- 1,738
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241
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What are some concrete applications of Grothendieck's six operations?
In Gallauer's An introduction to six-functor formalisms I read:
Indeed, the language and theory of six-functor formalisms permeates much of modern algebraic geometry and beyond, and has spawned ...
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0
answers
191
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Proof of the projection formula (for cohomology of $\mathbf{P}V$)
Let $V\to X$ be a vector bundle (over say a scheme).
Then the cohomology of its projectivisation is
$$\text{H}^*(\mathbf{P}V)\ =\ \text{H}^*(X)[t]/(t^{n+1}+c_1(V)t^n+\cdots+c_n(V))$$
as an algebra, ...
- 5,122
2
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0
answers
106
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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 ...
- 449
2
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0
answers
198
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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$ ...
- 4,994
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1
answer
209
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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$ ...
user480741
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1
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147
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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$ ...
- 125
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1
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400
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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^{\...
- 231
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0
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173
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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 ...
- 4,532
4
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0
answers
277
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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 ...
- 14.7k
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154
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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 ...
- 835
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0
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146
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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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1
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275
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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 ...
- 835
3
votes
1
answer
486
views
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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0
answers
107
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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}...
- 53
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0
answers
111
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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 ...
- 3,984
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0
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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 ...
- 428
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answers
151
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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 $ \...
- 695
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1
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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^\...
- 980
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1
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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 ...
- 428
2
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0
answers
134
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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)$?
- 428
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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 ...
- 2,139
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160
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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$), ...
- 587
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votes
1
answer
388
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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 ...
- 187
4
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2
answers
507
views
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 ...
- 435
3
votes
1
answer
232
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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}}\...
- 461
6
votes
1
answer
353
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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-...
- 381
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0
answers
165
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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 ...
- 1,393
2
votes
1
answer
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}(...
user290895
6
votes
1
answer
337
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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 ...
user335418