Questions tagged [etale-cohomology]
for questions about etale cohomology of schemes, including foundational material and applications.
329 questions with no upvoted or accepted answers
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Finiteness of etale cohomology for arithmetic schemes
By an arithmetic scheme I mean a finite type flat regular integral scheme over $\mathrm{Spec} \, \mathbb{Z}$.
Let $X$ be an arithmetic scheme. Then is $H_{et}^2(X,\mathbb{Z}/n\mathbb{Z})$ finite ...
- 21.3k
16
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0
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878
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L-Functions of Varieties, Zeta Functions of Their Models
Let $k$ denote a number field, with algebraic closure $\bar{k}$. Take a smooth, projective variety $X$ over $k$. If $\mathfrak{p}$ is a prime of $k$, and $l$ is a rational prime different to the ...
- 643
16
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0
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2k
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Can one compare integral structures on de Rham and crystalline cohomology?
Suppose $\mathfrak{X}$ is a smooth projective scheme of finite type over $\mathbb{Z}_p$, with generic fibre $X$. Then there are comparison theorems relating de Rham and crystalline cohomology,
$H^i_{\...
- 37k
15
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0
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439
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Applications of the Weight Monodromy conjecture
I think of the Weight Monodromy conjecture as an analogue of the Weil conjectures in the case of bad reduction. The Weil conjectures of course have lots of applications, from point counting to ...
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15
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665
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Étale cohomology of varieties in positive characteristic via singular cohomology
Suppose $X$ is a smooth scheme, not necessarily projective, over $\mathbb{Z}[1/N]$ for some integer $N\neq 0$. I would like to understand the cohomology groups $H^i_{ét}(X_{\overline{\mathbb{F}}_p}, \...
- 666
15
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517
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Zariski vs etale torsors over abelian varieties
Question. Let $A$ be an abelian variety (say, over the complex numbers), $G$ an algebraic group, $c$ a class in $H^1_{\rm et}(A, G)$. Denote the multiplication by $N$ map on A by $m_N:A\to A$. Does ...
- 16.1k
12
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0
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811
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Number field analog of Artin-Tate $\Rightarrow$ BSD?
What is the difference between the alternating product of the Hasse-Weil $L$-functions of the generic fiber of an arithmetic scheme $X\to\text{Spec}(\mathbf{Z})$ and the zeta function of $X$? (each ...
user113452
11
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721
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Degeneration of etale Hochschild--Serre exact sequence
Let $k$ be a field, $X$ a smooth $K$-variety and $\ell$ a prime not dividing the characteristic of $K$.
Then one can make sense of continuous $\ell$-adic etale cohomology (in the sense of Jannsen), ...
- 37k
10
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0
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479
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How do I produce a basis of cohomology?
Suppose I am discussing a smooth projective variety over an algebraically closed field with my friend on the phone and I want to make a statement about its $l$-adic cohomology (integral or rational). ...
user158636
10
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474
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Etale homotopy types of schemes over positive characteristic fields
The étale homotopy type is a construction due to Artin and Mazur that generalizes the étale fundamental group. If $X$ is a scheme over a separably closed field $k$, then the étale homotopy type of $X$ ...
- 2,670
10
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532
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Comparison of etale and formal etale cohomologies for l=p
Let $K$ be a finite extension of $\mathbb{Q} _p$ with a field of integers $\mathcal{O} _K$. Let $X$ be a semistable proper scheme over $\mathcal{O} _K$, and $\mathcal{X}$ the associated p-adic formal ...
9
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691
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In Mann's six-functor formalism, do diagrams with the forget-supports map commute?
One of the main goals in formalizing six-functor formalisms is to obtain some sort of "coherence theorem", affirming that "every diagram that should commute, commutes". In these ...
- 711
9
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300
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How did Jouanolou define the cup product with no finiteness hypotheses in SGA 5?
In SGA 5 Exposé VII, at the beginning of §2, Jouanolou lets $X$ and $Y$ denote two schemes, $f:X\rightarrow Y$ a morphism, and $A$ the ring $\mathbf{Z}/\nu\mathbf{Z}$ where $\nu$ is an integer prime ...
- 1,217
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352
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Does etale homotopy type see the existence of rational points?
Do there exist two smooth projective schemes over $\mathbb{Q}$ that are etale homotopy equivalent and only one of them has a $\mathbb{Q}$-point?
user137767
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703
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étale vs syntomic vs flat cohomology
Let $\mathscr{A}/X$ be an abelian scheme over $X$ of characterisitic $p$. The étale topology is not fine enough for the Kummer sequence for $\mathscr{A}$ to be (right) exact, but the syntomic and flat ...
user19475
9
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206
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Eilenberg-Moore spectral sequence in etale cohomology?
Let $X,Y \rightarrow S$ be schemes over an algebraically closed field $k$. (Actually I'm interested in the case where they are stacks, but I'll ignore that for now.) The vague form of my question is: ...
- 2,809
9
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201
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Etale maps and local intersection cohomology
Suppose that $f:(X,x) \to (Y,y)$ is etale at $x$, meaning that it induces an isomorphism $C_xX \to C_yY$ on tangent cones. Then $f$ induces an isomorphism from the cohomology of $IC_{X,x}$ (the stalk ...
- 2,537
8
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333
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Triple comparison of cohomology in algebraic geometry
Let $X$ be a smooth proper variety over $\mathbb{Q}$ and $p$ a prime number. For an integer $k$, we have:
a finitely-generated abelian group $H^k(X^{\mathrm{an}}(\mathbb{C});\mathbb{Z})$
a finitely-...
- 15.4k
8
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432
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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 ...
- 1,949
8
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680
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Stalks of limit sheaves
Let $\{\mathcal{F}_i\}_{i\in \mathbb{N}}$ be an inverse system of sheaves of abelian groups on a space $X$. Then for any $x\in X$ we have a natural map
$$\left(\lim_i \mathcal{F}_i\right)_x\rightarrow ...
- 4,408
8
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354
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Serre's examples in rigid analytic geometry
Over $\mathbb{C}$, we have the following phenomenon: there exist algebraic varieties whose etale homotopy types are isomorphic but the homotopy types of their analytifications are not. Such examples ...
user138661
8
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574
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Reference request: Motivic Cohomology and Cycle class maps
For a smooth projective variety $X$ over any field $K$, Voevodsky showed in his paper ``Motivic Cohomology Groups Are Isomorphic to Higher Chow Groups in Any Characteristic" that the motivic ...
- 592
8
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0
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636
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Weil pairing and Tate module for $p$-torsion in characteristic $p$
Let $A/\mathbf{F}_{p^n}$ be an Abelian variety with $\bar{A} = A \times_{\mathbf{F}_{p^n}} \bar{\mathbf{F}}_{p^n}$.
If $\ell \neq p$ is prime, there is a natural isomorphism $$\mathrm{H}^2_{\mathrm{...
user19475
8
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0
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157
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defining Selmer groups using étale cohomology
Concerning http://swc.math.arizona.edu/aws/1999/99RubinES.pdf, especially section I.5:
Can one define the Selmer groups and the unramified cohomology groups as étale
cohomology groups of certain ...
user19475
8
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0
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603
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A Generalization of the Tate-Shafarevich/Tate/Fontaine-Mazur Conjectures
Let $A$ be an abelian variety over a number field $k$. The Tate-Shafarevich conjecture says that the Tate-Shafarevich group of $A$ is finite.
A weakening of this conjecture states that the $\ell$-...
- 15.4k
8
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0
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286
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Functorial classes in Brauer group
For a smooth variety $X$ over a perfect field of characteristics $p$ the sheaf of differential operators is an Azumaya algebra(etale locally is isomorphic to endomorphisms of its center, which is ...
- 7,377
8
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301
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Intuition for local Lefschetz theory in SGA2
In SGA2,Grothendieck introduced two important examples（EXP X 2.1） which satisfy Grothendieck-Lefschetz condition.My question is what's the intuition for the "local" version of Lefschetz's theory ?
I ...
- 388
8
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980
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What is an example of a non-mixed $\ell$-adic sheaf?
$\def\FF{\mathbb{F}}\def\cG{\mathcal{G}}\def\QQ{\mathbb{Q}}\def\CC{\mathbb{C}}$I've been attending a reading seminar at Michigan on Kiehl and Weissauer's book Weil conjectures, perverse sheaves and l’...
- 156k
8
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565
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Gysin exact sequence for a singular subvariety
Let $k$ be an algebraically closed field (I'm interested in a characteristic $p>0$ specific example) and let $X$ be a (smooth if needed) algebraic variety.
Let $Y \subset X$ be a (possibly) ...
- 2,521
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318
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The Rappoport-Zink spectral sequence vs. the one of the complement of a normal crossing divisor
As far as I understand these matters, for a regular $\mathfrak{X}$ that is proper flat of finite type over $\operatorname{Spec}\mathbb{Z}_p$, the Rappoport-Zink spectral sequence relates the etale ...
- 16.9k
8
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1k
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Is there a deep relationship between models and étale cohomology ? If so, why, and is it made precise somewhere ?
Let me recall two theorems :
Let $K$ be a field, $\overline{K}$ be
a separable closure of $K$ with
absolute Galois group
$G_K:=Gal(\overline{K}/K)$, and let
$\ell$ be a prime that is ...
- 373
7
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148
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Is the $\ell$-adic cohomology ring of a cubic threefold a complete invariant?
The only interesting $\ell$-adic cohomology of a smooth cubic threefold $X$ is $H^3(X,\mathbb{Z}_{\ell}(2))$, which is isomorphic as a $\mathrm{Gal}_k$-module to $H^1(JX,\mathbb{Z}_{\ell}(1))^{\vee}$ ...
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7
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1
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549
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Finiteness of the Brauer group for a one-dimensional scheme that is proper over $\mathrm{Spec}(\mathbb{Z})$
Let $X$ be a scheme with $\dim(X)=1$ that is also proper over $\mathrm{Spec}(\mathbb{Z})$. In Milne's Etale Cohomology, he states that the finiteness of the Brauer group $\mathrm{Br}(X)$ follows from ...
user122877
7
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Invariants of etale topological type that are not homotopy invariants
Artin--Mazur theory attaches etale homotopy type to reasonable schemes. Associated to this homotopy type are certain invariants of the scheme, such as etale fundamental group and higher homotopy ...
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155
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Do residues commute with transverse base change?
Fix a number $n > 0$. Given a smooth $\mathbb{Z}[1/n]$-scheme $X$ (i.e., a smooth scheme such that $n$ is invertible in its ring of functions), we may consider the étale sheaf $\mu_n$ on $X$ which ...
- 9,481
7
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0
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832
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Difference between algebraic and etale K-theory
Due to the Quillen-Lichtenbaum conjecture (now proven by Rost, Voevodsky, and Weibel), the map $K_\ast(X,\mathbb{Z}/n)\rightarrow K_\ast^{et}(X,\mathbb{Z}/n)$ from algebraic K-theory to etale K-theory ...
- 889
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483
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independence of $\ell$ for $p$-adic cohomology of varieties over finite fields
Let $X/k$ be a smooth projective geometrically integral variety ($X = A$ an Abelian variety suffices) over $k = \mathbf{F}_q$ with absolute Galois group $\Gamma$, $\bar{X} = X \times_k \bar{k}$, $q = ...
user19475
7
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An example computation of etale cohomology
(edited for clarity)
In a comment on a response to this question, moonface states the following: "Even if you tried to compute H^2 [etale with Z/5Z-coefficients] of a surface fibered in genus 2 ...
- 13.1k
6
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0
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469
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Étale cohomology of the field with one element
In the function field - number field analogy, some expect progress on RH to come from reproducing various aspects of the Grothendieck program in a way where $\mathbb{Z}$ could be treated as a function ...
user30211
6
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0
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243
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Computing Hodge numbers by point counting
In the lecture note of Bhatt from Arizona winter school 2017, there is an exercise which claims if X is a proper smooth scheme defined over $\mathbb{Z}[1/N]$ and if there is a polynomial $P$ such that ...
- 1,093
6
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265
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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 ...
- 2,663
6
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574
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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 ...
- 1,364
6
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152
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Does an fppf cohomological class annihilated by an etale cover come from etale cohomological group?
Let $X$ be a scheme, $F$ a sheaf on the fppf site of $X$, and $\alpha\in H^i_{\mathrm{fppf}}(X,F)$ such that it is trivialized by an etale cover of $X$. Does $\alpha$ lie in the image of the canonical ...
- 997
6
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0
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611
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Poincare duality in etale cohomology
I am reading etale cohomology and a question about Poincare duality comes to me. Suppose $k$ is a number field and $X/k$ is a smooth variety with dimension $n$ and $Y$ is a possibly singular variety ...
- 2,971
6
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0
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366
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Galois invariants in étale cohomology
Suppose $X$ is a smooth projective variety over a field $k$, with separable closure $\overline{k}$, Galois group $G$, and let $\overline{X}$ be $X_{\overline{k}}$.
Do we have
$$(H^j(\overline{X},\...
user120812
6
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0
answers
247
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Torsors for discrete groups in the etale topology
Let $S$ be a smooth variety over $\mathbb C$ or a smooth quasi-projective integral scheme over Spec $\mathbb{Z}$.
Let $G$ be an (abstract) discrete group. For instance, $G =\mathbb{Z}^n$ or $G$ a ...
- 61
6
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0
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465
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Torsionfree crystalline cohomology implies torsionfree etale cohomology?
Let $X$ be a smooth projective variety over an algebraically closed field $k$ of characteristic $p>0$. Let $W=W(k)$ be the ring of Witt vectors of $k$.
Assume that the crystalline cohomology $H^2_{...
- 91
6
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0
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489
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A problem on universally locally acyclic
Let $k$ be an algebraically closed field of characteristic $p>0$. Let $X$ and $S$ be two smooth varieties over $k$ and $\mathcal F$ a constructible \'etale sheaf of $\mathbb F_\ell$-modules on $X$ (...
- 135
6
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0
answers
909
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Étale cohomology with support and functoriality
Suppose we have a scheme $X$ and a closed subscheme $Z$, with complement $U$. Then, for any étale sheaf $F$ on $X$, we get a long exact sequence in cohomology
$\cdots H^i(X,F) \to H^i(U,F) \to H^{...
- 4,185
6
votes
0
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466
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Purity and six operations?
The six operations $f_!,f^!,f_*,f^*,\otimes,\mathcal Hom$ have the property that they preserve estimates on weights in one direction.
For $f_!,f^!,f_*,f^*$ I can see, that they don't preserve purity ...
- 13.2k