Questions tagged [etale-cohomology]
for questions about etale cohomology of schemes, including foundational material and applications.
305
questions with no upvoted or accepted answers
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Does one need l to be invertible in S in order to consider the l-adic cohomology of S-schemes and motives?
When Ivorra defines the $l$-adic realization of $S$-motives (i.e. of Voevodsky's motives over a scheme $S$) he demands $l$ to be invertible in $S$. Is this condition really necessary? What happens ...
- 16.5k
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226
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Locally acyclic morphism which is not flat
Let $k$ be a closed field of characteristic $p \geqq 0$ and $\Lambda = \mathbf{Z}/\ell$, $\ell \neq p$. Recall that a morphism $f \colon X \to S$ of $k$-varieties is said to be locally acyclic if for ...
- 253
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329
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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 ...
5
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501
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Generalization of Weil Conjectures
is there a reference in English, besides Deligne's original publication: "La conjecture de Weil: II", not synthetic but complete that deals with the original argument of the generalization ...
- 51
5
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166
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Mod $l$ algebraic $K$-theory of product of an algebra with a complete algebra
By Gabber's rigidity the mod-$l$ $K$-theory of $k[[t]]$ and $k$ are isomorphic for a field $k$. Is there anything that we can say about the mod $l$ $K$-theory of $A\otimes_kk[[t]]$? Note that this is ...
- 5,851
5
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299
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Čech nerve $C(U)$ corresponds to $BG$ in same manner as a hypercover $\mathcal{H}(U)$ to
We can via the bar construction canonically associate to a monoid $A$ the nerve $N(B A)$, a simplicial set with $N(\mathbf{B}A)_k := \times^{k+1} A $ and canonical face maps and degeneracy maps ...
- 6,000
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402
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Étale cohomology and normalization?
I have an argument, which I wonder if someone could check:
Let $X$ be an irreducible reduced scheme over a field $k$. Then we have a normal scheme $X^{norm}$ with a finite birational $f:X^{norm}\...
- 4,246
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405
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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 ...
- 255
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231
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Cohomology groups on small fppf site and small etale site are not the same
Let $F$ be a quasi-coherent sheaf on a scheme $X$. Is there an example where cohomology groups of $F$ on small fppf site of $X$ and small etale site of $X$ are not isomorphic?
- 713
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610
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Comparison for cycle class maps for de Rham and etale cohomology via p-adic Hodge theory
Let $K$ be a p-adic local field, $X$ a smooth projective variety over $Spec(K)$, $CH^k(X)$ the Chow group of pure codimension $k$. Then there are cycle maps
$cl^X_{DR}:CH^k(X)\to H^{2k}_{DR}(X/K)$ ...
- 806
5
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381
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Geometric Frobenius on $\ell$-adic cohomology
Let $X$ be a smooth projective variety over a finite field $k$, and $F$ the geometric Frobenius on $H^*_{et}(X_{k^{sep}}, \mathbf{Z}_{\ell})$.
Is $F$ only rationally bijective, or integrally ...
user120812
5
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answers
1k
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Formal GAGA and étale cohomology
Let $\mathfrak{X}$ be a $p$-adic flat formal scheme over $\mathbf{Z}_p$, whose special fiber has an ample line bundle. Then $\mathfrak{X}$ is algebraizable, that is there exists an algebraic $\mathbf{...
user92332
5
votes
0
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480
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Poincaré duality for motivic cohomology
Is Poincaré duality for étale motivic cohomology known for projective regular (not necessarily smooth, just regular) schemes over Dedekind rings?
More precisely, two questions. Let $f: \mathcal{X}\to\...
user92332
5
votes
0
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281
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On rational points in an affine space
Let $X$ be a quasi-projective scheme over $\mathbb{F}_q$. If its base change to $\overline{\mathbb{F}}_q$ is an affine space $\mathbb{A}^n$, then it is known that $|X(\mathbb{F}_q)|=q^n$.
The only ...
- 1,534
5
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431
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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_{...
- 81
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404
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Is it possible to assume that an étale neighborhood is connected?
I am new to étale topology (though I've seen Grothendieck's sites before).
Let $S:=\mathcal{O}^\textrm{sh}_{X,x}$ be the strict local ring of a point $x$ of a scheme $X=\operatorname{Spec}R$ (over a ...
- 51
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273
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Tate's conjecture and symmetry of Hodge-Tate weights
I'm reading Bellaiche's notes on the Block-Kato conjecture (Hawaii summer school). Here is the link http://people.brandeis.edu/~jbellaic/BKHawaii5.pdf
At page 10 he claims that an indirect ...
- 231
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548
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What is the meaning of the cospecialization map?
This question comes from the same place as my other one. In reading SGA 4 1/2, but not SGA4 itself (at least, not the obvious sections xv + xvi), one can learn about the "cospecialization morphisms" ...
- 7,183
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1k
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Grothendieck monodromy theorem for l-adic sheaves
Hi,
Suppose that $F$ is a local field, $G_F$ its Galois group, $I$ the inertia subgroup, $k$ its residue field.
Let $X$ be a finite type scheme over $k$. Let $C$ be a constructible $l$-adic sheaf on $...
- 2,802
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688
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intersection cohomology and etale cohomology
Hello,
Can someone explain or give a reference on the comparison between intersection cohomology and l-adic etale cohomology of a variety over a field of characteristic zero?
Thanks!
- 2,802
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333
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Do 'change of coefficients' functors for sheaves commute with the four functors (formalism)?
For a morphism $f$ of varieties over a field of characteristic $\neq l$ I can consider the functors $Rf_*$, $f^\ast$, $f_!$, and $f^!$ both for the corresponding derived categories of 'all' (...
- 16.5k
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721
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Do all the main properties of constructible and perverse sheaves (in an 'arithmetic' situation) follow from results of Gabber?
This question is a continuation of Bad behaviour of perverse sheaves over 'general' bases?
Let $S$ (for example) be a finite type separated scheme over $\mathbb{Z}$. I would like: (1) to ...
- 16.5k
5
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211
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Which pure categories related with Weil cohomology theories are semi-simple?
As far as I understand, the category of pure polarizable Hodge modules is semi-simple, whereas the cohomology of the corresponding schemes is graded polarizable. Is it true that one doesn't have any ...
- 16.5k
4
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answers
191
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Is there a simple counterexample to étale proper base change on the unbounded derived category?
The best non-derived version of proper base change on the étale site of a scheme I know is that for $f : X \to Y$ proper and $g : Y' \to Y$ arbitrary, the base change morphism $g^{-1} R f_\star \...
- 545
4
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406
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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. ...
- 6,000
4
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388
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Henselization of normal rings (Milne's EC)
The usual way to define the Henselization $A^h$ of a local ring $(A, \mathfrak{m})$ is by taking the direct limit $\varinjlim (B, \mathfrak q)$ over all étale neighborhoods of $A$
(i.e. pairs $(B,\...
- 6,000
4
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325
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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 ...
- 15.1k
4
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319
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$\ell$-adic cohomology of a quotient by group action
Suppose $Y \to Y/G$ is the Galois cover induced from a finite group $G$ acting on a scheme $Y$ and that this is indeed a Galois cover with $Y/G$ a scheme. In my case $Y$ is the Drinfeld curve $\mathrm{...
- 469
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330
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Absolute purity for intersection cohomology
If $i:Z\hookrightarrow X$ is a closed embedding of codimension $c$, then
$$i^*k_X\ =\ k_Z , \ \ \ i^!k_X\ \stackrel{(\star)}{=}\ k_Z[2c]$$
where $(\star)$ is true when $i$ is in addition regular.
Here ...
- 5,505
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0
answers
180
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Sato-Tate over function fields
Suppose we have an elliptic surface $\pi: \mathscr E \to C$ over a curve over a finite field $\mathbb F_q$. We consider only the places on $C$ over which we have good reduction.
Over any point $x\in C$...
- 7,646
4
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264
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nearby cycles map for affine formal schemes
Assume that $X=Spf R$ is p-adic formal scheme over $O_{C_p}$ with generic fiber $X_{\eta}$. I want to know why the nearby cycles map $Ru^\star \mathbb{Z/p}$ is equal to $R\Gamma_{et}(spec R[1/p],\...
- 1,043
4
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0
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177
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Gysin map and $B\mathbf{G}_m$, confusion
Write $\text{Sh}(X)$ for the triangulated/stable $\infty$ category of $\ell$-adic sheaves on $X$, and $k\in\text{Sh}(X)$ fo the unit object. In playing around with $\text{Sh}(B\mathbf{G}_m)$ I've ...
- 5,505
4
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264
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Explicit linear object underlying $l$-adic cohomology for almost all $l$
If you are working with closed manifolds you can consider cohomology with any coefficients you like but ultimately everything is determined by the singular cohomology with $\mathbb{Z}$-coefficients.
...
user158636
4
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198
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Galois action of Weil restriction
Let $K/\mathbb{Q}$ be a quadratic field. Let $E$ be an elliptic curve defined over $K$ but not over $\mathbb{Q}$, and let $\bar{E}$ be the Galois conjugate of $E$. Then by the descent theory (for ...
- 451
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143
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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 ...
- 7,646
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116
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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 ...
- 16.5k
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303
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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 $i$, ...
user145520
4
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305
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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 ...
- 4,114
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381
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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)...
- 806
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297
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Etale cohomology of projective spaces in the rigid analytic setting
Take $K$ a complete non-archimedean field (maybe algebraically closed, to simplify the question), and $\mathbb{P}_K^d$ the rigid projective space over $K$. Can we compute the étale cohomology with ...
4
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150
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Traces of Frobenius Endomorphism on Etale Cohomology and $G$-torsors
I have a smooth, projective, and rigid Calabi-Yau threefold $X$ defined over $\mathbb{Q}$. Such spaces always have integral models. Let's assume we have an action on $X$ by a finite abelian group $G$...
- 1,701
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204
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What is the map induced by Frobenius on the cohomology group $H_{fppf}^1(X, \mathbb{G}_a)$?
Let $X$ be a projective curve of degree $p$ in $\mathbb{P}^2_k$, where $k$ is a field of charcteristic $p>0$. $X$ may not be reduced. I'm trying to compute $H_{fppf}^1(X, \alpha_p)$ using the ...
- 345
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257
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Motives up to homological equivalence
Let $X$ be a smooth projective variety over a field $k$ finitely generated over its prime field, and $M_{hom}(X)$ the category of motives modulo $\ell$-adic homological equivalence.
(1) Is $M_{hom}(...
user120812
4
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239
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Hard Lefschetz for cycles
Let $X$ be a smooth projective variety over a field $k$. It is known by work of Deligne, that the Lefschetz operator:
$$
L^k:H^{2n-2k}\left(X_{\overline{k}},\mathbf{Q}_{\ell}\right)\to H^{2n+2k}\left(...
user92332
4
votes
0
answers
422
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When is the etale cohomology of a surface free?
Let $X$ be a smooth variety over $\mathbb{F}_q$, $\ell$ a prime prime to $q$. I learned that if $X$ is a curve then $H_c^i(X,\mathbb{Z}_{\ell})$ is a free $\mathbb{Z}_{\ell}$-module.
When $X$ is of ...
- 1,534
4
votes
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421
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Reference request: sheaf-theoretic operations in the classical topology?
Like many graduate students before trying to learn something about étale cohomology and Deligne's proof(s) of the Riemann hypothesis part of the Weil conjectures, I am hunting for references detailing ...
- 273
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235
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Periodicity in Galois Cohomology
Let $k$ be a field which is not of characteristic $\ell$ and $\nu \geq 1$. Suppose for a moment that $\ell$ is odd. Even without $\ell$-th roots of $1$, then we have isomorphisms $H^*_{et}(k, \mu_{\...
- 2,277
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467
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Is the Gysin map in etale cohomology compatible with taking function fields?
Let $D\subset X$ be a smooth divisor on a smooth variety over $\mathbb{C}$.
Then we have Gysin maps in étale cohomology $H^2(X\backslash D,\mu_2)\rightarrow H^1(D,\mathbb{Z}/2)$ as well as $H^...
- 1,015
4
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0
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246
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Is there an analogue of the linking pairing in etale, crystalline, etc cohomology theories?
Let $M$ be a compact oriented manifold of dimension $n$. It is well-known that there is a perfect intersection pairing $$H_k(M;\mathbb{Z})_{torsion\,\,free}\otimes H_{n-k}(M;\mathbb{Z})_{torsion\,\,...
- 1,493
4
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0
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230
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Varieties with only top $\ell$-adic cohomology not vanish?
Let $X$ be a $d$-dimensional connected smooth variety over $\overline{\mathbb{F}}_q$.
It is well known that if $X$ is isomorphic to an affine space, then all the $\ell$-adic compactly supported ...
- 1,534