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113 views

integral hard Lefschetz

I am looking for examples $(X,\eta)$ where the integral hard Lefschetz is an isomorphism: $X/k$ is a smooth projective variety of dimension $d$ over an algebraic closure of a finite field and $\eta ...
3
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0answers
145 views

Cohomology of BG, algebraically

Let $k$ be a field (algebraically closed if you will) and $G$ be a connected reductive group over $k$. I would like to know a purely algebraic computation of the cohomology of $BG$, as the quotient ...
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1answer
136 views

Etale cohomology and restricted direct product

[migrated from math.stackexchange: http://math.stackexchange.com/questions/727896/etale-cohomology-and-restricted-direct-product] $\newcommand{\h}{\operatorname{H}}$ Let $k$ be a global field, $A$ an ...
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0answers
283 views

Why is this frobenius acting like that frobenius?

My question came up while reading this article by Nicholas Katz, specifically lemma 4.2. I don't think it's necessary to read the article to answer the question, but I'm including it anyways for ...
6
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1answer
228 views

Continuity of l-adic cohomology: is the cohomology of the generic point isomorphic to the completion of the limit of cohomology of open subvarieties?

Let $X$ be a variety over an algebraically closed field $k$. Denote by $\eta$ its generic point; it is the inverse limit of the open subvarieties $X_i$ of $X$. It is well known that the etale ...
2
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1answer
212 views

commutative diagram with Yoneda pairing, Weil pairing and edge morphism

Why does the following diagram commute?$\require{AMScd}$ \begin{CD} H^0(X,\mathscr{A}) \times \mathrm{Ext}^2_X(\mathscr{A},\mu_{\ell^n}) @>>> H^2(X,\mu_{\ell^n}) \\ @VVV @| \\ ...
3
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1answer
90 views

Relative flasqueness?

It is known that a flasque sheaf on a topological space has trivial cohomology. Suppose that we are in a relative situation of a smooth fibration $\pi: X \to S$ and $F$ is a sheaf on $X$. Is there ...
2
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1answer
100 views

When does the filtration in the limit of the Leray spectral sequence split?

Let $\ell$ be a prime, and $k$ a field of characteristic $\ne \ell$. Let $f \colon X \to Y$ be a proper map of smooth projective $k$-varieties. The Leray spectral sequence says $$ E_{2}^{pq} = ...
6
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0answers
166 views

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 ...
6
votes
1answer
188 views

'Cohomologically approximating' a $\mathbb{Q}[[t]]$-scheme by a one over the henselization of $\mathbb{Q}[t]$?

For certain matters the henselization $R$ of $\mathbb{Q}[t]$ at $0$ is a 'reasonable approximation' for $\mathbb{Q}[[t]]$ (Artin's approximation theorem and so on). Now, I would like to study certain ...
4
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0answers
134 views

Tame fundamental group

Let $X$ be a normal and flat scheme over $Spec(\mathbb{Z})$. We know a good way to compute the etale fundamental group of $X$. Can we say sth similar for the tame fundamental group. If so, what would ...
4
votes
1answer
108 views

Comparison of cycle maps

Let $X$ be an algebraic variety over $\bar{\mathbb{Q}}$ of dimension $d$, then there is the l-adic cycle map $\mathrm{cl}_{et}:\mathrm{CH}^i(X)\rightarrow\mathrm{H}^{2i}(X,\mathbb{Q}_\ell(i))$ from ...
1
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0answers
187 views

'etale topology

Could you recommend me please some basic, self-contained books on 'etale topology. I read Yoshida's article "local class field theory via lubin-tate theory" and some people said that it is somehow ...
3
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0answers
131 views

Regulator maps for ordinary varieties

Let $K$ be a finite extension of $\mathbf{Q}_p$ and $\mathcal{X}$ a smooth proper scheme over the ring of integers $\mathcal{O}_K$. For $i, j$ integers with $i \ne 2j$, there's a regulator map $$ ...
9
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0answers
209 views

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 ...
2
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0answers
79 views

integral stable conjugacy classes

Let $G$ be a semisimple simply connected group over $k$ algebraically closed field . Let $\gamma,\gamma'\in G(k[[\pi]])$ that are generically regular semisimple on $G(k((\pi)))$. We assume that ...
6
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1answer
381 views

Brauer groups of punctured affine lines over a base

Let $R$ be a torsion-free regular noetherian ring. The Brauer group $Br(R)$ of $R$, defined equivalently (by a theorem of Gabber) as the group of Morita equivalence classes of Azumaya $R$-algebras or ...
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0answers
86 views

topological invariance of direct image in the \'etale topology

Let $R$ be a complete local ring (even of dimension one if it helps) and write it as limit of artinian rings $R_n$. Let $X\rightarrow S=Spec(R)$ be proper, finite type even smooth outside the maximal ...
2
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0answers
72 views

Continuity of constructible derived category

Let $X_0$ be a variety over $\mathbb F_q$. Denote by $X_n$ its basechange to $\mathbb F_{q^n}$ and let $X=\lim X_n$ be its basechange to the algebraic closure $\overline{\mathbb F}_q$. Let ...
1
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0answers
73 views

Quasicoherent analogue of a theorem on fiberwise acyclicity for etale cohomology

I am interested in knowing what (if any) is the quasicoherent analogue of the following result that I have paraphrased from SGA 4, exposé xv, Théorème 1.15: Let $g \colon X ...
5
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1answer
364 views

Relation between Galois theory and Etale Cohomology

I am a graduate student working on category theory. I am familiar with categorical Galois theory, in the way developed by Janelidze - as described for example in "Galois Theories". I am now trying to ...
6
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0answers
156 views

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 ...
7
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1answer
206 views

Which sheaves satisfy cohomological purity?

The absolute cohomological purity theorem in étale cohomology is as follows. Let $X$ be a regular scheme over $\mathbb{Z}[1/n]$, and $i \colon Z \to X$ the inclusion of a regular closed ...
4
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0answers
183 views

Is first etale cohomology of a variety always (dual to) a Tate Module?

The two examples I have in mind are curves and abelian varieties. To be precise, if $C$ is a smooth projective algebraic curve over a number field $K$, then, for a prime $l$ ...
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1answer
479 views

Etale cohomology of the completion of a Henselian local ring

Let $\pi: R\to S$ be a local morphism of Henselian local rings. Let $f: R \to \hat{R}$ and $g: S \to \hat{S}$ be their completions. Let $\mathcal F$ be a constructible $l$-adic sheaf on $\operatorname ...
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0answers
274 views

Homotopy theory of schemes

I have seen the notion of Homotopy come up in several contexts in schemes. For example, the book "Lectures on Motivic Cohomology" by Mazza, Weibel and Voevodsky uses this language to some extent. I.e. ...
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2answers
244 views

Galois cohomology of the field of Laurent series

Let $k$ a separably closed field. Do we have that $k((t))$ is of cohomological dimension one?
4
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2answers
328 views

etale cohomology of an abelian variety and its dual

Let $A$ an abelian variety over a field $k$ and $A^{*}$ the dual abelian variety. How can we relate the étale cohomology of $A$ with etale cohomology of $A^{*}$?
4
votes
0answers
136 views

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" ...
4
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0answers
78 views

Fiberwise acyclic, locally acyclic morphisms

The quick definition of a map $f \colon X \to B$ of schemes being acyclic is that the natural unit of adjunction $\def\id{\operatorname{id}}\id \to f_* f^*$ is an isomorphism, where we take $f_*$ to ...
3
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1answer
310 views

Etale Cohomology of Punctured Spectra of Local Rings

Let $R=\mathbb{C}[[x,y]]$ be a power series ring in two variables (or maybe more generally a strictly Henselian local ring) with maximal ideal $\mathfrak{m}$. What is ...
2
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1answer
257 views

Frobenius weights on etale cohomology and purity

Let $X_0$ be a smooth variety (for simplicity I'm willing to assume that X is a curve) over a finite field $k$, $X$ its geometric base change, and $\mathcal{F}$ an $l$-adic etale sheaf on $X$ with ...
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0answers
83 views

The etale cohomology``ring" structure of torsion sheaves on varieties

For a topological manifold $M$, one can speak of the cohomology ring structure $H^*(M, k)$ where $k$ is a ring. If one replace $M$ by an arithmetic schemes $X$ over a base ring $S$, and replace $k$ by ...
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0answers
172 views

Pushforward on 1-dimensional etale cohomology

Background: For a smooth proper variety $X$ over an algebraically closed field $k$, we have the etale cohomology groups $H^i(X,\mathbb{Q}_{\ell})$ for $\ell \not= p$. We can use the Kummer exact ...
4
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1answer
657 views

what is Deligne's cohomological descent (and what are some examples)

As far as I understand Deligne's far reaching generalisation of Čech cohomology is called cohomological descent and is used to endow any variety with a (mixed) Hodge structure. Again, AFAIU, the idea ...
2
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1answer
256 views

Pullbacks of intermediate/middle extensions and Gabber's purity theorem

I am currently trying to understand intermediate extensions of perverse sheaves, specifically the proof of Gabber's purity theorem, which states that the intermediate extension of a pure perverse ...
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0answers
222 views

Is there excision for fppf cohomology?

I am wondering whether the analogue of III.1.27 in Milne's "Etale cohomology" holds true if one works with fppf cohomology with supports instead of etale cohomology with supports. More precisely, let ...
5
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0answers
397 views

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 ...
9
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1answer
424 views

Etale homology via étale cosheaves

Can one develop a theory of étale homology via étale cosheaves? The hope is that this would, for example, return the Tate module (and not its dual) for an elliptic curve, and it would return group ...
3
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0answers
224 views

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!
4
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1answer
468 views

Should the etale cohomology of a smooth projective variety (over rationals) be semi-simple; why?

$\DeclareMathOperator{\char}{char}\DeclareMathOperator{\gal}{Gal}$ Let $P$ be a smooth projective variety over a field $K$ (one may certainly assume that $K$ is perfect; the case $K=\mathbb{Q}$ ...
2
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0answers
179 views

Flat cohomology for finite infinitesimal group scheme over a perfect field

Let $G$ be a finite infinitesimal group scheme (e.g.$\mu_p,\alpha_p) $ over a perfect field $k$, how much is known about $H^1_{fppf}(k,G)$?
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0answers
87 views

A certain 'coniveau-like' filtration for cohomology: what can one say about the intersection of $Ker H^i(X)\to H^i(Z)$ for $Z$ running through subvarieties of $X$ of dimension $m$?

Let $X$ be a smooth variety (say, a complex one; denote its dimension by $n$). What can one say about the intersection of $Ker (H^i(X)\to H^i(Z))$ for $Z$ running through (closed, not necessarily ...
2
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0answers
106 views

Some 'weak proper and smooth base change' theorems for Nisnevich sheaves?

Among the most important tools for studying etale cohomology are the proper and smooth base change theorems. I suspect that these theorems are no longer true for Nisnevich cohomology (probably finite ...
0
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1answer
296 views

The fibres of smooth projective families over all geometric points have isomorphic cohomology; are these isomorphisms 'functorial'?

Let $p:P\to S$ (and $p':P\to S$) be proper smooth morphisms of 'nice' schemes (one may assume that $S$ is a complex variety). It is well-known that the fibres of $p$ (and $p'$) over all geometric ...
1
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1answer
282 views

Hodge-Tate weights of etale cohomology

Let $K/\mathbb Q_p$ be a local field, $X/K$ a proper scheme with semi-stable reduction. Question: What is the possible range of Hodge-Tate weights of the etale cohomology $H^i(X_{\overline K}, ...
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1answer
509 views

Points in sites (etale, fppf, … )

I asked a part of this in an earlier question, but that part of my question didn't receive precedence. Etale site is useful - examples of using the small fppf site? Let $X$ be a scheme (assume it ...
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1answer
820 views

Etale site is useful - examples of using the small fppf site?

Edit: After the answers and comments, I'm hoping for a little bit of elaboration (in the comment to the answer below.) Also, question 2 was discussed here: Points in sites (etale, fppf, ... ) There, ...
3
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1answer
137 views

Does the Hilbert polynomial determine the weight of the Galois representation associated to a variety

Let $X$ be a curve or an abelian variety (over a finite field). Then the Galois representation associated to $X$ via the etale cohomology of $X$ (in degree $1$) is integral of weight $1$ and its ...
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1answer
350 views

Fundamental Group and Etale Cohomology

I encountered the following statement without a reference many times. For a smooth variety $X$ over a perfect field $k$. $Hom(H^1_{et}(X, \mathbb{Z}/n), \mathbb{Z}/n) \cong \pi^{ab}_1(X)/n$ Is there ...