6
votes
1answer
266 views

$\ell$-adic monodromy theorems (over $\mathbb{C}$)

This question is about $\ell$-adic monodromy theorems for families over a number field. ($\ell$-adic analogues of Corollaries 6.2.8 and 6.2.9 in [BBD].) Notation $H$ denotes étale cohomology. Let ...
6
votes
2answers
319 views

Twist in identification with singular cohomology

Let $X$ be a smooth projective variety over $\mathbb{Q}$ and $$V = H^m(X(\mathbb{C}), \mathbb{Q} \cdot (2\pi i)^r)$$ Then I've seen people write the comparison with complex cohomology (an isomorphism ...
4
votes
1answer
230 views

Comparison of etale and singular cohomology for varieties over number fields

Whilst reading Hartshorne's appendix C I came across the comparison theorem for etale cohomology and singular cohomology: Let $X$ be a smooth projective variety over a number field $K$ and $\ell$ a ...
1
vote
0answers
80 views

Question about the specialization map for Etale Fundamental Groups

Let $A$ be a complete, discrete valuation ring, and let $s$ (resp $\eta$) be the special (resp. generic) point of $S=Spec(A)$. Let $\phi:X \rightarrow S$ be a proper morphism and fix geometric base ...
2
votes
0answers
194 views

vanishing of étale cohomology of affine surface

Let $U$ be an affine smooth surface over an algebraic closure of a finite field. Let $\mathscr{A}/U$ be an Abelian scheme and $\ell \neq \mathrm{char}(k)$ be prime. Are there vanishing results for ...
4
votes
1answer
185 views

Unravelling some hypotheses on a variety

In Le group de Brauer II, Grothendieck states Proposition 1.4.- Soit $X$ a préschéma noetherien. Supposon que les anneaux hensélisés stricts des anneaux locaux de $X$ soient factoriels, [...] ...
4
votes
1answer
209 views

etale cohomology of ${\mathbb P}^n_k$

Suppose $k$ is a number field. I want to compute $H^\ast({\mathbb P}^n_k,\mu_l^{\otimes r})$ where $l,r\in {\mathbb N}$. I know that Milne has some computations, but he assumes throughout that his ...
5
votes
1answer
252 views

Cohomology of a constant etale sheaf

Let $X$ be a smooth proper algebraic variety over $\mathbb{C}$. I know that in the analytic world, there is an isomorphism between the de Rham cohomology and the cohomology of the constant sheaf ...
3
votes
0answers
133 views

Reference request: construction of Chern classes

I am looking for a reference on splitting principle for etale cohomology of simplicial schemes (over arbitrary field k). I found a paper by Schechtman, "On the delooping of chern character and Adams ...
8
votes
1answer
208 views

Example of a variety over a number field with non-semisimple Galois representation on $\ell$-adic cohomology

This question is inspired by the question: Example of non-projective variety with non-semisimple Frobenius action on etale cohomology? Let $K$ be a number field (or finitely generated field of ...
10
votes
0answers
155 views

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), ...
5
votes
1answer
256 views

infinite grassmannian in algebraic geometry

Geometric realization of $B{\mathbb G}_{\mathfrak m}({\mathbb C})$ is ${\mathbb C}{\mathbb P}^\infty=\varinjlim_n~ {\mathbb C}{\mathbb P}^n_k$; what if one considers a separable field $k\neq ...
6
votes
0answers
256 views

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 ...
15
votes
1answer
691 views

Which algebraic surfaces have non-trivial H^1?

Informally, my question is the following: Is there an "inverse theorem" for the first cohomology group $H^1$ of (the projective completion of) an algebraic surface $S$? Namely, can we give a ...
2
votes
0answers
141 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 ...
2
votes
0answers
296 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
votes
1answer
255 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
votes
1answer
259 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
votes
1answer
96 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
votes
1answer
164 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
votes
0answers
183 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 ...
7
votes
1answer
194 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
votes
0answers
144 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 ...
10
votes
0answers
224 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 ...
5
votes
1answer
407 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 ...
1
vote
0answers
87 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 ...
1
vote
0answers
75 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 ...
8
votes
1answer
244 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
votes
0answers
222 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 $l$ ...
8
votes
1answer
513 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 ...
1
vote
0answers
308 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. ...
4
votes
2answers
348 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
145 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
votes
0answers
94 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
votes
1answer
324 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
votes
1answer
318 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 ...
1
vote
0answers
191 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
votes
1answer
709 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 ...
6
votes
0answers
240 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 ...
9
votes
1answer
449 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 ...
4
votes
1answer
504 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
votes
0answers
183 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 $H^1_{fppf}(k,G)$?
1
vote
0answers
90 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
votes
0answers
112 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
votes
1answer
312 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 ...
7
votes
1answer
539 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 ...
16
votes
1answer
873 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
votes
1answer
140 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 ...
1
vote
1answer
371 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 ...
4
votes
4answers
625 views

Hodge numbers of reduction mod $p$

Let $X$ be a projective variety defined over a number field $K$, and $p \in \textrm{Spec }\mathcal{O}_K$ a maximal ideal, so that reduction mod $p$ makes sense, and the resulting scheme (mod $p$) ...