**8**

votes

**1**answer

487 views

### Étale cohomology versus classical cohomology

Let $X$ be an algebraic variety over $\mathbb{C}$. If $X$ is smooth, the étale cohomology $H^p_{\textrm{ét}}(X,\mathbb{Z}/n)$ is isomorphic to the singular cohomology $H^p(X(\mathbb{C}),\mathbb{Z}/n)$...

**19**

votes

**1**answer

950 views

### When is “independence of l” known?

My question is for which varieties over local fields is "independence of l" known for
etale cohomology. Say $X/{\mathbb Q}_p$ is a complete non-singular variety and $W_l$ is the (complex) Weil-...

**0**

votes

**1**answer

139 views

### cohomology of an intermediate extension of a local system

Let $V$ be affine $n$-space over a field $k$; and $j \colon U \to V$ an open subscheme of $V$. Let $L$ be an $\ell$-adic local system on $U$.
Suppose the cohomology of $H^{\bullet}(U,L)$ does not ...

**4**

votes

**0**answers

202 views

### $H^2(S, f_* \mathbb{G}_m)$ in the fppf versus etale topology for proper $f$

Let $f\colon X \rightarrow S$ be a proper morphism of schemes. Is the cohomology group $H^2(S, f_* \mathbb{G}_m)$ the same regardless of whether it is computed in the etale or the fppf topology? And ...

**5**

votes

**1**answer

340 views

### “Weight-monodromy” for open varieties

Suppose that $X/\mathbb{Q}_p$ is a smooth, projective variety, and choose a prime $\ell\neq p$. Then the weight-monodromy conjecture says that the graded pieces $\mathrm{Gr}_k^M$ of the monodromy ...

**6**

votes

**1**answer

441 views

### Is the Weil–Deligne representations coming from $\ell$-adic cohomology independent of $\ell$?

Let $F$ be a $p$-adic field. Let $(G_{F}, W_{F}, I_{F})$ denote the (absolute Galois group, Weil group, inertia group) of $F$.
Let $X/F$ be a proper smooth variety. Let $\ell$ be a prime number $\ne p$...

**12**

votes

**2**answers

669 views

### Are there known cases of the Mumford–Tate conjecture that do not use Abelian varieties?

(For a formulation of the Mumford–Tate conjecture, see below.)
The question
As far as I know, all non-trivial known cases of the Mumford–Tate conjecture more or less depend on the Mumford–Tate ...

**2**

votes

**0**answers

115 views

### Hodge numbers of l-adic sheaves?

Assume first that $C$ is a curve, say over $\mathbb{Q}$ and $(E, \nabla)$ is a vector bundle with a flat connection. Assume further that $(E, \nabla)$ has regular singularities at $S=\overline{C}-C$. ...

**4**

votes

**0**answers

183 views

### 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 ...

**5**

votes

**1**answer

256 views

### Analogy between connections and $\ell$-adic sheaves: what happens with the residue?

There are many analogies between $\ell$-adic sheaves on varieties over finite fields and vector bundles with connections on varieties over fields of characteristic zero. I would like to know what is ...

**0**

votes

**0**answers

151 views

### Descent datum for a line bundle

Let $\pi:C \to \mathbb P^1$ be a double cover branched at $r$ points. To understand the theory of descent better, I would like, if possible, to construct by hands the descent datum of a line bundle ...

**2**

votes

**0**answers

234 views

### l-adic cohomology and perverse sheaves

Let consider the map $tr:\mathbb{G}_{m}^{n}\rightarrow\mathbb{A}^{1}_{\mathbb{F}_{q}}$ given by the sum of the coordinates and let $\psi:\mathbb{F}_{q}\rightarrow\mathbb{Q}_{l}^{*}$ a non trivial ...

**2**

votes

**1**answer

156 views

### Compute higher direct image for Gm under open embedding

Let $U \subset \mathbb P^1$ be an open subset of projective line (over $\mathbb C$) after removing $r$ points and $j: U\hookrightarrow \mathbb P^1$ an open immersion. How do I compute $R^1j_*\mathbb ...

**6**

votes

**1**answer

401 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 $...

**7**

votes

**2**answers

413 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

**1**answer

511 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

**0**answers

213 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

**0**answers

269 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

**1**answer

216 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

**2**answers

238 views

### an explicit weak equivalence between $B{\mathbb G}_m$ and ${\mathbb P}^\infty$

OK, so I asked a similar question before; $B{\mathbb G}_m$ is a simplicial presheaf over number field $k$. I see that there is some $A^1$-homotopy equivalence between the sheaf represented by ind-...

**4**

votes

**1**answer

244 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 ...

**7**

votes

**1**answer

449 views

### DG enhancements of $\ell$-adic derived categories

This question is similar in flavor to Existence of dg realization for 6 functors
Let $X$ be a complex variety and $D(X)$ the bounded derived category of constructible sheaves (the Euclidean topology ...

**7**

votes

**1**answer

438 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

**0**answers

169 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

**1**answer

331 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

**0**answers

235 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), ...

**6**

votes

**1**answer

366 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

**0**answers

355 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 H^{...

**19**

votes

**1**answer

1k 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

**0**answers

180 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 \...

**4**

votes

**0**answers

236 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 ...

**1**

vote

**1**answer

195 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 ...

**2**

votes

**0**answers

307 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

**1**answer

315 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

**1**answer

320 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 @| \\
H^1(...

**3**

votes

**1**answer

112 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

**1**answer

284 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} = H^{p}(\...

**6**

votes

**0**answers

226 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

**1**answer

214 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

**0**answers

232 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

**1**answer

171 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**

vote

**1**answer

310 views

### 'etale topology [duplicate]

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**

votes

**0**answers

156 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
$$ H^i_{...

**10**

votes

**0**answers

323 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**

votes

**0**answers

94 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**

votes

**1**answer

528 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

**0**answers

99 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**

votes

**0**answers

83 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 $D^b_c(X_n,...

**1**

vote

**0**answers

94 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**

votes

**1**answer

436 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 ...