**8**

votes

**1**answer

367 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 ...

**6**

votes

**1**answer

559 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 $l$
$H^1_{et}(C_{\bar{K}},\...

**8**

votes

**1**answer

763 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

**0**answers

432 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. ...

**3**

votes

**2**answers

323 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?

**5**

votes

**2**answers

534 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

**0**answers

209 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" ...

**5**

votes

**0**answers

193 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 ...

**5**

votes

**1**answer

400 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 $H^*_{et}(\operatorname{...

**5**

votes

**1**answer

586 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 $\...

**2**

votes

**0**answers

132 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 ...

**1**

vote

**0**answers

283 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 ...

**5**

votes

**1**answer

1k 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**

votes

**1**answer

528 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 ...

**6**

votes

**0**answers

315 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**

votes

**0**answers

625 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**

votes

**1**answer

605 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**

votes

**0**answers

324 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!

**5**

votes

**1**answer

634 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 $K=\mathbb{Q}$ ...

**3**

votes

**1**answer

303 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

**0**answers

100 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

**0**answers

141 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

**1**answer

364 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 ...

**3**

votes

**2**answers

553 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}, \...

**7**

votes

**1**answer

797 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

**1**answer

1k 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

**1**answer

157 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 ...

**2**

votes

**1**answer

444 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

**4**answers

720 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 $p$) $\...

**3**

votes

**2**answers

232 views

### $Pic(X)/l=0$ in terms of $H^*_{et}(X,\mu_{l^n})$?

I would like to calculate Picard groups of certain schemes over fields; I'm mostly interested in the question whether $Pic(X)$ is infinitely $l$-divisible, i.e. whether $Pic(X)/l=0$, $l$ is a prime ...

**7**

votes

**0**answers

587 views

### On the derived category of constructible étale sheaves

The derived category $D^{\flat}_{c}(X,R)$ of constructible sheaves of $R$-modules on $X_{et}$ is defined as the full subcategory of $D^b(X,R)$ whose cohomology sheaves are all constructible.
Clearly, ...

**4**

votes

**1**answer

269 views

### family of gerbes over smooth and proper algebraic varieties

Let $X$ be a smooth and proper variety over $\mathbb{C}$. Let $F$ be an $\mathbb{A}^1$ family of $\mathbb{G}_m$ gerbes over $X$. Suppose the fibers over every point away from 0 in $\mathbb{A}^1$ are ...

**1**

vote

**0**answers

88 views

### How can one compute the cohomology of $i'^*C$, for $i':\mathbb{A}^{N-1}\setminus \{0\}\to \mathbb{A}^{N}\setminus \{0\} $?

For an (etale or 'topological', constructible bounded) complex of sheaves $C$ on $X'=\mathbb{A}^{N}\setminus \{0\} $, $i'$ being the embedding $\mathbb{A}^{N-1}\setminus \{0\}\to \mathbb{A}^{N}\...

**7**

votes

**2**answers

651 views

### Interpreting $f^*f_*$

For a morphism of schemes $f: X\rightarrow Y$, one often considers the function $f^*f_*$ on sheaves. For example, this appears in adjunction for sheaves of $\mathcal{O}_X$-modules, the projection ...

**8**

votes

**2**answers

388 views

### Can one ignore primes lying over $l$ in the Fontaine-Mazur conjecture? Counterexamples?

The Fontaine-Mazur conjecture predicts that an $l$-adic Galois representation of a number field is 'geometric' if it is unramified outside a finite set of primes and is De Rham for primes lying over $...

**4**

votes

**2**answers

652 views

### Weights for etale cohomology: why does Deligne's definition work?

For a field $K$ and a variety $X/K$ (whose characteristic could be $0$) I need a 'simple' explanation for the (Deligne's) method of defining weights of the $l$-adic etale cohomology of $\overline{X}$ (...

**0**

votes

**1**answer

214 views

### Sommese's theorem (generalized Weak Lefschetz) in arbitrary characteristic?

Sommese's theorem is a natural generalization of the Weak Lefschetz; for a smooth projective (connected) $X$, an ample vector bundle $E/X$ of rank $e$, and a section $s:X\to E$ it states that the ...

**17**

votes

**1**answer

1k views

### Any algebraic substitute for Morse theory (and homology) in arbitrary characteristic?

As far as I know, Morse theory yields much information on the topology of smooth manifolds; in particular, it can be used to prove Artin's vanishing (that the singular cohomology of smooth complex ...

**5**

votes

**1**answer

502 views

### The main idea in the proof of Artin's vanishing

Does anybody know an easy explanation of the proof of Artin's vanishing theorem (that the etale cohomology of an affine variety of dimension $n$ over an algebraically closed field vanishes in degrees $...

**15**

votes

**2**answers

745 views

### Motivic generalisation of Neron-Ogg-Shaferevich criterion

Given a variety $X$ over $\mathbb{Q}$ with good reduction at $p$, proper smooth base change tells us that its $l$-adic cohomology groups are unramified at $p$ (and I'd guess some $p$-adic Hodge theory ...

**2**

votes

**1**answer

305 views

### Is There a Mayer-Vietoris Spectral Sequence of Motivic Cohomology for Closed Coverings?

For etale cohomology, there is a spectral sequence of the following form ("Mayer-Vietories spectral sequence for closed covers"):
$E_{1}^{p,q}=\oplus_{i_{0}< \cdots < i_{p}} H_{ Y_{i_{0} \cdots ...

**6**

votes

**1**answer

419 views

### motivic t-structure and realisations

Let $k$ be a field and $DM_k$ denote the triangulated category of geometric motives with $
\mathbb{Q}$ coeffients over $k$. Recall that there exists a motive functor $M: Var_k\rightarrow DM_k$, which ...

**0**

votes

**1**answer

400 views

### Lefschetz fixed point formula: an 'easy' proof; cohomology with compact support

I have two questions concerning the LFPF (for etale cohomology).
Is there an easy 'explanation' of this statements (that could be understood by students)? In particular, I would like to get away ...

**5**

votes

**1**answer

445 views

### Is there a “universal” cohomology theory for varieties over p-adic fields?

Let $K$ be a $p$-adic field, $X$ a smooth proper algebraic variety over $K$, and $0 \le i \le 2 \dim X$. For a prime $\ell \ne p$ one can consider the $\ell$-adic cohomology $H^i(\overline{X}, \mathbb{...

**7**

votes

**1**answer

390 views

### What is the importance of the conjectural semi-simplicity of the action of the Frobenius on the etale cohomology of a variety over a finite field ?

It is conjectured that the action of the Frobenius acting on the etale cohomology of an algebraic variety over a finite field is semisimple.
A first approximation of my question is :
What is the ...

**19**

votes

**1**answer

1k views

### Example of non-projective variety with non-semisimple Frobenius action on etale cohomology?

This question was motivated by a more general question raised by Jan Weidner here. In general one starts with a variety $X$ (say smooth) over an algebraic closure of a finite field $\mathbb{F}_q$ of ...

**8**

votes

**1**answer

859 views

### Semisimplicity of Frobenius operation on etale cohomology?

Let $X_0$ be a variety defined over a finite field of characteristic $p \neq l$.
Is it true, that the action of the frobenius on the l-adic cohomology $H_l^*(X)$ is semisimple (say for smooth $X_0$)? ...

**7**

votes

**1**answer

423 views

### Is the $\ell$-adic cohomology of a non-proper variety unramified at good primes?

Let $X$ be a smooth variety of finite type over a number field $k$. Let $\overline{X} = X \times_{k} \overline{k}$, and let $\ell$ be a prime. It's well known that if $X$ is proper, then the é...

**2**

votes

**0**answers

144 views

### Does mapping cylinder category have enough injectives?

Let $A, B$ be two abelian categories, and $\tau : A \to B$ a left exact functor.
We define a category $C$ as follows:
objects: triples $(M, N, \varphi)$ where $M\in A, N\in B$ and $\varphi: N\to \...

**1**

vote

**1**answer

221 views

### Are these connecting homomorphisms commutative?

Are the connecting homomorphism induced by Kummer sequence and that of localization sequence commutative?
In other words, is the following statement true?
If it is true, then, how can one prove it?
...