**4**

votes

**0**answers

382 views

### classify \mu_n torsors

Recently I read in Milne's book "etale cohomology" that the set $H^1(X,\mu_n)$ ($X$ a scheme, $n$ a nature number, the cohomology is flat cohomology) can be described as the set of pairs $(L,\phi)$, ...

**4**

votes

**1**answer

438 views

### is generically split Azumaya algebra locally split?

Let $A$ be an Azumaya algebra over a scheme $X$ (or maybe more specifically a scheme of finite type over a field). Suppose that the restriction of $A$ to $U=X\setminus Z$ (where $Z$ is a closed set) ...

**1**

vote

**2**answers

210 views

### Detecting zero morphisms via an open subscheme and its complement.

In the setting described in Bernstein, Beilinson, and Deligne, associated to a scheme $X$, a closed subscheme $i: Z \to X$ and its open complement $j: U \to X$ we have six functors between the ...

**2**

votes

**1**answer

423 views

### Poincare pairing and polarization of Hodge structure. Kuga-Satake construction.

If $X$ is a K3 surface over the complex (algebraic) then I wonder if the poincare pairing induces a polarization on the Hodge structure $H^2(X,\mathbb Z)$? The point is that I see that when ...

**4**

votes

**1**answer

673 views

### A Kunneth formula for the etale cohomology of the product of ('simple') varieties over not (necessarily) algebraically closed field

If $X$ and $Y$ are varieties over an algebraically closed field, then in the corresponding derived category of complexes we have $RH_{et}(X\times Y,\mathbb{Z}/l^n\mathbb{Z})\cong ...

**1**

vote

**1**answer

185 views

### How can one bound 'the lower perverse degree' for a constant sheaf on a variety that is smooth in high codimension?

Let $V$ be a variety (or a Whitney stratified space); $C$ is a constant etale ('topological') sheaf on it. Let $t$ denote the middle perverse t-structure for the corresponding derived category (of ...

**4**

votes

**1**answer

794 views

### The Gysin long exact sequence for the complement of the zero section of a line bundle over a (possibly) singular base

Let $pr:X\to Z$ be a line bundle of (possibly) singular varieties (that could be reducible) over a characteristic $p$ field ($p$ could be zero); $U$ is the complement to the zero section $Z\to X$. ...

**1**

vote

**0**answers

204 views

### What can one say about a smooth variety whose lower cohomology is trivial?

Let $X$ be a smooth (quasi-affine) complex variety; suppose that its cohomology (say, with integral coefficients) is trivial in degrees $0 < i\le s$ (for some $s>0$). What can one say about such ...

**7**

votes

**1**answer

868 views

### Is there an integral version of Faltings' isomorphism in p-adic Hodge theory between etale and Hodge cohomologies

Let $K$ be a $p$-adic field, that is a complete discrete valuation ring of characteristic $0$ with a perfect residue field $k$ of characteristic $p > 0$ (to simplify one could also take $K$ to be a ...

**19**

votes

**1**answer

708 views

### Status of conjectures in Serre's 1969 expose on Galois representations on l-adic cohomology

In
[S]: Serre, Jean-Pierre. Facteurs locaux des fonctions zeta des varietes algebriques (definitions et conjectures), Seminaire Delange-Pisot-Poitou, 1969-70
Serre presents nine conjectures ...

**6**

votes

**1**answer

403 views

### How to compute the étale cohomology of the quotient of a surface by a finite group of automorphisms ?

Let $S$ be a smooth surface defined over a finite field $K$ of char. $p$. Let $G$ be a finite group of automorphisms of $S$. Let $Z\to S/G$ be the minimal resolution of the quotient of $S$ by $G$. ...

**3**

votes

**2**answers

494 views

### Can we decide if an abelian variety is simple by knowing its Zeta function ?

Let $A$ be an Abelian variety defined over the finite field with $q$ elements. Let $P_i(T)$ be the characteristic polynomial of the action of the Frobenius on the $i^{th}$ étale cohomology group.
Is ...

**4**

votes

**1**answer

276 views

### Reference wanted - etale sheaves on $X$ versus on $\overline{X}$

Hello,
Let $X$ be a scheme of finite type over a field $k$. Let $l$ be an Galois extension of $k$ with Galois group $\Gamma$, and $\overline{X}$ be the base change of $X$ from $k$ to $l$. Then If I ...

**5**

votes

**0**answers

316 views

### 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' ...

**1**

vote

**1**answer

379 views

### vanishing of cohomology sheaves with supports and values in the multiplicative group

Let $X$ be a locally noetherian regular scheme and $Y$ be a closed subscheme of codimension $d > 0$ in every point. Why does it "immédiatement" (Grothendieck, Groupe de Brauer III, §6, p. 133 f.) ...

**3**

votes

**0**answers

201 views

### limit of étale cohomology with supports

I would like to know if the following generalisation of Milne Étale Cohomology, Lemma III.1.16, p. 88 holds: (all cohomology groups with respect to the étale topology)
Let $Y \hookrightarrow X$ ...

**1**

vote

**1**answer

252 views

### The cohomology of a $G_m$-bundle

Let $X$ be a smooth variety over an algebraically closed field (whose characteristic could be positive), $Y\to X$ is a $G_m$-bundle ($G_m=\mathbb{A}\setminus \{0\}$). Then I want to have a long exact ...

**5**

votes

**0**answers

293 views

### Do there exist weak Lefschetz-type statements that were proved for varieties over complex numbers, but were not proved in finite characteristic?

As far as I understood the situation (reading section 3.5B of Lazarsfeld's "Positivity in algebraic geometry" and also some of the references), some of weak Lefschetz-type statements known rely on ...

**1**

vote

**1**answer

189 views

### combination of two duality theorems

For a variety $X$ over a finite field $k$, one combines étale Poincaré duality for $X \times_k \bar{k}$ with duality for $k$ ($H^0(k,M) \times H^1(k,M^\vee) \to H^1(k,\mathbf{Z}/n) = \mathbf{Z}/n$) to ...

**6**

votes

**1**answer

703 views

### $\ell$-adic Weil cohomology theory

I have a reference or counterexample request. Suppose $k$ is a field and $\ell\neq char(k)$. There are several common references that show that $H^i_{et}(-, \mathbb{Q}_\ell )$ is a Weil cohomology ...

**2**

votes

**1**answer

296 views

### Equivalent forms of the proper base change isomorphism

$\DeclareMathOperator{\Nat}{Nat}$In a current project, I am trying to "commute" $!$ and $*$ functors that are both upper or both lower. (Sheaf-theoretic context: constructible étale sheaves.) ...

**16**

votes

**2**answers

1k views

### Etale cohomology with coefficients in the integers

Here is a basic question. When does $H^1_{et}(X,\mathbb{Z})$ vanish? Using the exact sequence of constant etale sheaves ...

**6**

votes

**0**answers

1k views

### project proposal: English translation of Deligne's “La conjecture de Weil : II” [closed]

First of all, I hope this "question" is appropriate here. If not, please delete it.
I would like to propose a translation project of Deligne's "La conjecture de Weil : II" ...

**3**

votes

**0**answers

269 views

### Does Artin's vanishing hold for '$E_2$-weight pieces' for (torsion) cohomology of affine varieties?

Let $X$ be an affine variety of dimension $n$ (say, over complex numbers). Then Artin's vanishing theorem yields: both singular and etale cohomology $H^i(X):=H^i(X,\mathbb{Z}/l\mathbb{Z})=0$ for ...

**6**

votes

**3**answers

596 views

### étale cohomology with values in an abelian scheme is torsion?

Let $A/X$ be an abelian scheme. Is $H^n(X,A)$ torsion for $n > 0$?
Perhaps this can be proved analogously as Proposition IV.2.7 of Milne's Étale cohomology (where it is proved that the ...

**1**

vote

**0**answers

341 views

### General cohomology groups and motives

Let $X$ be a variety over $\mathbb{Q}$. Let $\mathcal{F}$ be a sheaf on $X$. Then we have an action of $Gal(\mathbb{Q})$ on $H_{et}^i(X,\mathcal{F})$. In certain cases we can say a lot about this ...

**3**

votes

**1**answer

327 views

### vanishing of étale cohomology groups with small support with values in an abelian scheme

Let $S/k$ be a smooth variety and $A/S$ be an abelian scheme. Let $Z \hookrightarrow S$ be a reduced closed subscheme of codimension $\geq 2$.
I want to show that in this situation, $H^i_Z(S, A) = 0$ ...

**1**

vote

**1**answer

727 views

### Etale cohomology in the $p$-adic setting

Can we hope for application of Etale cohomology techniques in proving results concerning semialgebraic subsets of $\mathbb{Q}_p^n$?
Recall that semialgebraic subsets are obtained from $p$-adic ...

**7**

votes

**1**answer

394 views

### is the presheaf of automorphisms a sheaf?

In Chapter III,$\S 4$ of Milne's Etale cohomology a correspondence between twisted forms and Cech cohomology cocycles is described.
Fix some Grothendieck topology, say, etale, and let $A$ be a ...

**21**

votes

**5**answers

6k views

**1**

vote

**0**answers

153 views

### Galois cohomology of generic points of formal completions (of components of a hypercovering of the subvariety): examples or general statements?

Let $Y$ be a closed smooth subvariety in a (smooth) affine variety $X$. What can one say about the etale cohomology of the generic points of the formal completion of $X$ along $Y$ i.e. about the ...

**10**

votes

**3**answers

679 views

### Is the Gelfand-Graev character isomorphic to a cohomology group for some sheaf on a Deligne-Lusztig variety?

Deligne-Lusztig theory
is awesome. You take a maximal torus $T$, you take a character $\theta$, construct a variety $X_T$$^*$, take etale cohomology, get a virtual character $R_T^\theta$, maybe it's ...

**8**

votes

**1**answer

1k views

### Why does a group action on a scheme induce a group action on cohomology?

This is probably totally obvious but I have no clue how this is done: Say you have an endomorphism $f:X \rightarrow X$ of schemes. Why (if true, perhaps some additional assumptions are necessary!) do ...

**4**

votes

**1**answer

568 views

### The etale fundamental group and etale cohomology with compact support

Before me, the following was asked:
etale fundamental group and etale cohomology of curves
However, that question dealt only with projective curves.
Question
Let $X$ be any scheme (or if you prefer ...

**6**

votes

**1**answer

543 views

### Basic properties of Nisnevich cohomology; $l'$-topology?

I would like to know more about Nisnevich cohomology (especially, on its properties that could be easily formulated). In particular, I would like to know which of the following statements are true, ...

**1**

vote

**0**answers

112 views

### reading off invariants of a scheme $X$ from $D^b_c(X, \bar{\mathbf{Q}}_\ell)$

Which invariants of a scheme $X$ can be read off from $D^b_c(X, \bar{\mathbf{Q}}_\ell)$ (the bounded derived category of $\bar{\mathbf{Q}}_\ell$-sheaves on $X$, see e.g. [Kiehl-Weissauer])?

**2**

votes

**0**answers

241 views

### The restriction of the Gersten resolution (for etale cohomology) onto a closed subvariety.

There is a very important result of Bloch and Ogus: for any smooth variety $X$ and fixed $r\in \mathbb{Z}$, $r\ge 0$, $l$ is prime to the residue field characteristic, the Zariski sheafification of ...

**0**

votes

**0**answers

247 views

### Ordered Cech(-like) complexes that compute etale cohomology (of fields!)

It is well known (cf. Equivalence of ordered and unordered cech cohomology. ) that for 'usual' topologies one can compute the cohomology of sheaves either using unordered Cech complexes or ordered ...

**1**

vote

**0**answers

212 views

### On inverse images with respect to Zariski-etale topology.

For a variety $X$ I define its Zariski-etale site as follows: the category is the category of etale $X$-schemes, and the coverings are Zariski ones. Note that this topology is more coarse than the ...

**2**

votes

**0**answers

312 views

### What are the easiest cases of base change (for sheaves on sites)?

I have a closed embedding of schemes $i:X'\to X$, and for each of them I consider three Grothendieck topologies for the category of the corresponding (relatively) \'etale schemes: the \'etale one, the ...

**2**

votes

**0**answers

266 views

### Do inverse images respect flabby sheaves?

Let $i:Y\to X$ be a closed embedding of varieties, and let $S$ be a flabby \'etale (or Nisnevich) sheaf of abelian groups on $X$. Is $i^*S$ flabby also? I am mostly interested in the case when ...

**8**

votes

**0**answers

393 views

### Comparison of etale and formal etale cohomologies for l=p

Let $K$ be a finite extension of $\mathbb{Q} _p$ with a field of integers $\mathcal{O} _K$. Let $X$ be a semistable proper scheme over $\mathcal{O} _K$, and $\mathcal{X}$ the associated p-adic formal ...

**4**

votes

**1**answer

380 views

### Functoriality properties of the perverse $t$-structure for torsion (constructible complexes of) sheaves

I would like to apply the usual 'functoriality properties' of the perverse $t$-structure to torsion (constructible complexes of) sheaves (I am in the algebraic setting, so these are etale sheaves, ...

**0**

votes

**0**answers

182 views

### On 'special properties' of various 'sheaf image' functors for a local complete intersection morphism

Let $f:X\to Y$ be a local complete intersection morphism (of schemes or varieties) of (relative) dimension $c$ everywhere. Is it true that $f^!\cong f^*[2c]$ (as a functor between the derived ...

**1**

vote

**2**answers

549 views

### Finiteness of étale Cohomology Groups

Mr. Milne, in "Étale Cohomology", gives the following proposition (p.224, Corollary VI.2.8):
Proposition: Let $F$ a constructible sheaf on $X_{et}$, the small étale site of $X$, $X$ proper over a ...

**4**

votes

**0**answers

217 views

### On (the cohomology of) Hensel pairs

I would like to study the cohomology of the Henselization $H_X(Z)$ of a closed subvariety $Z$ of a variety $X$.
I would like the following facts to be true (and to make sense!:)).
a.) The motivic ...

**1**

vote

**0**answers

225 views

### Can etale $X$-schemes be lifted to $Y$, where $X$ is closed in $Y$?

For a closed embedding (of varieties) $X\to Y$ let $U/X$ be etale. Is is true that there necessarily exists an etale $U'/Y$ such that $U'_X=U$? If this is wrong in general, are there any assumptions ...

**2**

votes

**1**answer

530 views

### Does one need to sheafify when defining the inverse image of a sheaf with respect to an embedding?

This seems to be a rather simple (stupid?:)) question; yet I was not able to find an answer quickly.
For a morphism $f:X\to Y$ of schemes (or topological spaces) and an (etale or topological) sheaf ...

**1**

vote

**0**answers

302 views

### Does n-multiplication maps of cohomology groups vanish if it vanishes at the 0th cohomology?

In general, we know that a morphism $f=(f ^ {q})$ between universal (cohomological) $\delta$ functors $S=(S ^ {q}),T=(T ^ {q})\ $vanishes if and only if $f ^ {0} \ \colon \ S^{0} \to T^{0}$ ...

**3**

votes

**1**answer

618 views

### Principal bundles in the etale and Zariski topology and extensions of the structure group

Say $G$ is a reductive group over a field $k$. I usually take $k = \mathbb{C}$ so assume what you want about the field except maybe that its finite. If $X$ is a scheme over $k$ then a principal $G$ ...