**1**

vote

**2**answers

558 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

220 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

534 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

304 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 $f ^ {0} \ \colon \ S^{0} \to T^{0}$ vanishes....

**3**

votes

**1**answer

636 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 $G$ ...

**3**

votes

**1**answer

795 views

### Cohomologie Etale

Is there an English translation available for Deligne's Cohomologie Etale (Arcata) that is now part of the SGA 4 1/2 ?? Atleast for the first two sections - Grothendieck Topologies and Etale Topology.

**8**

votes

**2**answers

993 views

### locally constant constructible sheaves and finite etale coverings

Maybe it is well known to experts or maybe it is just a stupid idea, but I will ask any way.
We know that if $X$ is a topological space, then there is an equivalence of categories between the ...

**24**

votes

**2**answers

1k views

### The category of l-adic sheaves

I'm currently trying to understand the construction of the category of l-adic constructible sheaves as in SGA5, and it seems that quite a lot of machinery (the MLAR condition, localization of the ...

**2**

votes

**0**answers

243 views

### Galois action on the derived category of $l$-adic sheaves

Hi,
Suppose that $G$ is a group acting on a scheme $X$, and $F$ is an $l$-adic sheaf on $X$ EDIT: with an action of $G$ (thanks Torsten).
Is it true that $R\Gamma(X,F)$ is well-defined as an object ...

**1**

vote

**2**answers

320 views

### global complete intersection and independence of $l$

Hello,
I remember reading that if $X/\mathbf F_q$ is a projective smooth global complete intersection, then the characteristic polynomial of the $\mathbf F_q$-linear Frobenius of $X$
on $H^i_{et}(X\...

**2**

votes

**1**answer

331 views

### Additive form of Hilbert 90 for schemes?

First, I am by no means well-versed on cohomology so I apologize if this is too elementary.
I have been going through some basics of etale cohomology, with my ultimate goal being an understanding of ...

**7**

votes

**1**answer

945 views

### Can proper-smooth base change be used to show that varieties cannot be lifted to characteristic zero?

Recall the following corollary to the proper and smooth base change theorems:
Let $\pi: X \to S$ be a proper, smooth morphism. Then the direct images $R^i \pi_* \mathcal{F}$ are locally constant ...

**15**

votes

**1**answer

1k views

### Serre and Tate's conjectures on étale cohomology

In the appendix of Serre and Tate "Good Reduction of Abelian Varieties" [Annals of Mathematics 88 (1968), 492-517], the authors make the following conjectures.
Suppose that $X$ is a smooth proper ...

**3**

votes

**1**answer

646 views

### Special case of Leray spectral sequence

I am looking for a reference for what is stated in Srinivasan's book "Representations of Finite Chevalley Groups", which is apparently a special case of Leray spectral sequence. I'll quote the ...

**2**

votes

**1**answer

278 views

### For an l-adic sheaf (F_n), why is the complex F_n of finite Tor dimension?

Let $X$ be a variety and let $\mathcal{F}=(\mathcal{F}_n)_{n\geq 0}$ be a (constructible) $\ell$-adic sheaf on $X$. Let $K_n$ be the object in the derived category $D(X,\mathbf{Z}/\ell^{n+1})$ of ...

**2**

votes

**0**answers

192 views

### structure of $T_\ell A$ for $A/\mathbf{F}_q$ an abelian variety

Can someone give me references for the structure of the $G_{\mathbf{F}_q}$-module $T_\ell A$, $A/\mathbf{F}_q$ an abelian variety?

**9**

votes

**1**answer

688 views

### Galois descent for K-groups (or for étale cohomology groups)

Let $F/K$ be a Galois extension of number fields with Galois group $G$. Let $\mathcal{O}_F$ and $\mathcal{O}_K$ be the associated rings of integers, and let $n\geq 1$.
When is
$$
K_{2n-1}(\...

**24**

votes

**1**answer

2k views

### How is etale cohomology of integer rings related to Galois cohomology?

In the paper of Bloch and Kato in the Grothendieck Festschrift, and some other papers relating to the Bloch-Kato conjecture and the ETNC, the cohomology groups
$H^i_{\mathrm{et}}(\operatorname{Spec} ...

**4**

votes

**2**answers

524 views

### Twisted forms and $\check{H}^1$

I am reading Milne's Étale cohomology, III.4.
A twisted form of an object $Y$ (a scheme, a sheaf of modules, of algebras...) over a scheme $X$ is an object $Y'$ such that there exists a covering in ...

**21**

votes

**3**answers

1k views

### how to find the varieties whose cohomology realizes certain representations?

The cohomology of Shimura varieties and Drinfeld shtukas is conjectured to realize the representations sought for in the Langlands programme/conjectures, the cohomology of Deligne-Lusztig varieties ...

**6**

votes

**0**answers

628 views

### Definition of etale neighbourhood

In Milne's EC, he defines an etale neighbourhood of a point $x \in X$ by a pair $(Y,y)$ with an etale morphism $f: Y \to X$ where $f(y) = x$, such that $k(x) = k(y)$.
What I don't understand is this ...

**4**

votes

**1**answer

812 views

### Galois cohomology groups given by étale cohomology

What are cases when Galois cohomology groups are given by étale cohomology?
Example: $S = Spec(K)$ the spectrum of a field, $F \in Sh(K)$, then $H^p(K, F) = H^p(G_K, F_{\bar{K}})$.
What if $G = \...

**1**

vote

**1**answer

602 views

### prove statement in Galois cohomology by étale cohomology

According to Milne's Arithmetic Duality Theorems, Proposition I.6.4: $0 \to H^1(G_S, A[m]) \to H^1(K, A[m]) \to \oplus_{v \not\in S}H^1(K_v, A)$ for an abelian variety $A$ and a nonempty set of primes ...

**2**

votes

**0**answers

149 views

### When inverse image is conservative; a reference or a generalization?

I am interested in the following question: for $f$ being a morphism of schemes, which conditions ensure that $Rf^*_{et}$ is conservative? This is true if $f$ has a section or if $f$ is an \'etale ...

**5**

votes

**0**answers

577 views

### Do all the main properties of constructible and perverse sheaves (in an 'arithmetic' situation) follow from results of Gabber?

This question is a continuation of Bad behaviour of perverse sheaves over 'general' bases?
Let $S$ (for example) be a finite type separated scheme over $\mathbb{Z}$. I would like: (1) to ...

**3**

votes

**1**answer

502 views

### Stalks of etale sheaves

Let $\pi:X \rightarrow Y$ be a finite morphism of schemes and $\mathfrak{F}$ be an etale sheaf on $X$. Then for a $y \in Y$ we have the stalk $(\pi_{*}\mathfrak{F})_{y}=\prod_{\pi(x)=y}\mathfrak{F}_{x}...

**10**

votes

**1**answer

694 views

### Bad behaviour of perverse sheaves over 'general' bases?

Could one define $\mathbb{Q}_l$-perverse etale sheaves over more or less general (excellent, separated) base scheme by combining the results of Gabber and Ekedahl? Would their functoriality properties ...

**9**

votes

**2**answers

440 views

### Concrete interpretations of higher (sheaf) cohomology groups

$H^1$ has an interpretation as torsors. But what about the higher $H^i$ (in the setting of algebraic geometry, and étale or flat cohomology)? For example $H^2(X, \mathbf{G}_m)$ is (often) isomorphic ...

**4**

votes

**2**answers

475 views

### If one wants to work with $Q_l$-adic sheaves, should the scheme be of finite type over a 1-dimensional one?

In section 6 of his 'Adic Formalism' T. Ekedahl states that $l$-adic sheaves 'behave nicely' for finite type separated schemes over $S$ that is regular of dimension $\le 1$. Is the dimension ...

**1**

vote

**2**answers

305 views

### Could the Kunneth decomposition of a motif depend on the choice of $l$?

Suppose that over some (algebraically closed) field $K$ of characteristic $p>0$ we have: numerical equivalence of cycles coincides with homological one with respect to ${\mathbb{Q}}_{l}$ and ${\...

**8**

votes

**1**answer

929 views

### If the numerical equivalence of cycles coincides with the homological one, does the Hodge standard conjecture follow?

Suppose that over an algebraically closed field $K$ of finite characteristic the numerical equivalence of cycles relation (for algebraic cycles of smooth projective varieties) coincides with the ...

**3**

votes

**1**answer

256 views

### Do $Q_l$-etale Euler characteristics of Chow motives coincide for all $l$?

I am interested in (Chow) motives over (algebraically closed) characteristic $p>0$ fields. For $H$ being $\mathbb{Q}_l$-adic cohomology, one can consider $Ch_l(M)=\sum (-1)^i\dim_{\mathbb{Q}_l} H^i(...

**12**

votes

**2**answers

2k views

### Locally constant sheaves for the étale topology, lack of intuition about “étale-localness”

I have started studying some étale cohomology and I am trying to build up some intuition about the concept of local for the étale topology. I can understand some nice examples (like Kummer exact ...

**5**

votes

**1**answer

524 views

### Questions on standard (motivic) conjectures

Over an (algebraically closed) characteristic $p$ field, is it known that the cohomological equivalence of cycles relation (with respect to $\mathbb{Q}_l$-adic \'etale cohomology) does not depend on ...

**6**

votes

**0**answers

280 views

### Does one need l to be invertible in S in order to consider the l-adic cohomology of S-schemes and motives?

When Ivorra defines the $l$-adic realization of $S$-motives (i.e. of Voevodsky's motives over a scheme $S$) he demands $l$ to be invertible in $S$. Is this condition really necessary? What happens ...

**14**

votes

**1**answer

828 views

### Motivic cohomology with finite coefficients for singular varieties

Let $X$ be a smooth variety over a field $K$ whose characteristic does not divide a positive integer $m$. Then the motivic cohomology of $X$ with coefficients in $\mathbb Z/m(j)$ can be computed in ...

**14**

votes

**2**answers

2k views

### comparison of de Rham cohomology and etale cohomology

I have a basic question concerning comparison of different cohomology theories. Let $X$ be a projective smooth (or just proper smooth) variety over a separably closed field $k$ of characteristic $p,$ ...

**15**

votes

**1**answer

1k views

### Universal homeomorphisms and the étale topology

Let $f:X\to S$ be a universal homeomorphism of schemes. Assume $X(S')\neq\emptyset$ for some étale surjective $S'\to S$. Does $f$ have a section?
The answer is yes if $S$ is reduced, by descent. ...

**21**

votes

**0**answers

1k views

### Is there a Grothendieck-Riemann-Roch type of theorem generalizing Grothendieck's Lefschetz trace formula

Grothendieck deduced that the L-function of a (constructible) $\ell$-adic sheaf on a variety over $\mathbf{F}_p$ is rational from the generalized trace formula.
My first question is based on the ...

**16**

votes

**0**answers

683 views

### Can one compare integral structures on de Rham and crystalline cohomology?

Suppose $\mathfrak{X}$ is a smooth projective scheme of finite type over $\mathbb{Z}_p$, with generic fibre $X$. Then there are comparison theorems relating de Rham and crystalline cohomology,
$H^i_{\...

**4**

votes

**1**answer

515 views

### A silly question: is the number of points on a Jacobian (of a curve, over a finite field) known?

In a cryptography book I read that people does not known how to compute the number of points on a Jacobian of a hyperelliptic curve $C$ over a finite field $F_q$? Is this true? It seems easy to ...

**2**

votes

**0**answers

133 views

### On 'graded polarizable' triangulated categories; are there any mixed Galois module analogues known? Also on mixed realizations

There is a construction by Beilinson (in section 3 of "Notes on absolute Hodge cohomology") of the derived category of graded polarizable mixed Hodge complexes; he also proved that this category is ...

**1**

vote

**0**answers

352 views

### Another stupid question on l-adic sheaves: does a generically zero constructible sheaf vanish on an open subvariety?

Suppose that the stalk of a constructible l-adic ($\mathbb{Z}_l$-adic or $\mathbb{Q}_l$-adic) etale sheaf $S$ over a generic (Zariski) point of a variety $V$ is zero. Does this imply that $S$ vanishes ...

**1**

vote

**1**answer

834 views

### Zariski sheaves lifted to etale topology

Let $X$ be a "reasonable" scheme (I am particularly interested in smooth algebraic varieties over a field). Let $Zar_X$ denote the (small) Zariski site of (open subschemes of) $X$ and $Et_X$ denote ...

**5**

votes

**0**answers

198 views

### Which pure categories related with Weil cohomology theories are semi-simple?

As far as I understand, the category of pure polarizable Hodge modules is semi-simple, whereas the cohomology of the corresponding schemes is graded polarizable. Is it true that one doesn't have any ...

**3**

votes

**0**answers

865 views

### Etale cohomology of regular local rings

Let $R$ be a regular local ring (I am particularly interested in the case when $R$ is the local ring of a point on a smooth scheme of finite type over a field). Let $G$ be the etale fundamental group ...

**3**

votes

**1**answer

591 views

### Behaviour of euler characteristics in characteristic p for finite etale covers

Let $k$ be an algebraic closure of a finite field of characteristic $p$. Fix an integer $l\neq p$. For a separated $k$-scheme $X$ of finite type, we define the (compactly supported) Euler ...

**2**

votes

**0**answers

362 views

### cohomological dimension of the push-forward functor

Let $\ell$ be a prime number and let $f:X \to Y$ be a morphism of schemes of finite type over the complex numbers (or a regular scheme of dimension at most 1, in which $\ell$ is invertible). How to ...

**5**

votes

**1**answer

539 views

### Characteristic Complexes in Iwasawa theory

For all that follows, $p$ is a fixed odd prime. In the formulation of the Noncommutative Main Conjecture of Iwasawa theory one uses étale cohomology to define an algebraic object analogous to Iwasawas ...