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5
votes
2answers
738 views

loaclly 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 ...
23
votes
2answers
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
0answers
214 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
2answers
285 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 ...
2
votes
1answer
302 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 ...
6
votes
1answer
691 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 ...
14
votes
1answer
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
1answer
588 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
1answer
260 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
0answers
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?
6
votes
1answer
578 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 $$ ...
19
votes
1answer
1k 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
2answers
466 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 ...
20
votes
3answers
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
0answers
484 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 ...
3
votes
1answer
724 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
1answer
596 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 ...
1
vote
0answers
140 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
0answers
504 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
1answer
449 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 ...
10
votes
1answer
607 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 ...
8
votes
2answers
420 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
2answers
454 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
2answers
283 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 ...
7
votes
1answer
866 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
1answer
244 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} ...
10
votes
2answers
1k 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 ...
4
votes
1answer
478 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
0answers
266 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 ...
11
votes
1answer
686 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 ...
10
votes
2answers
1k 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 $p,$ ...
14
votes
1answer
908 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. ...
17
votes
0answers
840 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
0answers
570 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, ...
4
votes
1answer
455 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
0answers
130 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
0answers
347 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
1answer
703 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
0answers
195 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
0answers
776 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
1answer
502 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
0answers
326 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 ...
4
votes
1answer
470 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 ...
18
votes
2answers
1k views

Comparing cohomology over ${\mathbb C}$ and over ${\mathbb F}_q$

I have the following (probably well-known) question: let $X$ be a regular scheme over $\mathbb Z$. Let $p$ be a prime and Let us denote the reduction of $X$ mod $p$ by $X_p$. Let also $X_{\mathbb ...
21
votes
2answers
1k views

When do two elliptic curves have equivalent small etale toposes?

Let $X$ and $Y$ be elliptic curves over an algebraically closed field $K$. If the characteristic of $K$ is nonzero, assume both curves are ordinary or both are supersingular. Does it follow that $X$ ...
3
votes
0answers
890 views

When is the base change morphism an isomorphism?

This is a rewrite of a previous question, which was in turn a follow up question to Leray-Hirsch principle for étale cohomology The motivation is to clarify some points in Torsten Ekedahl's ...
4
votes
1answer
381 views

Exact sequence in étale cohomology related with proper birational morphism

Let f:X→Y be proper birational morphism between two quasi-projective varieties over an algebraically closed field $k$. I am particularly interested in the case where the characteristic of $k$ is ...
4
votes
0answers
328 views

Étale cohomology of linear groups

This is in a sense a follow up question to the answer here Analytic tools in algebraic geometry Let $k$ be an algebraically closed field of positive characteristic and let $R$ be the result of ...
4
votes
2answers
486 views

Relationship between topological cohomology and $\ell$-adic cohomology

Let $\Delta \subset \mathbb{R}^n$ be an $n$-dimensional integral polytope, let $f$ be a Laurent polynomial in $n$-variables with coefficients in an extension of the integers and Newton polytope ...
6
votes
1answer
2k views

A nice explanation of what is a smooth (l-adic) sheaf?

I would like to understand this concept. It seems to be important (for the theory of perverse sheaves), yet I don't know any nice exposition of the properties of smooth sheaves.