All Questions
1,114 questions
7
votes
1
answer
1k
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, ...
- 1,553
23
votes
2
answers
4k
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 $0\rightarrow\mathbb{Z}\rightarrow\mathbb{Q}\rightarrow\mathbb{Q}/\mathbb{Z}\...
- 1,553
16
votes
1
answer
1k
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 ...
- 1,553
8
votes
1
answer
1k
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 ...
- 9,229
3
votes
2
answers
1k
views
roots of analytic functions
Let $z$ be a complex variable and $f(z)$ be a formal power series with rational coefficients (an element in $\mathbb Q[[z]]$), with a finite radius of convergence, and assume $f(z)$ has a meromorphic ...
- 4,265
14
votes
1
answer
1k
views
Frobenius splitting of Fano varieties
Dear MO,
Question 1. Do you know of an example of a Fano variety which is not Frobenius split?
Background
(1) A variety $X$ in characteristic $p$ is called Frobenius split if there is a "$p$-th ...
CommunityBot
- 1
0
votes
1
answer
526
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 with ...
- 23k
7
votes
1
answer
700
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 ...
- 5,050
19
votes
1
answer
1k
views
Is there a connected $k$-group scheme $G$ such that $G_{red}$ is not a subgroup?
I've been trying a learn a little more about group schemes by working through a set of exercises on Brian Conrad's website. Exercise 8.3 of http://math.stanford.edu/~conrad/papers/gpschemehw1.pdf ...
- 6,543
10
votes
6
answers
2k
views
Proofs in the same vein as Ax-Grothendieck
I would like to see other examples of (ideas of) proofs and results in the same vein as the proof of the Ax-Grothendieck theorem. To explain what I mean by "in the same vein", I will quote from the ...
- 38.6k
7
votes
0
answers
207
views
Unicritical rational functions on curves in characteristic $p$
Let $k$ be an algebraically closed field of positive characteristic $p$, and let $X_{/k}$ be a smooth projective connected curve. Let $x_0$ be a point of $X(k)$.
How precisely can one describe the ...
- 1,199
9
votes
1
answer
983
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...
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2
votes
0
answers
175
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 \...
- 945
4
votes
1
answer
918
views
Heisenberg group in characteristic two
I have seen two constructions called the Heisenberg group. If $k$ is a field of characteristic not equal to $2$ and $V$ is a $2n$-dimensional vector space over $k$ with symplectic form $\omega : V \...
- 45.7k
1
vote
1
answer
885
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 ...
- 12.9k
4
votes
2
answers
1k
views
Is the absolute Galois group of the field of Laurent series in positive characteristic finitely generated?
If $K$ is an algebraically closed field of characteristic $p>0$, then $K((t))$, the field of Laurent series with coefficients in $K$, has infinitely many Galois extensions of degree $p$. Indeed, ...
- 18.7k
1
vote
1
answer
245
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?
...
- 23k
1
vote
0
answers
238
views
why is the local field Q_l not an etale sheaf over a scheme X?
I would like to know the reason why the local field Q_l is not an etale sheaf over a scheme X while its ring of integers Z_l can be regarded as a constant etale sheaf?
- 11
1
vote
0
answers
238
views
Classification of fibres in pencils of curves of genus two
For the case of characteristic positive, there exist some clasification of families of curves of genus two over a elliptic curve?.
- 11
7
votes
1
answer
2k
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 ...
- 5,670
4
votes
1
answer
648
views
Lemmas on etale cohomology with compact support from the book 'Arithmetic Duality Theorems'
I was reading Milne's book "Arithmetic Duality Theorems". On page 166 there are a lot of useful lemmas on the etale cohomology with compact support on S-integers. However, I get confused when I tried ...
- 3,867
2
votes
0
answers
279
views
deRham cohomoloy of CM liftings of Jacobians
Let $k$ be field of characteristic $p>0$ and $W=W(k)$ the ring of Witt vectors of $k$. We call a smooth curve over $k$, ordinary, when the Jacobian of $J(X)$ of $X$ is an ordinary abelian variety. ...
- 637
6
votes
0
answers
423
views
Formality of algebraic varieties via l-adic cohomology?
The cohomology with say real coefficients of any smooth projective algebraic variety is formal, by
Deligne, Griffiths, Morgan, Sullivan: Real homotopy theory of Kähler manifolds, Inv. Math. 29, 245-...
- 13.2k
4
votes
1
answer
734
views
etale homotopy and Adams conjecture
I was reading Quillen's paper on "Some remarks on etale homotopy theory and a conjecture of Adams". Up to some fact about etale version of spherical fibrations which was later proved by Friedlander, ...
- 9,283
3
votes
0
answers
742
views
Explicit description of O^{cris}_n in Fontaine/Messing
Let $k$ be a perfect field of characteristic $p$, $W(k)$ the Witt ring and $K$ its quotient field. In their article "$p$-adic periods and $p$-adic etale cohomology" Fontaine and Messing give in II.1.4 ...
2
votes
1
answer
2k
views
Long exact sequence of cohomology and fibration
In an article I'm reading, the author is stating :
$O$ is isomorphic to the complement of a zero section of a line bundle over $X$. We have a long exact sequence of (étale) cohomology associated to ...
- 40.3k
26
votes
2
answers
2k
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 ...
- 1,576
2
votes
1
answer
510
views
hyperalgebras (positive characteristic)
The question is about commutator in integral forms. Let $A$ an associative algebra over a field of characteristic zero, $x\in A$ and $k\in \mathbb Z$, we denote $x^{(k)}=\frac{x^{k}}{k!}$.
How to ...
- 829
4
votes
0
answers
791
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 $(L,\phi)$, ...
- 271
12
votes
3
answers
1k
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 ...
- 686
4
votes
1
answer
715
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) ...
- 4,195
1
vote
2
answers
233
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 ...
- 16.2k
2
votes
1
answer
669
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 ...
8
votes
1
answer
982
views
Is there a really big ring of differential operators in characteristic p?
$k$ is a field of characteristic $p$.
$k[t]$ has canonical first-order differential operator $\partial$
As an endomorphism of $k[t]$, $\partial^p=0$.
First way to fix it:
Use the divided power ...
CommunityBot
- 1
1
vote
1
answer
244
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 ...
- 686
5
votes
1
answer
2k
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 $Z\to X$. ...
CommunityBot
- 1
5
votes
1
answer
2k
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 RH_{et}(X,\mathbb{Z}/...
CommunityBot
- 1
5
votes
0
answers
336
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' (...
- 16.9k
1
vote
0
answers
232
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 ...
- 16.9k
7
votes
1
answer
742
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 $G$. ...
- 35.2k
6
votes
2
answers
802
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 ...
- 156k
4
votes
1
answer
309
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 ...
CommunityBot
- 1
14
votes
3
answers
2k
views
Representations in characteristic p
Let G be a finite group and let F be an algebraically closed field. If the characteristic of F is 0, then the number of irreducible F-representations of G is given by the number of conjugacy classes ...
- 6,357
3
votes
0
answers
230
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 $Y \hookrightarrow X$ quasi-...
CommunityBot
- 1
3
votes
1
answer
440
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, $H^i_Z(S, A) = 0$ ...
CommunityBot
- 1
1
vote
1
answer
596
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.) ...
- 3,032
1
vote
1
answer
472
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 ...
- 27k
5
votes
2
answers
556
views
Existence of certain identities involving characteristic 2 "thetas"
Let l=2m+1 be prime. In my previous MO question, "What are the polynomial relations between these characteristic 2 thetas?", I defined a subring of Z/2[[x]] as follows:
The subring, S, is generated ...
- 5,422
6
votes
0
answers
436
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 '...
- 156k
5
votes
1
answer
1k
views
Minimal Model Program for surfaces over algebraically closed fields of characteristic p
Let $k$ be an algebraically closed field of characteristic $p>0$.
I have been trying to find out unsuccessfully if there is a mmp for algebraic surfaces over $k$. I know minimal surfaces are ...
- 656