All Questions
1,114 questions
0
votes
0
answers
188
views
constructibility for pushforward
Let consider a quasicompact open $j:U\rightarrow\mathbb{A}^{\mathbb{N}}$ over a field $k$, Is there an example where $Rj_{*}\mathbb{Z}/n\mathbb{Z}$ is not constructible, where $n$ is prime to the ...
- 3,472
5
votes
1
answer
667
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 ...
- 16.9k
0
votes
2
answers
2k
views
non discrete valuation ring [closed]
Hi,
I am looking for examples of non-discrete valuation rings. Could you help me?
Thanks
- 141
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 $G$. ...
- 486
1
vote
0
answers
566
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. ...
- 33
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, ...
- 1,003
11
votes
1
answer
615
views
Do Richardson varieties have rational singularities in arbitrary characteristic?
The title basically asks the question. I'll review the relevant terminology and explain what I have and haven't found in the literature.
Let $G$ be a reductive group. Let $v \leq w$ be elements of ...
- 156k
4
votes
0
answers
209
views
Partial differential Equation over characteristic p
I want some references on partial differential equations over characteristic $p$. If we have a first order partial differential equation, how can we check whether there exists polynomials or rational ...
2
votes
0
answers
20
views
Cohomology of $\mathbf{G}_a\otimes_{\mathbf{Z}}\mathbf{G}_a$
Let $X$ be a smooth projective variety over a field.
Is $$H^p(X_{Zar},\mathbf{G}_a\otimes_{\mathbf{Z}}\mathbf{G}_a)$$
at all related to $H^p(X_{Zar},\mathbf{G}_a) = H^p(X,\mathcal{O}_X)$ via tensor ...
user92332
6
votes
2
answers
507
views
Concerning the dimension of a complex variety modulo a prime
Let V be a complex affine variety given as the vanishing set of a set of polynomials with integral coefficients. I have 3 questions.
1)
Under what assumption will the dimension of V over C remain ...
- 63
2
votes
1
answer
267
views
On Non F-pure ideal and Sharp F-Purity for a pair $(X, \Delta)$ where $K_X+\Delta$ is NOT $\mathbb{Q}$-Cartier
Suppose $(X,\Delta\ge 0)$ is a pair such that $(p^g-1)(K_X+\Delta)$ is an Integral Weil Divisor for some $g>0$ and $X$ is a normal variety. Define $\mathcal{L}_{e,\Delta} = \mathcal{O}_X( (1-p^g)(...
- 165
5
votes
1
answer
535
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, ...
- 16.9k
2
votes
0
answers
606
views
Kawamata-Viehweg Vanishing Theorem for Excellent Surfaces
Is there any version of Kawamata-Viehweg vanishing theorem that holds for excellent surfaces? I will be very glad if there is one such, but if not then is it at least true for a surface over a non ...
- 165
8
votes
0
answers
318
views
The Rappoport-Zink spectral sequence vs. the one of the complement of a normal crossing divisor
As far as I understand these matters, for a regular $\mathfrak{X}$ that is proper flat of finite type over $\operatorname{Spec}\mathbb{Z}_p$, the Rappoport-Zink spectral sequence relates the etale ...
- 16.9k
2
votes
0
answers
698
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 ...
- 317
4
votes
2
answers
694
views
Ample line bundle and Frobenius morphism on smooth toric variety
Let $k$ be an algebraically closed field of $\mathrm{char}(k)=p>0$, $X$ a smooth toric projective variety of $\dim X=n$, $F_X:X\rightarrow X$ the absolute Frobenius morphism of $X$. Then for any $\...
- 85
2
votes
1
answer
528
views
Is there an easy proof of the fact that the intermediate image functor respects weights?
It was proven in BBD (see Corollary 5.3.2) that for an open immersion $j$ the functor $j_{!*}$ preserves weights of mixed sheaves. The proof relies on several previous results; it is especially ...
- 16.9k
2
votes
1
answer
303
views
Lifting vector fields to its resolution in char $p$
In this paper 4.7, the authors showed that on a normal variety $X$, if there is a tangent vector field on its smooth locus, then it can be lifted as a logarithmic tangent vector field on a log ...
- 656
6
votes
0
answers
763
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 $...
- 5,019
5
votes
0
answers
578
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" ...
- 7,273
10
votes
2
answers
393
views
Counting points on varieties of low codimension
The graduate students here at MIT have been thinking about questions like the following: Over $\mathbb{F}\_q$, how many symmetric matrices are there with nonzero determinant and $0$'s on the diagonal? ...
- 156k
18
votes
1
answer
594
views
Does every hyperbolic curve over a finite field have an etale cover with a real Frobenius eigenvalue?
More precisely: let X/F_q be a smooth projective algebraic curve of genus at least 2. Does there always exist a curve Y/F_{q^d} with a finite etale projection Y -> X, such that one of the Frobenius ...
- 19.2k
8
votes
1
answer
1k
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 $...
- 16.9k
2
votes
1
answer
615
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 ...
- 945
5
votes
0
answers
725
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!
- 2,842
1
vote
0
answers
132
views
Picard sequence for sujective morphisms
Given $\phi:X\rightarrow Y$ a surjective morphism of $k$-algebraic varieties ($k$ separably closed), I wanted to find how the write an exact sequence involving Pic(X) and Pic(Y). We can use the long ...
- 101
13
votes
1
answer
1k
views
Analogue of Shimura curves in the symplectic case?
My understanding is this: one can attach 2-d Galois representations to classical modular eigenforms because one can look in the etale cohomology of modular curves. For Hilbert modular forms the naive ...
- 41.4k
6
votes
0
answers
466
views
Purity and six operations?
The six operations $f_!,f^!,f_*,f^*,\otimes,\mathcal Hom$ have the property that they preserve estimates on weights in one direction.
For $f_!,f^!,f_*,f^*$ I can see, that they don't preserve purity ...
- 13.2k
8
votes
1
answer
1k
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}(\...
- 13k
2
votes
0
answers
173
views
Hodge numbers of l-adic sheaves?
Assume first that $C$ is a curve, say over $\mathbb{Q}$ and $(E, \nabla)$ is a vector bundle with a flat connection. Assume further that $(E, \nabla)$ has regular singularities at $S=\overline{C}-C$. ...
- 71
7
votes
1
answer
799
views
Liftability of Enriques Surfaces (from char. p to zero)
Let $k$ be an algebraically closed field of characteristic $p > 0$, $X$ a variety over $k$.
We say $X$ lifts to characteristic zero, if there exists a local ring $R$ containing $\mathbb Z$ with ...
- 839
6
votes
0
answers
2k
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" 52_137_0">http://www.numdam....
user19475
4
votes
2
answers
339
views
Is the pre-image of a regular subscheme with respect to a universal homeomorphism of regular schemes regular?
Let $f:X\to Y$ be a universal homeomorphism of regular (excellent finite-dimensional) schemes, $Z\subset Y$ be a regular subscheme. Is $f^{-1}(Z)$ necessarily regular?
- 16.9k
2
votes
1
answer
588
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 ...
- 801
10
votes
2
answers
1k
views
Does a universal Frobenius map exist?
For any prime p, one has the Frobenius homomorphism Fp defined on rings of characteristic p.
Is there any kind of object, say U, with a "universal Frobenius map" F such that for any prime p and any ...
- 2,399
7
votes
1
answer
690
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 ...
- 16.9k
1
vote
1
answer
342
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 ...
- 16.9k
2
votes
1
answer
332
views
Ample bundle under Frobenius morphism
Let $k$ be an algebraically closed field of char($k$)=p>0, $X$ a smooth projective variety over $k$, $F:X\rightarrow X^{(1)}$ the relative Frobenius morphism. Let $E$ be an ample vector bundle on $X$. ...
- 85
2
votes
2
answers
2k
views
Reference of primitive root mod p
Can any body give me a reference of the result about primitive root mod p for a class of prime number p.
The result that I am looking for is something along this line:
$2$ is a primitive root mod $p$...
- 73
5
votes
0
answers
287
views
Nori fundamental group and etale fundamental group in positive characteristic
Let $X$ be a smooth projective surface over an algebraically closed field of char $p > 0$. Suppose that $\pi_{1}^{et}(X) = \{1\}$. Can Nori fundamental group scheme of $X$ be non-trivial?
- 163
3
votes
1
answer
805
views
Finite connected groups over a perfect field of characteristic p
In 14.4 of "Introduction to Affine Group Schemes" it is proved (!) that if $A$ represents a finite connected group scheme over a perfect field $k$ of characteristic $p$ then $A$ has the form $k[X_{1}, ...
- 163
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.) ...
user19475
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?
...
- 945
7
votes
0
answers
236
views
Invariant theory of $SL_2$ over a field of positive characteristic
Let $k$ be an algebraically closed field of characteristic $p>0$. Let $W$ be a finite dimensional $SL_2$-module over $k$. Let $V$ be the natural representation of $SL_2$.
What can be said - in ...
- 605
6
votes
2
answers
790
views
Do quotients of representable sheaves represent quotients?
Here's the context for the question: Proposition 4.6 of Freitag and Kiehl's book on etale cohomology shows that a sheaf (of sets) $\mathcal{F}$ (on the site Et(X)) is constructible if and only if it ...
- 1,988
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 ...
- 16.9k
3
votes
1
answer
249
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 ...
- 295
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 ...
- 1,160
3
votes
1
answer
914
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 ...
- 311
3
votes
2
answers
733
views
If the morphism of root data induced by an isogeny of a reductive group is a Frobenius, is then the isogeny itself a Frobenius?
Let $G$ be a reductive (or just semisimple) algebraic group over an algebraically closed field $k$ of characteristic $p > 0$, let $T$ be a maximal Torus and let $f:G \rightarrow G$ be an isogeny. ...
- 5,243