All Questions
1,114 questions
2
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0
answers
146
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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 ...
- 16.9k
1
vote
0
answers
424
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 ...
- 16.9k
11
votes
0
answers
1k
views
Do the Standard Conjectures imply parts of the "Weil II" Riemann Hypothesis?
It is known that Grothendieck's Standard Conjectures on algebraic cycles imply the Riemann Hypothesis of the original Weil Conjectures. However, do they also say something about the version of the ...
- 1,764
5
votes
1
answer
2k
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 ...
- 27k
5
votes
0
answers
213
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 ...
- 16.9k
3
votes
0
answers
1k
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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 ...
- 15.6k
22
votes
2
answers
3k
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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 $X_{\mathbb C}$...
- 1,806
21
votes
2
answers
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 $X$ ...
- 43.2k
2
votes
0
answers
600
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 ...
- 21k
3
votes
0
answers
2k
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 ...
CommunityBot
- 1
2
votes
0
answers
321
views
Dimension of fibres of moment maps in characteristic $p$
Suppose $G$ is a connected semisimple linear algebraic group with Lie algebra $\mathfrak{g}$ and $X$ is a homogeneous $G$-space with isotropy subgroup $H$ (associated Lie algebra $\mathfrak{h}$) that ...
- 3,545
10
votes
1
answer
1k
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Leray-Hirsch principle for étale cohomology
Let $p:E\to B$ be a continuous map of topological spaces and set $F_x=p^{-1}(x)$ for an $x\in B$. Take a commutative ring $A$ and assume for simplicity that each $H^\*(F_x,A)$ is a free $A$-module. ...
- 22.6k
6
votes
1
answer
894
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 ...
- 22.6k
23
votes
1
answer
2k
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Wanted: Quadratic Space in Characteristic 2 as a Counterexample to a Theorem of Arf
Hi. Peter Roquette sent me an email asking for an example of a quadratic space in characteristic 2 having certain features. I have no idea on this, but maybe someone reading this does.
He would ...
- 50.7k
4
votes
0
answers
571
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 ...
CommunityBot
- 1
4
votes
2
answers
622
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 $\...
- 22.6k
17
votes
1
answer
7k
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.
- 35.2k
7
votes
3
answers
3k
views
congruent to 1 mod p
This is a somewhat vague question: for a prime number p, we often see that various counts come out to be 1 modulo p. What are the possible reasons for this?
Here are some I've encountered:
For some ...
- 39.9k
16
votes
3
answers
2k
views
On Category O in positive characteristic
Let $G$ be a semisimple algebraic group over an algebraically closed field $k$. In the case that $k$ has characteristic 0, there has been intensive study of the BGG category O of representations of ...
- 52.9k
9
votes
1
answer
777
views
Geometric (or intuitive) interpretation of additional derivatives in characteristic p > 0
In characteristic $p > 0$ there are "extra" differential operators, i.e., ones that are outside the algebra generated by first-order derivations.
Is there any interpretation of these operators in ...
- 12.4k
4
votes
2
answers
923
views
What is the correct formulation of the CDE triangle?
The CDE triangle with C Cartan matrix and D decomposition matrix is well-known for finite groups. I have not seen a full account of this for finite dimensional algebras which I find surprising as ...
- 44.7k
25
votes
1
answer
3k
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Are all Galois cohomology groups also étale cohomology groups?
Let $K$ be a field and $K^s$ a separable closure of $K$, and let $\mathcal{F}$ be a sheaf on $\mathrm{Spec}(K)$ (in the étale topology).
By Grothendieck's Galois Theory, we have the isomorphism
$$H_{...
- 5,540
2
votes
1
answer
278
views
What is the family derived from the absolute Frobenius on the Hilbert scheme?
Let $f$ be a Hilbert polynomial, and $X := Hilb_h(P^d_{F_p})$ a Hilbert scheme defined over $F_p$. Then there is an absolute Frobenius map $F: X \to X$. I'm even interested in the case $f \equiv 1$, ...
- 1,411
7
votes
0
answers
491
views
Alterations of regular varieties
Let $X$ be a regular quasi-projective variety over a perfect field $k$. The existence of a "good compactification" of $X$, i.e. a regular projective variety $\bar{X}$ with an embedding $X\...
- 4,450
2
votes
1
answer
515
views
Reference for comparison of complex and étale cohomology
Hello everybody!
I am looking for a nice DETAIL account of the comparison of étale cohomology and complex cohomology, an alternative reference instead of SGA. Especially on this stuff of "Artin ...
- 71
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
8
votes
0
answers
1k
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Is there a deep relationship between models and étale cohomology ? If so, why, and is it made precise somewhere ?
Let me recall two theorems :
Let $K$ be a field, $\overline{K}$ be
a separable closure of $K$ with
absolute Galois group
$G_K:=Gal(\overline{K}/K)$, and let
$\ell$ be a prime that is ...
- 373
4
votes
1
answer
830
views
2d Weil conjecture
Does there exist a two variable analogue of the Weil conjecture?
What I mean is that the usual Weil involves a one-variable zeta-function which you get by using numbers $V_n = V ( GF(p^n))$ of points ...
- 2,971
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 ...
- 13.6k
15
votes
2
answers
814
views
Can the failure of the multiplicativity of Euler factors at bad primes be corrected?
Warning: This one of those does-anyone-know-how-to-fix-this-vague-problem questions, and not an actual mathematics question at all.
If $X$ is a scheme of finite type over a finite field, then the ...
- 9,418
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 ...
- 22.6k
14
votes
1
answer
2k
views
Why does the naive definition of compactly supported étale cohomology give the wrong answer?
Illusie's article about étale cohomology available here (in French) mentions that the standard definition of compactly supported cohomology (and higher direct images with compact support) does not ...
- 8,296
48
votes
5
answers
15k
views
Algebraically closed fields of positive characteristic
I'm taking introductory algebraic geometry this term, so a lot of the theorems we see in class start with "Let k be an algebraically closed field." One of the things that's annoyed me is that as far ...
CommunityBot
- 1
12
votes
0
answers
716
views
Lifting abelian varieties in (the closed fiber of) a fixed Neron model
Suppose that $R$ is a dvr with field of fractions $K$ and residue field $k$ and that $A_K$ is an abelian variety over $K$ with Neron model $A$ over $R$. Then the closed fiber $A_k$ is a smooth ...
- 1,609
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. ...
- 7,108
5
votes
3
answers
739
views
Smoothness of hyperplane sections
Suppose $X\subset \mathbb{P}^n$ is a smooth hypersurface defined over $\mathbb{Q}$. For a "generic" prime $p$, what can be said about the set of hyperplanes $H$ in $\mathbb{P}^n(\mathbb{F}_p)$ for ...
- 16.9k
9
votes
2
answers
656
views
How does the order of a pole of a zeta function indicate any geometric information?
Here, I'm primarily concerced about zeta functions of hypersurfaces over fields of finite characteristic.
Assume $F_q$ to be a finite field with q elements. Consider the zeta function of the ...
- 65.4k
6
votes
0
answers
456
views
On periods of algebraic integers modulo rational primes
I run, somewhat indirectly, into the following problem and I have no hints where to look in the literature in search for answers or clues.
Let $K$ be a number field, which we may assume Galois if it ...
- 65.4k
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$...
- 65.4k
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 ...
- 65.4k
4
votes
1
answer
3k
views
etale fundamental group and etale cohomology of curves
Given a curve $C$. Is there any relation between the etale fundamental group $\pi_1(C)$ and the first etale cohomology of the constant sheaf , say $Z/nZ$, on $C$ ?
For example, if $C$ is a complex ...
- 4,487
7
votes
4
answers
736
views
Simply connected quasi-projective varieties in positive characteristic
I am looking for examples of non-projective (quasi-projective) varieties $X$ defined over a field of positive characteristic, which have trivial étale fundamental group.
It is well known that the ...
CommunityBot
- 1
16
votes
1
answer
1k
views
Motivation for the étale topology over other possibilities
In the search for a Weil cohomology theory $H$ over a field $K$ (with $\text{char}(K)=0$) for varieties in characteristic $p$, a classical argument by Serre shows that the coefficient field cannot be ...
- 5,540
9
votes
1
answer
763
views
Restriction theorems over finite fields
A short while ago, Dvir proved the Kakeya conjecture over finite fields. Does this have any implications for restriction theorems over finite fields? I am aware only of implications going in the ...
- 65.4k
25
votes
3
answers
2k
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product of all F_p, p prime
Let $R$ be the ring $$R = \prod_{p\ \text{prime}} \mathbb{F}_p$$ where $\mathbb{F}_p$ is the field having $p$ elements.
Is it true that $R$ has a quotient by a maximal ideal which is a field of ...
- 65.4k
10
votes
3
answers
2k
views
Is $Sym^n (V^*) \cong Sym^n (V)^\ast$ naturally in positive characteristic?
Background/motivation
It is a classical fact that we have a natural isomorphism $Sym^n (V^*) \cong Sym^n (V) ^\ast$ for vector spaces $V$ over a field $k$ of characteristic 0. One way to see this is ...
- 65.4k
8
votes
2
answers
8k
views
What does "supersingular" mean?
Are supersingular primes and supersingular elliptic curves related?
(this was essentially a subquestion in my earlier question, but still looks sufficiently different to me to deserve a separate post)...
CommunityBot
- 1
13
votes
0
answers
943
views
Beilinson-Bernstein localization in positive characteristic
This is a follow-up to this question; in particular, I'm wondering if anyone can expand upon the interesting answers given by Kevin McGerty and David Ben-Zvi there. (In particular, in this question I'...
CommunityBot
- 1
15
votes
1
answer
4k
views
Frobenius Descent
Let $S$ be a scheme of positive characteristic $p$ and $X$ a smooth $S$-scheme. Let $F:X\rightarrow X^{(p)}$ denote the relative Frobenius. A result by Cartier (often called Cartier descent or ...
- 57.6k
12
votes
2
answers
1k
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Weil Conjectures for nonprojective algebraic varieties
If we replace projective variety with algebraic variety in the statement of the Weil conjectures what happens? To me it seems the statement still makes sense. But is it still true?
- 65.4k