All Questions
1,114 questions
4
votes
1
answer
627
views
Can the projection (tensor algebra) -> (symmetric algebra) be forced to split in char. p by factoring out p-th powers?
Question 1 (the weak and simple statement, which, I think, already is wrong): Let $p$ be a prime. Let $k$ be a field with characteristic $p$.
For any $k$-vector space $V$, consider the canonical ...
CommunityBot
- 1
32
votes
10
answers
3k
views
Which 'well-known' algebraic geometric results do not hold in characteristic 2?
A smooth curve $X$ in $\mathbb{P}^n$ is strange if there is a point $p$ which lies on all the tangent lines of $X$.
Examples are $\mathbb{P}^1$ is strange and so is $y=x^2$ in characteristic $2$. ...
- 22.6k
2
votes
0
answers
352
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 ...
- 647
4
votes
1
answer
221
views
Do permutation modules of solvable groups have self-dual socle in characteristic 2?
I was searching through the small groups database in GAP to find counterexamples to a certain conjecture (which is not important here). I was checking non-nilpotent solvable groups and noticed that ...
- 44.4k
1
vote
2
answers
438
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\...
- 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 ...
CommunityBot
- 1
19
votes
6
answers
2k
views
weight 4 eigenforms with rational coefficients---is it reasonable to expect they all come from Calabi-Yaus?
A weight 2 modular form which happens to be a normalised cuspidal eigenform with rational coefficients has a natural geometric avatar---namely an elliptic curve over the rationals. It seems to be a ...
5
votes
0
answers
234
views
Modular reduction of exceptional complex reflection groups
I am interested in reducing reflection representations of complex reflection groups modulo a prime $p$. For the infinite family $G(m,r,n)$, it is straightforward to get "good reduction" provided that $...
- 10.7k
7
votes
1
answer
2k
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 ...
- 25.6k
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 ...
- 40.3k
13
votes
2
answers
1k
views
Is the Characteristic of a Field Detectable from the Topology of a Topological Vector Space?
Motivation
A topological vector space is a vector space over a (topological) field, K, that carries a topology such that addition and scalar multiplication are continuous maps, e.g., all normed vector ...
CommunityBot
- 1
3
votes
1
answer
464
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 ...
- 9,497
13
votes
1
answer
651
views
Help wanted with Chebotarev condition in characteristic 2
Having promised a longtime collaborator that I would clear my plate to finish up some joint work of ours, I am swallowing my pride and tossing up the following technical point of function field ...
- 50.7k
4
votes
2
answers
604
views
Adem-Wu relations from Bullett-Macdonald identities
Question. Let $p$ be a prime. Let $q$ be a power of $p$. Let $P^0$, $P^1$, $P^2$, ... be elements of some associative $\mathbb F_q$-algebra $A$. (Here, $P^i$ does not mean $P$ to the $i$-th power; ...
- 20.7k
2
votes
0
answers
214
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?
- 417
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
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 $\...
- 16.2k
1
vote
1
answer
633
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 ...
CommunityBot
- 1
4
votes
2
answers
906
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 ...
- 3,032
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 $X$. ...
- 35.2k
8
votes
1
answer
1k
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 ...
CommunityBot
- 1
6
votes
0
answers
1k
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 ...
- 371
9
votes
4
answers
1k
views
cohomology of moduli spaces
Does anyone know if there's any reference on the $\ell$-adic cohomology of some simple moduli spaces/Shimura varieties, like Siegel moduli varieties $A_{g,N}$ of genus $g$ and level $N,$ for small $g$ ...
- 4,265
0
votes
0
answers
352
views
Liftability in positive characteristic
What clsses of algebraic varieties over field of positive characteristic can be lift to $W_2(k)$?
- 85
7
votes
2
answers
513
views
Tameness for the Galois closure of a map of curves
Say we are working over some $K=\overline{K}$, of characteristic $p>0$. Let $\phi: Y\rightarrow X$ be a nonconstant map of smooth projective curves. To this map we can associate a map $\psi: Z\...
- 839
5
votes
1
answer
1k
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 = \...
- 431
0
votes
0
answers
524
views
DeRham cohomology
The Poincare lemma fails in positive characteristic, since pth powers vanish under differention. My question is : is there still some kind of resolution of the local system k by considering some ...
- 1
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?
- 20.5k
6
votes
1
answer
825
views
More on universal homeomorphisms
I would like to understand this notion better; where could I find some examples? In particular, I am interested in the following questions (and references for the answers).
Is a universal ...
- 27k
2
votes
0
answers
220
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 ...
- 16.9k
6
votes
1
answer
393
views
finite quotients of fundamental groups in positive characteristic
For affine smooth curves over $k=\bar{k}$ of char. $p,$ Abhyankar's conjecture (proved by Raynaud and Harbater) tells us exactly which finite groups can be realized as quotients of their fundamental ...
CommunityBot
- 1
5
votes
0
answers
735
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 ...
CommunityBot
- 1
4
votes
1
answer
774
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}...
- 1,673
11
votes
2
answers
863
views
Valuations and separable extensions
Let $R$ be a valuation ring containing a field $k$, with residue field $F$ and quotient field $K$. Assume $F/k$ is separable. Is $K/k$ separable?
I have convinced myself that (for a positive answer) ...
- 423
9
votes
2
answers
523
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 ...
- 35.5k
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 ...
- 1,576
4
votes
2
answers
402
views
lower bound for torsion of abelian varieties
Let $A$ be an abelian variety defined over a field $K$ of characteristic $p>0$. Let $A[\ell]$ be the group of $\ell$-torsion points, $\ell\neq p$ a prime. Are there positive constants $C, \eta$ ...
- 5,050
4
votes
2
answers
570
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 ...
- 1,576
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
- 91
1
vote
2
answers
393
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 ${\...
- 1,576
3
votes
1
answer
288
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(...
- 10.9k
18
votes
2
answers
4k
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 ...
CommunityBot
- 1
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 ...
- 176
0
votes
1
answer
175
views
An inseparable lift of a regular variety.
Let $X$ be a variety over an (imperfect) field $k$, that is regular as a scheme. Let $k'/k$ be an algebraic inseparable extension (I am interested in $k'$ being the perfection or the algebraic closure ...
- 19.6k
13
votes
2
answers
843
views
When can cohomology be calculated on the coarse moduli space?
Suppose $\cal{X}$ is a DM-stack, and X its coarse moduli space. Let F be a sheaf on $\cal{X}$, and $\pi : \mathcal{X} \to X$ the projection. In all examples I have seen, it has been true that
$H^i(\...
- 24.1k
31
votes
4
answers
5k
views
The Frobenius morphism
I found the following list on the "Frobenius Page" by David Ben-Zvi, described by the author as "an outdated collection of intuitive ways to think about raising to the p-th power".
Generates a ...
- 10.8k
6
votes
0
answers
305
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 ...
- 16.9k
2
votes
1
answer
646
views
Quotient by p-th roots of unity in characteristic p
Let $X$ be a variety over $k$ of characteristic $p>0$ (you can assume $k$ algebraically closed and $X$ normal) with an action of the group scheme of $p$-th roots of unity $\mu_p = {\rm Spec}\ k[\...
- 12.4k
16
votes
1
answer
2k
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. ...
- 27k
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 ...
- 66.3k