All Questions
1,114 questions
7
votes
2
answers
827
views
$p$-torsion of an abelian variety of $p$-rank $0$
Let $k$ be an algebraically closed field of characteristic $p > 0$ and let $A$ be an abelian variety over $k$ such that $A[p](k) = 0$, i.e., such that $A$ has $p$-rank $0$. If I am not mistaken, ...
- 2,663
7
votes
2
answers
506
views
Algebraic proof without using comparison theorem for étale cohomology
Let $X$ be some smooth scheme over $\mathbf C$ equipped with an action of $\mu_n$ (the group of $n$th roots of unity).
The étale cohomology groups of X are therefore equipped with an action of $\mu_n$...
- 2,521
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 ...
- 5,019
7
votes
2
answers
1k
views
Hochschild Kostant Rosenberg theorem for varieties in positive characteristic?
Is there is a known version of the HKR theorem as proved in say Swan's paper "Hochschild Cohomology of Quasiprojective Varieties" in positive characteristic? I assume something is known about this as ...
- 3,758
7
votes
1
answer
999
views
Generalized Behrend version for Grothendieck-Lefschetz trace formula
[MOVED HERE FROM MSE.]
The statement of the Grothendieck-Lefschetz fixed point theorem is well-known. For a proper algebraic variety $X$ over $\mathbb F_q$,
$$\#X(\mathbb F_q) =\sum_i (−1)^i Tr(Fr_X, ...
- 455
7
votes
1
answer
532
views
Vector space objects in schemes - confusion
Let $R$ be the ring $\mathbf{C}\times\mathbf{C}$, and consider the affine line $\mathbf{A}^1_R$.
$\mathbf{A}^1_R$ can be given the structure of additive group scheme over $R$, denoted $(\mathbf{G}_a)...
user128448
7
votes
1
answer
808
views
Etale and Algebraic K-theory with rational coefficients
We know that the Quillen-Lichtenbaum conjecture, now proved by Rost, Voevodsky, and Weibel, says that for smooth finite type $k$-schemes $X$, etale and algebraic $K$-theory with finite coefficient $\...
- 71
7
votes
1
answer
442
views
Is anything known about de Rham $K(\pi,1)$'s?
Let $X$ be a connected qcqs scheme. We say that $X$ is a (étale) $K(\pi,1)$ if for every locally constant constructible abelian sheaf $\mathscr{F}$ on $X$ and every geometric point $\overline{x}$ the ...
- 773
7
votes
1
answer
388
views
On a quasi-separated assumption in a lemma for the homotopy exact sequence of the etale fundamental group
Background:
I've seen two versions of the homotopy exact sequence for etale fundamental groups. One from Stacks:
Stacks 0BTX: Let $k$ be a field with algebraic closure $\overline{k}$. Let $X$ be a ...
- 701
7
votes
1
answer
1k
views
"Weight-monodromy" for open varieties
Suppose that $X/\mathbb{Q}_p$ is a smooth, projective variety, and choose a prime $\ell\neq p$. Then the weight-monodromy conjecture says that the graded pieces $\mathrm{Gr}_k^M$ of the monodromy ...
- 1,838
7
votes
1
answer
2k
views
Frobenius weights on etale cohomology and purity
Let $X_0$ be a smooth variety (for simplicity I'm willing to assume that X is a curve) over a finite field $k$, $X$ its geometric base change, and $\mathcal{F}$ an $l$-adic etale sheaf on $X$ with $\...
- 469
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\...
- 801
7
votes
1
answer
689
views
Two motivic complexes, compared
Bloch defines the motivic complexes $\mathbf{Z}(n)$ in his paper "Algebraic Cycles and Higher K-Theory" (1986).
Some references (that I currently am unable to track down) use $$\check{\mathbf{Z}}(n) :...
user120812
7
votes
1
answer
1k
views
Specialization map étale cohomology
Let $R$ be a henselian dvr, $s,\eta\in\text{Spec}(R)$ the closed and generic points, and $f : X\to \text{Spec}(R)$ a proper smooth scheme.
For a prime $\ell$ invertible on $R$, is there a ...
user95222
7
votes
1
answer
1k
views
Brauer groups of punctured affine lines over a base
Let $R$ be a torsion-free regular noetherian ring. The Brauer group $Br(R)$ of $R$, defined equivalently (by a theorem of Gabber) as the group of Morita equivalence classes of Azumaya $R$-algebras or ...
- 25.6k
7
votes
2
answers
1k
views
How does one compute induced representations for modular representations?
The set-up is this: Let $G$ be a finite group, and $H$ a subgroup. We are given an irreducible representation of $H, \rho: H\rightarrow GL_n(K)$ (I will notationally identify $\rho$ with its character)...
- 1,386
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
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 ...
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 ...
- 6,920
7
votes
1
answer
5k
views
Chevalley's Theorem on Constructible Sets
I'm having a hard time understanding the theorem in the title, more specifically the proof of the related fact that the image of a dominant morphism contains a dense open set of it's closure. (My ...
- 75
7
votes
1
answer
718
views
Ways to characterize supersingular primes?
I've read the definition, and it basically says p is a supersingular prime iff
the fundamental domain of a group generated by \Gamma(p) and a matrix ((0, 1), (-p, 0)) is rational.
And there's a ...
- 15.1k
7
votes
1
answer
263
views
'Cohomologically approximating' a $\mathbb{Q}[[t]]$-scheme by a one over the henselization of $\mathbb{Q}[t]$?
For certain matters the henselization $R$ of $\mathbb{Q}[t]$ at $0$ is a 'reasonable approximation' for $\mathbb{Q}[[t]]$ (Artin's approximation theorem and so on). Now, I would like to study certain ...
- 16.9k
7
votes
2
answers
775
views
actions of the absolute Galois group and the motivic Galois group on étale cohomology
Let $K$ be a field of characteristic $0$; let $\ell$ be any prime; and let $\mathrm{Mot}(K, \mathbb{Q}_{\ell})$ be a Tannakian category of motives over $K$ with coefficients in $\mathbb{Q}_{\ell}$. So,...
- 1,298
7
votes
2
answers
796
views
Comparison between analytic etale cohomology and algebraic etale cohomology for affinoids
Let $\mathcal{A}$ be an affinoid algebra over a complete non-archimedean field $K$. We have two objects we can investigate, namely the algebro-geometric spectrum $X = \operatorname{spec} \mathcal{A}$ ...
- 892
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
7
votes
1
answer
498
views
Weyl algebra as an Azumaya algebra over its centre
Assume that $k$ is an algebraically closed field of positive characteristic $p$. On page 3 (page 6 of the PDF file) of Bezrukavnikov, Mirković, and Rumynin - Localisation of Modules for a semisimple ...
- 111
7
votes
1
answer
181
views
What is the etale homotopy type of the Witt group of braided fusion categories?
The Witt group $\mathcal{W}$ of braided fusion categories (see also the sequel paper) can be defined over any field; I am happy to restrict to characteristic $0$ if it matters.
Is $\mathbb k \...
- 54.6k
7
votes
2
answers
841
views
What is the need for torsion in the definition of lisse sheaves?
I am studying the basics of constructible and lisse sheaves, and am trying to understand SGA 4, IX. As Grothendieck himself observes at the beginning of the chapter, one is forced to work with torsion ...
- 5,670
7
votes
1
answer
428
views
Reference result: proof of theorem of Kazhdan-Margulis on monodromy group of a Lefschetz pencil of odd fiber dimenion is "as big as possible"
In Deligne's paper on his first proof of the Weil conjectures, we have the following result.
Theorem 5.10 (Kazhdan-Margulis). L'image de $\rho: \pi_1(U, u) \to \text{Sp}(E/(E \cap E^\perp), \psi)$ ...
user97565
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
7
votes
1
answer
513
views
Is there any theory of "étale cohomology" with algebraic coefficients?
For simplicity, I will restrict attention to untwisted coefficients.
Let $k$ be a finite field of characteristic $p$, and $\ell\ne p$ prime. Can one define a cohomology theory with $\mathbb{Q}_\ell^{\...
- 371
7
votes
1
answer
540
views
Mod 2 modular forms in levels 5 and 25--how to account for this Hecke isomorphism?
The space $P1$ of my earlier question 203755 "Two spaces attached to mod 2 level 9 modular forms...", is essentially the space of mod 2 level 3 modular forms. That such a space should appear inside ...
- 5,422
7
votes
2
answers
2k
views
Weights for etale cohomology: why does Deligne's definition work?
For a field $K$ and a variety $X/K$ (whose characteristic could be $0$) I need a 'simple' explanation for the (Deligne's) method of defining weights of the $l$-adic etale cohomology of $\overline{X}$ (...
- 16.9k
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, ...
- 16.9k
7
votes
1
answer
454
views
The weight filtration on etale cohomology and Berkovich analytic geometry
If $X$ is a smooth projective curve over $\mathbb C_p$, then its first etale cohomology $\mathrm H^1_{et}(X,\mathbb Q_\ell)$ (with $\ell\neq p$) carries a certain weight filtration $W_\bullet$ -- also ...
- 1,790
7
votes
2
answers
536
views
What are the polynomial relations between these characteristic 2 "thetas" ?
Suppose $\ell=2m+1$, $m>0$. Define $[i]$ in $\mathbb{Z}/2\mathbb{Z}[[x]]$ to be $$\sum_{n\equiv i\mod l} x^{n^2}.$$ Note that $[0]=1$, and that $[i]=[j]$ whenever $\ell$ divides $i+j$ or $i-j$.
...
- 5,422
7
votes
0
answers
148
views
Is the $\ell$-adic cohomology ring of a cubic threefold a complete invariant?
The only interesting $\ell$-adic cohomology of a smooth cubic threefold $X$ is $H^3(X,\mathbb{Z}_{\ell}(2))$, which is isomorphic as a $\mathrm{Gal}_k$-module to $H^1(JX,\mathbb{Z}_{\ell}(1))^{\vee}$ ...
- 679
7
votes
1
answer
549
views
Finiteness of the Brauer group for a one-dimensional scheme that is proper over $\mathrm{Spec}(\mathbb{Z})$
Let $X$ be a scheme with $\dim(X)=1$ that is also proper over $\mathrm{Spec}(\mathbb{Z})$. In Milne's Etale Cohomology, he states that the finiteness of the Brauer group $\mathrm{Br}(X)$ follows from ...
user122877
7
votes
0
answers
156
views
Invariants of etale topological type that are not homotopy invariants
Artin--Mazur theory attaches etale homotopy type to reasonable schemes. Associated to this homotopy type are certain invariants of the scheme, such as etale fundamental group and higher homotopy ...
- 723
7
votes
0
answers
155
views
Do residues commute with transverse base change?
Fix a number $n > 0$. Given a smooth $\mathbb{Z}[1/n]$-scheme $X$ (i.e., a smooth scheme such that $n$ is invertible in its ring of functions), we may consider the étale sheaf $\mu_n$ on $X$ which ...
- 9,491
7
votes
0
answers
834
views
Difference between algebraic and etale K-theory
Due to the Quillen-Lichtenbaum conjecture (now proven by Rost, Voevodsky, and Weibel), the map $K_\ast(X,\mathbb{Z}/n)\rightarrow K_\ast^{et}(X,\mathbb{Z}/n)$ from algebraic K-theory to etale K-theory ...
- 889
7
votes
0
answers
484
views
independence of $\ell$ for $p$-adic cohomology of varieties over finite fields
Let $X/k$ be a smooth projective geometrically integral variety ($X = A$ an Abelian variety suffices) over $k = \mathbf{F}_q$ with absolute Galois group $\Gamma$, $\bar{X} = X \times_k \bar{k}$, $q = ...
user19475
7
votes
0
answers
374
views
Arbitrarily non-degenerate Hodge to de Rham spectral sequence
It is true that for any $n$ there exists a compact complex manifold which Frolicher spectral sequence does not degenerate at the $n$-th page(https://arxiv.org/pdf/0709.0481.pdf).
Does the analogous ...
- 7,377
7
votes
0
answers
355
views
Are curves over imperfect fields defined over a smaller field?
Let $C$ be regular projective curve defined over a field $K$. Let $K/L$ be a totally inseparable finite extension. Does there exist a regular projective curve $C'$ over $L$ such that that the pullback ...
- 149k
7
votes
0
answers
294
views
Picard scheme of varieties over imperfect fields
Let $k$ be a field and $X$ a proper $k$-scheme. It is a theorem of Murre and Oort that the Picard functor is representable by a $k$-group scheme $\operatorname{Pic}_{X/k}$ which is locally of finite ...
- 4,450
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
7
votes
0
answers
286
views
Level p characteristic 2 modular forms and thetas
BACKGROUND
Let p be an odd prime. An element of Z/2[[x]] is "modular of level p" if it is the mod 2 reduction of a g in Z[[x]] with g the Fourier expansion of a modular form for gamma_0(p). In ...
- 5,422
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
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
7
votes
0
answers
2k
views
An example computation of etale cohomology
(edited for clarity)
In a comment on a response to this question, moonface states the following: "Even if you tried to compute H^2 [etale with Z/5Z-coefficients] of a surface fibered in genus 2 ...
- 13.1k