All Questions
1,114 questions
2
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0
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47
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Characters of simple $\mathfrak{sl}_n$-modules in positive characteristic with subregular nilpotent central character
Consider representations of $\mathfrak{sl}_n$ in positive characteristic with a subregular nilpotent central character $\chi$ (i.e. $\chi$ is a nilpotent matrix whose Jordan normal form has two blocks ...
- 63.9k
8
votes
0
answers
433
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Intuition for de Rham comparison theorem in $p$-adic Hodge theory
The de Rham comparison theorem from $p$-adic Hodge theory compares the etale cohomology of a variety with the de Rham cohomology of that variety. It says the following:
Let $K/\mathbf{Q}_p$ be a ...
- 1,949
4
votes
1
answer
580
views
Etale cohomology and Kummer theory
If $K$ is a field and $n \geq 1$ is such that $n \in K^{\times}$, then $H^1_{et}(\mathrm{Spec}(K),\mu_n)=K^{\times} / (K^{\times})^n$. This is easy to prove, see for instance Tamme, Etale Cohomology, ...
- 63.1k
4
votes
0
answers
64
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An analog of a BGG resolution in subregular case in positive characteristic
Consider representations of $\mathfrak{sl}_n$ in positive characteristic with a subregular nilpotent central character $\chi$. For every regular weight $\lambda$ of $\mathfrak{sl}_n$, we have the ...
1
vote
0
answers
220
views
Cohomology of a curve and its Jacobian over an algebraic closure of a number field
In this MathOverflow post, the smooth projective curve $C$ was defined over $\mathbb{C}$ and we have an isomorphism of de Rham cohomology groups
$$H^1(C, \mathbb{C}) \cong H^1(J_C, \mathbb{C}),$$
...
- 895
3
votes
0
answers
120
views
Resolving the "wild" singularities of $\mathbb A^n/C_n$
Let the cyclic group on $n$ elements, $C_n$, act on $\mathbb A^n$ by permuting the co-ordinates (over a field $k$). If $n \neq 0 \in k$, we can resolve the singularities of $X = \mathbb A^n/C_n$ by ...
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27
votes
4
answers
3k
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Why do we care about the eigenvalues of the Frobenius map?
The Riemann hypothesis for finite fields can be stated as follows: take a smooth projective variety X of finite type over the finite field $\mathbb{F}_q$ for some $q=p^n$. Then the eigenvalues $\...
- 47.4k
3
votes
0
answers
299
views
What are some concrete applications of Grothendieck's six operations?
In Gallauer's An introduction to six-functor formalisms I read:
Indeed, the language and theory of six-functor formalisms permeates much of modern algebraic geometry and beyond, and has spawned ...
- 91
1
vote
1
answer
149
views
When is $R$ a direct summand of Frobenius pushforwards?
Let $(R,\mathfrak m)$ be a reduced Noetherian local ring of prime characteristic $p$. For integer $e>0$, let $F^e_* R$ denote the $R$-module which is $R$ as an abelian group, but the $R$-module ...
- 20.5k
2
votes
0
answers
177
views
How do characters of representations in cohomology depend on the (positive-characteristic) field?
The following sentence appears in Jantzen - Representations of algebraic groups, 2nd edition, p. x, where $G$ is a reductive group over an algebraically closed field $k$, $B$ is a Borel subgroup, $T$ ...
- 12.9k
2
votes
0
answers
246
views
Proof of the projection formula (for cohomology of $\mathbf{P}V$)
Let $V\to X$ be a vector bundle (over say a scheme).
Then the cohomology of its projectivisation is
$$\text{H}^*(\mathbf{P}V)\ =\ \text{H}^*(X)[t]/(t^{n+1}+c_1(V)t^n+\cdots+c_n(V))$$
as an algebra, ...
- 5,711
10
votes
1
answer
1k
views
$\ell$-adic Weil cohomology theory
I have a reference or counterexample request. Suppose $k$ is a field and $\ell\neq char(k)$. There are several common references that show that $H^i_{et}(-, \mathbb{Q}_\ell )$ is a Weil cohomology ...
- 2,821
4
votes
1
answer
227
views
Compute de Rham-Witt sheaves
I am really new to this, but I am having a hard time understanding all the de Rham-Witt construction.
It seems to be really difficult to compute anything with those beasts: like I cannot find any ...
9
votes
0
answers
1k
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Ample vector bundles, $H^1=0$ and global generation in characteristic $p$
This is a follow up from Ample vector bundles on curves question, in characteristic $p>0$. First, some background. It is known that in characteristic $0$, a vector bundle $E$ on say a projective ...
- 12.9k
2
votes
0
answers
136
views
Is there the specialisation map of etale K theory?
Take a smooth proper morphism of schemes $X\to S$. Fix a point $t\in S$ and a point $s\in \overline\{s\}$. For a prime $l$ which is invertible in $S$, is there the natural specialization map of etale ...
- 519
6
votes
1
answer
771
views
A regular, geometrically reduced but non-smooth curve
Can anyone give an example of a projective, regular, geometrically reduced but non-smooth curve ?
Of course, the base field should be imperfect.
In Exercise 4.3.22 of Qing Liu's book Algebraic ...
- 149k
4
votes
0
answers
197
views
Inequalities between numerical invariants of nonsingular projective Varieties in positive Characteristic
It is well-known that Miyaoka and Yau-type inequalities do not hold in positive characteristic. In "a note on Bogomolov-Gieseker’s inequality in positive characteristic", however, we can ...
- 2,821
2
votes
0
answers
253
views
Künneth formula for algebraic de Rham cohomology
Let $X$ and $Y$ be finite type schemes over a field $k$, and let $H^i(X/k)$ denote the $i$-th algebraic de Rham cohomology group of $X$ over $k$. I'm interested in the extent to which a Künneth ...
- 63.9k
13
votes
2
answers
1k
views
Which sheaves satisfy cohomological purity?
The absolute cohomological purity theorem in étale cohomology is as follows.
Let $X$ be a regular scheme over $\mathbb{Z}[1/n]$, and $i \colon Z \to X$ the inclusion of a regular
closed subscheme ...
- 371
3
votes
1
answer
642
views
Decomposition theorem for polarized abelian varieties in positive characteristic
In characteristic zero we have the following decomposition theorem for polarized abelian varieties: it gives an isomorphism between a PPAV and a product of PPAV's of lower dimension and is valid (as ...
CommunityBot
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1
vote
1
answer
229
views
Purity for proper varieties
Let $X$ be a proper, geometrically connected, geometrically integral variety over $\mathbf{F}_q$. There exists a finite field extension $k/\mathbf{F}_q$ of degree $d$ and an alteration $X'\to X_k$ ...
- 149k
3
votes
1
answer
283
views
Is the theta characteristic attached to an etale double cover of a plane quintic arising from a cubic threefold even in characteristic two?
We work over an algebraically closed field $k$ of characteristic $2$. Let $X$ be a cubic threefold realized as a conic bundle via $f: X \to \mathbb{P}^2$ (after blowing up some line). Let $C \to \...
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1
vote
1
answer
156
views
$p$-adic étale cohomology groups are not $\mathbb{C}_p$-admissible
It is stated in Caruso - An introduction to $p$-adic period rings (the remarks following equation (2)) that the $p$-adic étale cohomology groups of an algebraic variety $X$ over a finite extension $K$ ...
- 149k
4
votes
0
answers
284
views
modularity of elliptic curves over function fields in positive characteristic
Let $F$ be a global function field over a finite field, and $E$ be an elliptic curve over $F$. Much like the number field case, it is natural to study the Galois representation on the Tate module of $...
- 685
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^{\...
CommunityBot
- 1
0
votes
0
answers
633
views
"Cohomology with compact support isomorphic to homology" theorems
I am collecting theorems throughout different fields which say roughly something of the form "Cohomology with compact support isomorphic to homology".
I'm studying this situation (and its ...
- 12.9k
4
votes
0
answers
342
views
Voevodsky's motives and Deligne's systems of realizations
$\newcommand{\gm}{\mathrm{gm}}$Let $\mathbf{DM}_{\gm}(\mathbb{Q},\mathbb{Z})$ be Voevodsky's category of geometric motives over $\mathbb{Q}$ with coefficients in $\mathbb{Z}$ (e.g. as on p.124 of ...
- 63.9k
0
votes
0
answers
208
views
Cohomology map induced by inclusion of curves
Let $C$ be a smooth affine geometrically integral curve of genus $\geq 1$ over an algebraically closed field $k$, and let $\iota: C \rightarrow C'$ denote the inclusion into its smooth ...
- 895
2
votes
0
answers
218
views
Borel-Weil-Bott theorem for wonderful compactification in characteristic p
Are there any known results for a Borel-Weil-Bott theorem for the wonderful compactifications over characteristic $p$ (i.e., theorems that classify the cohomologies of all line bundles on a wonderful ...
- 41
4
votes
3
answers
795
views
Quotients of Tate modules
Let $p$ be a prime number, let $K$ denote a finite extension $\mathbb{Q}_{p}$ and let
$\overline{K}$ be an algebraic closure of $K$. Let $A$ be an ellitpic curve over
$K$ and denote by $T_{p}A$ its ...
- 1,949
10
votes
1
answer
1k
views
Are there "reasonable" criteria for existence/non-existence of Levi factors or their conjugacy in prime characteristic?
Classical theorems attributed to Levi, Mal'cev, Harish-Chandra for a finite
dimensional Lie algebra over a field of characteristic 0 state that it has a Levi decomposition (semisimple subalgebra plus ...
- 2,821
3
votes
0
answers
168
views
Symmetrical monoidal $2$-category of cohomological correspondences
My question is whether a symmetric monoidal $2$-category of ``cohomological correspondences'' has been been rigorously constructed anywhere in the literature.
Let me be more precise about what I mean.
...
- 2,923
4
votes
0
answers
245
views
Hard Lefschetz for cycles
Let $X$ be a smooth projective variety over a field $k$. It is known by work of Deligne, that the Lefschetz operator:
$$
L^k:H^{2n-2k}\left(X_{\overline{k}},\mathbf{Q}_{\ell}\right)\to H^{2n+2k}\left(...
- 828
4
votes
0
answers
296
views
de Rham Witt complex vs. de Rham complex of the Witt ring
I am reading the paper "Revisiting the de Rham-Witt complex" by Bhatt-Lurie-Mathew and I am a bit confused about the difference between $W\Omega_R^*$ and $\hat{\Omega}^*_{W(R)}$.
Let $\...
- 218
4
votes
0
answers
204
views
Explicit description of wonderful compactification for PGL_3
Let $k$ be an algebraically closed field of positive characteristics. Let $X$ be the wonderful compactification of $PGL_3$ (see for example section 6 of "Frobenius Splitting Methods in Geometry ...
- 163
2
votes
0
answers
147
views
Automorphism groups of "reductive" Lie algebras in positive characteristic
I put "reductive" in quotes because, of course, in positive characteristic one should speak of Lie algebras of reductive groups, not of reductive Lie algebras.
Let $G$ be a reductive group ...
- 63.9k
2
votes
1
answer
473
views
Torsors over elliptic curves
Let $G$ be a finite abelian etale group scheme over a number field $k$. Let $E$ be an elliptic curve over $k$ and $C := E\backslash \{O\}$ its affine model of the same equation.
Recall that for a ...
- 23k
4
votes
1
answer
647
views
A comparison theorem between crystalline cohomology and étale cohomology
Suppose $X/\mathbb F_q$ is a smooth projective variety. Katz-Messing (eudml) shows that the characteristic polynomial of the Frobenius on $H^i_{et}(\overline{X},\mathbb Q_\ell)$ and $H^i_{crys}(X)$ ...
CommunityBot
- 1
2
votes
0
answers
114
views
Two natural morphisms of sheaves with the same source and target; do they agree?
Suppose we have a diagram
$\require{AMScd}$
\begin{CD}
A @>a>> B\\
@V b V V @VV c V\\
C @>>d> D @>e>> E \\
@VfVV @VVgV @VVhV \\
F @>>i> G @>>j> H
\end{CD}...
- 125
1
vote
0
answers
126
views
Strict henselianization of complete intersections
As far as I understand (and tbh for my purposes), one of the main points of strict henselisation of a local ring is that it computes the stalk at a point of a scheme in the étale topology. In the ...
- 4,428
1
vote
0
answers
190
views
Does the Gross-Hopkins period map have a natural interpretation coming from derived algebraic geometry?
The Gross-Hopkins period map is a map on the $W(k)$-points of $LT_n \to P^{n-1}$, where $k = F_{p^n}$. It sends a lift $G$ of the Honda formal group to the 1-$d$ subspace $\omega_G$ of the Dieudonn'e ...
- 448
20
votes
5
answers
4k
views
Equivalent statements of the Riemann hypothesis in the Weil conjectures
In the cohomological incarnation, the Riemann hypothesis part of the Weil conjectures for a smooth proper scheme of finite type over a finite field with $q$ elements says that: the eigenvalues of ...
- 4,713
2
votes
1
answer
361
views
Lie algebroid in algebraic geometry
When I did net-surfing at home, I met some geometric backgrounds of Lie algebras and encountered the concept of Lie algebroids. In differential geometry, a Lie algebroid seems to be defined as ...
- 4,008
3
votes
0
answers
175
views
Deligne's integrality theorem in the setting of $ \mathbb{F}_{\ell}((t)) $-adic cohomology
Let $ \mathbb{F}_{q} $ be a finite field of characteristic $ p $ and $ \overline{\mathbb{F}_{q}} $ be an algebraic closure of $ \mathbb{F}_{q} $. Let $ X $ be a smooth projective variety over $ \...
- 863
9
votes
1
answer
356
views
Reference request for indecomposable representations of $\mathfrak{sl}(2)$ over an algebraically closed field of characteristic $p > 0$
Is there a good reference on the classification of indecomposable representations of the Lie algebra $\mathfrak{sl}(2)$ over an algebraically closed field of characteristic $p > 0$ (of course, in ...
CommunityBot
- 1
2
votes
0
answers
97
views
Non-noetherian Cartier Isomorphism
A result in positive characteristic is that if $R/\mathbb{F}_p$ is a smooth ring, then we have a Cartier isomorphism
$$\Omega_{R}^\bullet\cong H^\bullet(\Omega_R^\bullet)$$
which is essentially ...
- 4,428
1
vote
1
answer
293
views
Every rank 1 local system $L$ satisfying $m^*L=L\boxtimes L$ comes from the Lang torsor? The same holds for D-modules?
Let $G$ be a commutative connected algebraic group over a field $k$, with group operation $m:G\times G\to G$. If $k=\mathbb{F}_q$, we may use a character $\varphi:G(k)\to\overline{\mathbb{Q}}_\ell^\...
- 149k
2
votes
1
answer
545
views
Computation of cohomology of Eilenberg-Maclane spaces
$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Ext{Ext}\DeclareMathOperator\Spf{Spf}$Background:
If $E$ is a complex-oriented spectrum, then $E^*(K(\mathbb{Z}/p^k,1))$ sits inside a long exact ...
- 56.9k
2
votes
0
answers
156
views
When is map of $E_{\infty}$-ring spectra etale iff certain condition is fullfilled
When is it true that a map of $E_{\infty}$-ring spectra $R \to S$ is etale (in Lurie's sense) if and only if, $\operatorname{TAQ}^R(S) = 0$ and $ \pi_*(R)\otimes_{\pi_0(R)} \pi_0(S) = \pi_*(S)$?
- 2,866
2
votes
1
answer
367
views
Groupoid of points, shape and stratified shape of $\operatorname{Sh} (X_\text{pro-ét})$
$\DeclareMathOperator\Sh{Sh}\DeclareMathOperator\Pt{Pt}$Maybe this is well-known or even a stupid misunderstanding of something very basic. It's well-known that the groupoid of points (i.e., groupoid ...
- 12.9k