All Questions
2,543 questions
12
votes
1
answer
2k
views
Replacement for derivations in characteristic p?
Let $k$ be a field.
If $f \in k[x]$ is a polynomial, and $d/dx\ f = 0$, then either
$f$ is constant, or
$char\ k = p$ and $f \in k[x^p]$.
So "annihilated by all derivations" is perhaps not the right ...
- 27.9k
21
votes
4
answers
2k
views
Simplest example of jumping of cohomology of structure sheaf in smooth families?
Using Hodge theory (and the ill-defined Lefschetz principle), one can show that in characteristic 0, given a proper smooth family $X \rightarrow B$, the cohomology groups of the structure sheaf of the ...
- 3,857
7
votes
1
answer
561
views
How does the right regular of GL(n, R) and GL(n,Qp) decompose?
The question is contained in the title. I would guess that this question is already answered in the literature.
Given the reductive group $GL(n)$ over a complete local field, how does the right ...
- 11.2k
0
votes
0
answers
2k
views
Book on linear algebraic groups in scheme language
Is there a book on linear algebraic groups using the scheme language (i.e. not Springer or Borel, but like Waterhouse, but more in-depth)?
The book should discuss topics like Borel subgroups etc.
(...
- 417
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 ...
- 33.8k
9
votes
0
answers
1k
views
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 ...
- 2,976
4
votes
1
answer
2k
views
Why is the double cover of $Sl(2,\mathbb{R})$ not algebraic?
Today in a talk, it has been mentioned that there exists algebraic groups over the local field $\mathbb{R}$ such that the finite central extension can not be defined algbraically over $\mathbb{R}$ or ...
- 11.2k
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 $2$. ...
- 2,215
21
votes
0
answers
588
views
p-groups as rational points of unipotent groups
Is it true that every finite p-group can be realized as the group of rational points over $\mathbb{F_p}$ of some connected unipotent algebraic group defined over $\mathbb{F_p}$? For abelian p-groups, ...
- 221
14
votes
0
answers
466
views
Analog of Peter-Weyl theorem for $G[[t]]$
Let $G$ be a reductive group over ${\mathbb C}$ and let $G[[t]]$ denote the corresponding
group over the formal power series ring ${\mathbb C}[[t]]$. This is a group scheme, so one
can speak about its ...
- 7,037
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 ...
- 143
7
votes
2
answers
578
views
Does the action of an affine group scheme preserve the nilradical of an algebra?
Let $k$ be a commutative ring and let $G$ be a flat affine algebraic group scheme over $k$.
Let $G$ act by algebra automorphisms on the commutative $k$-algebra $A$. So $G(R)$ acts by $R$-algebra
...
- 3,504
5
votes
2
answers
556
views
Existence of certain identities involving characteristic 2 "thetas"
Let l=2m+1 be prime. In my previous MO question, "What are the polynomial relations between these characteristic 2 thetas?", I defined a subring of Z/2[[x]] as follows:
The subring, S, is generated ...
- 5,422
4
votes
0
answers
409
views
About ℓ-adic and perverse stuff and ℓ-adic cohomology with compact support
Maybe this question is trivial.
We know from this paper at Inv. Math 1976 (DOI link), T. A. Springer constructed representation of the Weyl group $W$
on the cohomology of the Springer fibre. Also, ...
- 93
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
8
votes
2
answers
3k
views
Is there a Levi decomposition for Lie group and algebraic group?
Let $G$ be a Lie group and $R$ be the largest connected solvable
normal subgroup of $G$.
Question 1
Is there a Lie subgroup $S$ such that: (1) $G=SR$; (2)
every real representation of $S$ is ...
- 491
3
votes
1
answer
613
views
Union of disjoint Vitali Sets...
We say $X$ is a Vitali set if there exists a countably dense subgroup, $\Gamma$, of the additive group $\mathbb{R}$, such that $X$ is a selector of the partition of $\mathbb{R}$ canonically associated ...
- 463
13
votes
2
answers
3k
views
Literature on the Springer resolution
Could you suggest me a basic reading list on the Springer resolution? Is there a textbook I can refer to? Or do I need to start with the original paper?
Unfortunately googling for "Springer" and "...
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
19
votes
2
answers
3k
views
Bertini theorems for base-point-free linear systems in positive characteristics
Suppose that $X$ is a smooth algebraic variety over an algebraically closed (uncountable if it helps) field of characteristic $p > 0$. Suppose that $L$ is a line bundle, probably ample or at least ...
- 20.5k
5
votes
4
answers
809
views
Countable Dense Sub-Groups of the Reals...
Can countable dense additive subgroups of the reals be well-ordered up to isomorphism by inclusion?
If so, is $\mathbb{Q}$ the smallest (up to isomorphism) countable dense subgroup of the reals, and ...
- 463
14
votes
1
answer
1k
views
Uniform proof of dimension formula for minimal special nilpotent orbit?
Given a simple Lie algebra over an algebraically closed field of good characteristic such
as $\mathbb{C}$, its subvariety $\mathcal{N}$ of nilpotent elements has dimension $2N$ (where $N$ is the ...
- 52.9k
7
votes
1
answer
658
views
Kempf Vanishing theorem and Representation of Lie algebras.
Let $G$ be a reductive connected algebraic group and let $B$ a Borel subgroup. One of central themes of the representation theory of $G$ is the study of the induction functor $H^0$ from $B$ ...
- 829
9
votes
2
answers
2k
views
Is the category of affine fppf groups closed under normal quotients?
Let $S$ be a scheme and let $N$, $G$ be affine flat group schemes of finite presentation over $S$.
If we assume that $N$ is a closed normal subgroup of $G$, we may form the fppf quotient sheaf $G/N$, ...
- 1,538
12
votes
4
answers
2k
views
Finite subgroups of $PGL_2(K)$ in characteristic $p$
Let $K$ be a field of characteristic $p$. What are the finite subgroups of $PGL_2(K)$ whose orders are divisible by $p$? And if $G$ and $H$ are two such subgroups that are isomorphic, can one say when ...
- 1,199
6
votes
2
answers
1k
views
Maximal torus and parabolic subgroups in reductive groups over finite fields
Let $G$ be a reductive group, $B_0$ a $F$-stable Borel subgroup and $T_0$ a fixed $F$-stable maximal torus contained in $B_0$. ($F$ = Frobenius morphism).
Let $r$ be the semisimple $\mathbb{F}_q$-rank ...
- 311
2
votes
1
answer
690
views
Restricted universal enveloping algebra of Abelian p-Lie algebra
Question: Let $p$ be a prime. Let $k$ be a commutative ring such that $p=0$ in $k$.
Let $\mathfrak g$ be an abelian $p$-restricted Lie algebra over $k$. In other words, let $\mathfrak g$ be a $k$-...
- 33.8k
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 ...
4
votes
0
answers
385
views
idelic closures of units of number fields
Let $K$ be a number field, $\mathcal O _K^\times$ its group of integral units and $\mathcal O _{K,+} ^\times$ its group of totally positive units. Denote further by $\widehat{\mathcal O }_K^\times$ ...
- 2,029
3
votes
1
answer
984
views
Borel subgroups contained in a fixed parabolic subgroup
The question is asked in the context of (connected) reductive groups.
In the article i'm working on, the author states the following fact (well it's not word to word exact, I simplified it a little) :...
- 311
3
votes
2
answers
489
views
Hopf algebra of Chevalley group from the root system
Has anyone worked out a uniform way of constructing the Hopf algebra of a Chevalley group out of the root system (or, more precisely out of the root datum for reductive groups).
By "uniform", I mean ...
- 778
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 ...
- 65.4k
15
votes
0
answers
779
views
Lifting varieties from char. $p$ to char. 0 after alterations
The question is related to this MO question:
Lifting varieties to characteristic zero.
Let $X$ be a projective smooth variety over $k$ alg. closed field of char. $p.$ Does there always exist an ...
- 4,265
3
votes
2
answers
1k
views
Connectedness of Centralizers in $GL_n$
I was wondering if there is any obvious reason or quick proof that for every $g\in GL_n$ the centralizer $Z_{GL_n}(g)$ is connected. Also I wanted to see why for any semisimple $s\in Sp_{2n}$ the ...
- 2,108
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; ...
- 33.8k
3
votes
0
answers
379
views
Maximal compact subgroup of a real semisimple Lie group of "quasi-adjoint" type.
Let $\mathfrak{g}$ be a semisimple complex Lie algebra, and $G$ the adjoint group of $\mathfrak{g}$.
Let $\sigma:\mathfrak{g}\rightarrow \mathfrak{g}$ be a complex conjugation and $\mathfrak{g}^{\...
- 59
2
votes
5
answers
1k
views
Product of two algebraic subgroups of a (solvable) group = another algebraic subgroup?
Let $G$ be a linear algebraic group over a field $K$. (Say $K=\mathbb{F}_q$ or
$K=\mathbb{C}$; do not assume $K$ is algebraically closed or of characteristic $0$.) Let $H_1$, $H_2$ be algebraic ...
- 20.2k
3
votes
1
answer
351
views
Is the union of Cartan subgroups over $k$ dense?
Let $k$ be a finite field and $G$ a connected, reductive linear algebraic group defined over $k$. It is well-known that the union of the maximal tori of $G$ is dense in $G$ (more generally, if $G$ is ...
- 33
5
votes
1
answer
461
views
Given a branched cover with branch cycle description $(g_1,...,g_r)$, does $g_i$ generate some decomposition group?
Classically:
Let $a_1,...,a_r$ be points in $\mathbb{P}^1_{\mathbb{C}}$, and let $\alpha_1,...,\alpha_r$ be simple loops around the $a_i$, all counterclockwise, and none touching (so $\alpha_1...\...
- 5,929
8
votes
2
answers
3k
views
Lie algebras of algebraic groups
Where can i find material about the definition of the exponential morphism from the Lie algebra of an algebraic affine group to the group?
- 143
30
votes
7
answers
5k
views
Shuffle Hopf algebra: how to prove its properties in a slick way?
Let $k$ be a commutative ring with $1$, and let $V$ be a $k$-module. Let $TV$ be the $k$-module $\bigoplus\limits_{n\in\mathbb N}V^{\otimes n}$, where all tensor products are over $k$.
We define a $k$...
- 33.8k
21
votes
2
answers
5k
views
State of resolution in positive characteristic?
Heisuke Hironaka's coming talk makes me wonder how the state of the work on that theme is. So far, I noticed (but didn't read) these papers:
Kawanoue, Hiraku, Toward resolution of singularities over ...
- 10.8k
2
votes
1
answer
249
views
unipotent group and translation invariant metric
Let $U$ be a unipotent upper triangluar group over a local field $K$ of characteristic
zero. Can we guarantee that there is a right translation invariant metric on $U$ such
that any ball of finite ...
- 853
9
votes
2
answers
1k
views
Action on the highest weight vector of a representation of a semisimple linear algebraic group
Let $G$ be a semisimple linear algebraic group, $V$ a $G$-representation and $v \in V$ a vector of highest weight $\lambda$. Is it true, that for any positive root $\alpha \in R^+$ the one dimensional ...
- 783
2
votes
2
answers
329
views
Existence of proper invariant subset in an irreducible action
Let $G<\rm{GL}_n(\mathbb{k})$ be a linear group, where $\mathbb{k}$ is an algebraically closed field. Assume that the linear action of $G$ on $\mathbb{k}^n$ is strongly-irreducible (i.e. there are ...
- 215
2
votes
1
answer
828
views
Is a double centralizer type theorem ( encountered in semisimple algebras) true for algebraic groups ?
Let, $G(k^{al})$ be an algebraic group, over an algebraically closed field, and $\Gamma_{G}$ is the set of all closed subgroups of $G(k^{al})$.
Then is the map $Z_{G}: \Gamma_{G} \rightarrow \Gamma_{...
- 419
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
9
votes
1
answer
756
views
Is every reductive group scheme etale locally trivial?
Let $S$ be a scheme over a field $k$, and let $G$ be a reductive group scheme over $S$. Let us call it trivial, if it is a pull-back of a group scheme over $k$ via the structure morphism $S\to k$. Is ...
- 1,590
3
votes
2
answers
708
views
$k$ structures on $K$ vector spaces
The following statement is made in Borel's Linear Algebraic groups, section 11 on $k$ structures.
Let $V$ and $W$ be $K$ vector spaces with $k$ structures. If $f:V\to W$ is $K$ linear, then $f$ is ...
- 1,563