All Questions
2,543 questions
4
votes
1
answer
412
views
Applications of Chevalley groups theory for dummies
As an algebraist i frequently receive questions from my friends-mathematicians and non-mathematicians about applications of my topic "in real life". I study algebraic groups in the stream of ...
- 96.4k
9
votes
0
answers
560
views
Function fields of characteristic p modular curves, and mod p reductions of the classical modular equation
Let l and p be distinct primes, l>2. There are "characteristic p modular curves" X_0(l) and X(l), defined over an algebraic closure, K, of Z/p, solving moduli problems for elliptic curves with some ...
- 5,422
0
votes
1
answer
203
views
Action of $SL_{n+1}$ on couples of linear spaces in $\mathbb{P}^n$.
Let $SL_{n+1}$ act on $\mathbb{P}^n$ in the natural way. Suppose I take two linear subspaces $\mathbb{P}^m$ and $\mathbb{P}^{n-m}$, with $m < n$, that intersect in one point. Is the action of $SL_{...
- 1,810
2
votes
1
answer
406
views
Isomorphism between pull-backs of an F-crystal by different liftings of Frobenius
This might be a naive question. But since I haven't seen this in any reference, I'll try to ask it here. Let $T$ be a smooth scheme over the algebraically closed field $k$ of characteristic $p>0$ (...
- 3,032
5
votes
1
answer
446
views
More questions involving characteristic 2 theta series identities
In my answer to my earlier question, "Existence of certain identities involving characteristic 2 thetas", I established some curious identities when the thetas have prime "level" congruent to 1 mod 4 ...
- 5,422
20
votes
2
answers
2k
views
Frobenius splitting and derived Cartier isomorphism
Let $X$ be a smooth algebraic variety over an algebraically closed field $k$ of characteristic $p>\dim X$. The motivation for my question comes from the following results.
1. If $X$ is Frobenius ...
CommunityBot
- 1
9
votes
1
answer
903
views
Principal congruence subgroups in higher rank
I don't seem to have seen any explicit generators for the principal congruence subgroups of $SL(n, \mathbb{Z}),$ for $n>2,$ although it is known (Sury+Venkataramana) is that the number of ...
- 96.4k
6
votes
0
answers
304
views
How to decide if two surfaces occurring in Springer theory are isomorphic?
In the study of a simple algebraic group (say over $\mathbb{C}$) and related geometry of its flag variety associated with the Springer correspondence, one encounters pairs of surfaces which have some ...
- 52.9k
6
votes
1
answer
804
views
Del pezzo surfaces in positive characteristic
For me a Del Pezzo surface $X$ over an algebraically closed field of characteristic $p$ is an algebraic surface where the anticanonical bundle $\omega^{-1}_X$ or $-K_X$ is ample. (I prefer the second ...
- 2,215
3
votes
1
answer
121
views
Lower bound on the degree of a product of elements in a hyperalgebra/enveloping algebra
Background:
Fix a linear algebraic group $G$ over an algebraically closed field $k$ of arbitrary characteristic and let $B \subseteq G$ be a Borel subgroup with unipotent radical $N$. Let $\Delta^+$ ...
- 3,702
2
votes
1
answer
710
views
Iwahori for PGL_2
What is the Iwahori subgroup for $PGL_2(F)$ where $F$ is a local field? I am also looking for the Levi subgroups but it seems that there is only 1 levi subgroup been the identity but this seems odd to ...
- 20.8k
21
votes
1
answer
2k
views
When does the relative differential $df=0$ imply that $f$ comes from the base?
Let $A \to B$ be a map of commutative rings, and $d : B \to I/I^2$ be
defined by $df = f\otimes 1 - 1\otimes f$, where $I$ is the kernel of
$B \otimes_A B \to B$, as in [Hartshorne II.8].
If $df=0$,...
- 11.1k
7
votes
3
answers
3k
views
Longest element of a Weyl group
Let $G$ an algebraic (reductive) group. $T$ a maximal torus, $B$ a Borel subgroup containing $T$, and $w_0$ the longest element of the Weyl group.
I'm looking for a reference explaining why when you ...
- 52.9k
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 "...
- 30.3k
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 ...
CommunityBot
- 1
1
vote
2
answers
879
views
closure of orbit of a group action on a variety
Let $X$ be a (smooth) algebraic variety (over $\mathbb{C}$). Let $G \subset \operatorname{Aut}(X)$ be a subgroup of automorphisms of $X$. Is it true that for any $x\in X$ the closure $\overline{O_x}$ ...
- 7,338
11
votes
1
answer
615
views
Do Richardson varieties have rational singularities in arbitrary characteristic?
The title basically asks the question. I'll review the relevant terminology and explain what I have and haven't found in the literature.
Let $G$ be a reductive group. Let $v \leq w$ be elements of ...
- 156k
2
votes
1
answer
304
views
Connected extensions of finite by connected algebraic groups
Let $1\to H\to E\to G\to 1$ be a short exact sequence of algebraic groups defined over an algebraically closed field $k$ of characteristic $p$. Suppose $H$ is a finite group, and $G$ and $E$ are ...
- 22.6k
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...\...
CommunityBot
- 1
5
votes
0
answers
261
views
Lattices in Hermitian spaces over local fields
Let $F$ be a $p$-adic field, $E / F$ a quadratic extension, and $n \ge 1$. Let $V = E^n$ with the obvious diagonal Hermitian form,
$$ \langle (u_1, \dots, u_n), (v_1, \dots, v_n) \rangle = \sum_{i = ...
- 37k
9
votes
1
answer
1k
views
Central extensions of group schemes
In the category of groups, it is elementary that all central extensions of a cyclic group are abelian. Is the same true, in the category of (finite?) group schemes over a field $k$, for central ...
- 22.6k
5
votes
1
answer
556
views
Does the Zariski closure of a maximal subgroup remain maximal?
Let $k$ be an algebraically closed field and let $G\leq\rm{GL}_n(k)$ be a linear group. Assume that $M< G$ is a maximal subgroup (in the abstract group sense). Denote by $\bar{G}^Z$ the Zariski ...
- 156k
3
votes
0
answers
281
views
What would be a characteristic-$p$ analogue for $C^{\infty}$-fiber bundles?
I'd like to know a notion for a morphism between algebraic varieties in characteristic $p$ that plays the role of a $C^{\infty}$-fiber bundle. It should be, in particular, flat. I'm not assuming the ...
- 4,265
1
vote
0
answers
300
views
When does a group action on $X$ preserve the reduction $X_\text{red}$? [duplicate]
Possible Duplicate:
Does the action of an affine group scheme preserve the nilradical of an algebra?
Let the group scheme $G$ act on the scheme $X$. I labored for a time under the misapprehension ...
CommunityBot
- 1
1
vote
0
answers
313
views
two different properties for the quotient
(Updated)
I have looked the draft of Ch4 of the book "Abelian Varieties" by Gerard van der Geer and Ben Moonen. It looks like in order to see the group scheme structure on G/H, one should consider ...
- 1,109
13
votes
1
answer
690
views
Obstructions to formally integrating vector fields in characteristic p?
Let $M$ be a smooth scheme over some field $k$ of characteristic $p$, and $\vec X$ a vector field on it. Equivalently, $\vec X$ gives a map $Spec\ k[\epsilon]/\langle \epsilon^2 \rangle \times M \to M$...
CommunityBot
- 1
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 ...
CommunityBot
- 1
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 ...
CommunityBot
- 1
2
votes
1
answer
939
views
Weyl group Invariants
What are the generators of $\mathbb C[V^m]^W$, where $W$ is the Weyl group
of type $E_6, E_7, E_8$, V^m denote 'm' (m > 1) copies of the Cartan subalgebra
and the action is the diagonal action?
Is ...
- 396
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 ...
- 27.9k
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.
(...
CommunityBot
- 1
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 ...
CommunityBot
- 1
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
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 ...
- 44.4k
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
...
- 27k
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
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 ...
- 73.2k
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 $G/N$, ...
- 4,492
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 $B$ ...
- 52.9k
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
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
30
votes
5
answers
4k
views
Deformation theory of representations of an algebraic group
For an algebraic group G and a representation V, I think it's a standard result (but I don't have a reference) that
the obstruction to deforming V as a representation of G is an element of H2(G,V&...
- 1,180
6
votes
2
answers
1k
views
Cartan subgroups of p-adic groups.
Does anyone know about any reference describing the structure of Cartan subgroups in the case of connected p-adic reductive (or let's say semi-simple) groups?
I would like to know how different is ...
- 366
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 ...
- 1
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