All Questions
2,543 questions
1
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1
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224
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Taking quotient of a variety by the additive group
1. Let $X$ be a smooth irreducible $\Bbb C$-variety,
on which the algebraic $\Bbb C$-group $G={\bf G}_{a,{\Bbb C}}$
(the additive group) acts freely on the right:
$$ X\times _{\Bbb C} G\to X,\quad (x,...
- 14.2k
1
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1
answer
264
views
Is the Borel subgroup the only closed double coset?
Let $G$ be a quasisplit connected reductive group over a $p$-adic field $k$. Identify $G$ with its rational points. Let $B$ be a Borel subgroup of $G$ containing a maximal torus $T$, both defined ...
- 6,180
1
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1
answer
209
views
Use of generic points to show that $k$-tori are split over $k_s$
Let $k$ be a field of characteristic $p$, and let $T$ be a $k$-torus. To say that $k$-split is to say that every character $T \rightarrow \mathbf{G}_m$ is defined over $k$.
The following result is ...
- 6,180
1
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2
answers
205
views
Dimension of maximal subgroups
Let $G$ be an almost-simple classical real algebraic group, and $H$ a maximal proper closed subgroup. Assume that the $n$ in $G := A_n,B_n,C_n,D_n$ is sufficiently large.
Is it true that dim$(G)$ - ...
- 601
1
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1
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528
views
Inner forms of $GL_n(F)$
If $F$ is a non-Archimedean local field, then any inner form of $GL_n(F)$ is isomorphic to $GL_m(D)$, for a central $F$-division algebra of dimension $d^2, md=n$. Why is this true? I looked in ...
1
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2
answers
326
views
Examples of quotients by infinitesimal group schemes
I'm looking for examples of explicit actions of the infinitesimal group schemes $\alpha_{p^n}$ on schemes (maybe as simple as the affine plane) in characteristic $p$ or mixed characteristic, and their ...
- 1,841
1
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2
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404
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A question on the effective cone
Let $X$ be a projective variety and $G$ a finite group acting on $X$. We consider the quotient $\pi:X\rightarrow Y :=X/G$.
I'm interested in the relation between $Eff(X)$ and $Eff(Y)$. In particular,...
user68440
1
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2
answers
271
views
Duality for group variety
For any abelian variety $A$, there is a dual abelian variety $\hat{A}$ which parametrizes degree zero line bundles.
Is it possible to expect similar duality for group varieties (suppose over $\mathbb{...
- 3,472
1
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2
answers
289
views
Reference request: Lusztig's symmetries
Let $W$ be the Weyl group of a simple algebraic group $G$. The Artin braid group $Br_{\mathfrak{g}}$ is generated by the $T_i$ , $i \in I$ such that for all $i, j \in I$,
\begin{align}
\underbrace{...
- 6,211
1
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1
answer
242
views
Smooth map to the stack of G-bundles
Let $G$ a semisimple group and $B$ a Borel subgroup.
We denote by $Bun_{G}$ the stack of G-bundles.
Is it true that a certain open subset $Bun_{B,r}$ maps smoothly to $Bun_{G}$?
My question comes ...
- 3,472
1
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1
answer
347
views
Reference on elements of finite order in principal congruence subgroups of symplectic groups
We should start with the definition of the symplectic group for an arbitrary ring $R$.
The symplectic group $Sp(g,R)$ is the subgroup of $SL(2g,R)$ such that all elements satisfy $M=J_g^t M J_g$ with $...
- 85
1
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1
answer
331
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Questions about multiplicative homomorphism of $\mathbb{R}$
Regard $K=\mathbb{R}-\lbrace{0\rbrace}$ as a multiplication group. Let $f:K\to K$ be a multiplication homormorphism.
Question 1. Whether that $f$ is surjective implies that $f$ is injective?
...
- 435
1
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2
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350
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Finding lectures PDF "Four lectures on simple groups and singularities"
I would be very interested to find the PDF "Four lectures on simple groups and singularities" by Peter Slodowy, especially the lecture 4. I used to print them but lost it. Does anyone has ...
1
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1
answer
165
views
Character group functor of an exact sequence of algebraic groups
Let $k$ be a number field and $\mathbb{G}_m$ be the multiplicative group sheaf. For an algebraic group $G$, we define the character group $\widehat{G}:= \mathrm{Hom}_{\bar{k}}(\bar{G},\mathbb{G}_{m,\...
- 895
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1
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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 ...
- 413
1
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1
answer
185
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General centralizer of algebraic group
Perhaps there is a simple answer, but I'm very puzzled by the following question:
Question: Does there exist a (smooth, connected) algebraic group $G$ such that the general centralizer (i.e. the ...
- 507
1
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1
answer
383
views
$ S_4 $ subgroups and $ \operatorname{SO}_3(\mathbb{R}) $
$\DeclareMathOperator\SO{SO}$I posted this on MSE 10 days ago and it got 3 upvotes but no answers or comments, so I'm cross-posting to MO.
Background: The group of rotations $ \SO_3(\mathbb{R}) $ has ...
- 4,081
1
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1
answer
258
views
Picard groups of determinantal varieties
Consider a general $4\times 4$ matrix:
$$
X:=\left(
\begin{array}{cccc}
X_0 & X_1 & X_2 & X_3 \\
X_4 & X_5 & X_6 & X_7 \\
X_8 & X_9 & X_{10} & X_{11} \\
X_{12} &...
user124234
1
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1
answer
154
views
Lifting Lang-Steinberg to DVR's in Characteristic 0
Let $A$ be a compact DVR in characteristic $0$, uniformizer $\pi$ and residue field $k$. Let $A\subset B$ be a complete DVR with the same uniformizer $\pi$ and algebraicly closed residue field $F$. ...
- 273
1
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1
answer
280
views
Norms of elements in Artin-Schreier extensions
The following is claimed in the proof of Theorem 7.5 of Auslander, Goldman, "The Brauer group of a commutative ring":
Let $k$ be a nonperfect field of positive characteristic $p$, let $K := k(x)$ ...
- 428
1
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1
answer
437
views
Zariski-closed subgroups of ${\mathbf G}_{\mathbf a}^n$
Let's work over an algebraically closed field $K$. A $1$-dimensional Zariski-closed connected subgroup of ${\mathbf G}_{\mathbf a}^n$ is isomorphic to ${\mathbf G}_{\mathbf a}^1$. If $K$ has ...
- 1,555
1
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1
answer
645
views
Is a semisimple conjugacy class closed?
Let $G$ be any algebraic subgroup of $\mathrm{GL}_n$ over an algebraically closed field of any characteristic.
If $s$ is a semisimple element of $G$, can the $G$-conjugacy class of $s$ fail to be ...
1
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1
answer
450
views
Equivariant fibre product
Let $G$ be an algebraic group. Let $X$ and $Y$ be $S$-schemes such that $X$, $Y$ and $S$ are $G$-schemes and the structural morphisms are equivariant. My question is: Can the fiber product $P:=X\...
- 4,902
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1
answer
2k
views
About isomorphism of $PGL(2)$ and $SO(3)$ [closed]
I need to prove that $PGL_2(\mathbb{R})\cong SO_3(\mathbb{R})$. Abstract considerations show that both can be identified with the group of projective motions of a conic curve. But maybe there is more ...
- 125
1
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1
answer
700
views
CM liftings of abelian varieties and liftings of Frobenius
It is well-known that if $A$ is an ordinary abelian variety over a finite perfect field $ k$ of characteristic $ p>0$ and $ W=W(k)$ is the ring of Witt vectors over $ k$, then the canonical ...
- 395
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1
answer
405
views
Does the semi-stable set determine the linearization of a GIT quotient?
Suppose I have a morphism $f:X\to Y$ which is a GIT quotient of $X$ with respect to some reductive, linear group.
Does the semistable $X^{ss}$ and stable locus $X^s\subset X$ determine completely the ...
- 3,779
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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 $\overline{O_x}$ ...
1
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1
answer
282
views
Spectral decomposition of parabolic induced for GL2(Zp)
Let $F$ be a number field and let $o$ be its ring of integers. Let $o_p$ resp. $F_p$ be the completion at a prime ideal $p$ in $o$. Let $B$ be the group of upper triangular matrices in $GL_2$. Let $\...
- 11.2k
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2
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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 ${\...
- 16.9k
1
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1
answer
212
views
Lie algebras and pulled back group schemes
Suppose I have an extension of fields $L/K$, a group scheme $G_K$ over $\operatorname {Spec} K$. Let $G_L$ denote the pullback of $G_K$ to $\operatorname{Spec} L$. Then, under what conditions on the ...
- 13
1
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1
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88
views
Generic finite subgroups, associated to small finite fields, of reductive algebraic groups
Theorem 1 of [LS] Liebeck and Seitz - On the subgroup structure of exceptional groups says:
Let $X = X(q)$ be a quasisimple group of Lie type in characteristic $p$, and suppose that $X < G$, where ...
- 13k
1
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1
answer
128
views
Koszul complex of equations defining a stabilizer
Very specific question. We work over $\mathbb{C}$, although really just want alg. closed of char. 0.
Suppose that $G$ is an algebraic group and $V$ is a finite-dimensional $G$-module, meaning that we ...
- 1,077
1
vote
1
answer
211
views
Notation for the restriction map in Galois cohomology
My coauthors and I are writing a paper based on MO questions and answers:
Friedrich Knop's answer,
my answer 1
and
my answer 2.
For a linear algebraic group $G$ over a perfect field $k$, I consider a ...
- 14.2k
1
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1
answer
277
views
Linear subspaces under the action of $PGL_n$
A pair of ordered collections of linear subspaces $\Lambda_1, \ldots, \Lambda_k$ and $\Lambda'_1, \ldots, \Lambda'_k$ of $\mathbb{P}^n$ are called projectively equivalent if there exists a regular ...
- 197
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1
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343
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Constructing groups of Type E^{66}_{7,1} having non trivial Tits algebra
This can be considered as a continuation of my last useful question:
Constructing groups of Type E7 with certain Tits Index
It is known that a quadratic form $q$ of dimension $12$, having splitting ...
- 1,479
1
vote
1
answer
147
views
Maximal split torus of universal chevalley group
Let $G$ be simply connected chevalley group over a field $K$. I am following the notations as in 'Lectures on Chevalley group' by Steinberg (Yale lectures). Let $H$ be the subgroup generated by $\{h_{\...
- 685
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1
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243
views
Universal Chevalley group associated to $D_l$
Consider the simple Lie algebra $D_l$. Consider the universal Chevalley group $G$ over a field $K$ associated to it. Then $G$ is a subgroup of the orthogonal group $O_{2l}(K, f)$ where $f$ is the ...
- 11
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1
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543
views
Fixed points of group action
Let us consider the group $PGL(2,\mathbb{R})$ as the group of automorphisms of real projective line and $H\subset PGL(2,\mathbb{R})$ is a subgroup of prime order $> 2$. Is it true that there always ...
- 13
1
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1
answer
509
views
Extension of unipotent algebraic groups
Let $G$ be an algebraic group with closed normal subgroup $N$. Suppose that $N$ and $G/N$ are both unipotent. Does it imply that $G$ is also unipotent?
- 97
1
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1
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234
views
Describing a matrix group (with integer coefficients) through conditions on the coefficients.
I'm wondering if there's always a (not too complicated?) way to characterize a matrix group by conditions on the coefficients.
I know if I'm dealing with matrix groups over a field, then it's sort of ...
- 10.7k
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1
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660
views
Centralizer of elliptic elements in $GL(2)$
Consider a global field $F$ and the group $\Gamma =GL(2,F)$. An element $\gamma \in \Gamma$ is called elliptic, if its eigenvalues do not lie in $F$. Now consider a completion $F_v$ of $F$ and $G_v = ...
- 11.2k
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1
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434
views
Tori acting on vector spaces
Let $T$ be a torus defined over a field $K$ of characteristic $p>0$. Suppose that $T$ acts (algebraically) on some vector space $V$ (over the same field $K$). Let $W$ be a subspace of $V$. Now ...
- 11.2k
1
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1
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493
views
Rational points
Let $G$ be an affine algebraic group defined over a field of characteristic zero $K$. Suppose $G$ has only one single $K$-point, can we conclude that $G$ does not have more points?
- 143
1
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1
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435
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Definition of congruence subgroup for non-matrix groups
For an algebraic group that cannot be embedded into $GL_n$, is there a nice definition for congruence subgroup? Do we just define it as the compact open subgroup of $G(A_f)$, where $A_f$ is the finite ...
- 1,091
1
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1
answer
820
views
Comparing Iwahori Decompositions
Let G be a p-adic group, U a (n appropriate) unipotent subgroup and I an Iwahori subgroup. Then there are Iwahori decompositions I\G/I=U\G/I=W where W is the affine Weyl group. I suspect that
$$...
- 8,866
1
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2
answers
197
views
What are the finite étale coverings of a quasi-hyperelliptic surface?
Let $X$ be a quasi-hyperelliptic surface in characteristic 3 where the canonical bundle $K_X$ is trivial.
Question: Is there a finite étale covering $Y \rightarrow X$ such that
$Y$ is an abelian ...
- 9,501
1
vote
1
answer
293
views
Subgroups of $\operatorname{PGL}_n$
As algebraic groups over an algebraically closed field $K$ of characteristic not $2$, $\operatorname{GO}_{2n}$ is a closed normal subgroup of the conformal orthogonal group $\operatorname{CO}_{2n}$. ...
- 501
1
vote
1
answer
109
views
Inclusion of flag varieties and Schubert decomposition
$\newcommand\Fl{\mathrm{Fl}}$Let $G$ be a connected, reductive algebraic group over $\mathbb{C}$. Fix a maximal torus $T$ and Borel subgroup $B$. Let $L$ be a generalized Levi ($L = Z_G(s)^\circ$, for ...
- 57
1
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1
answer
158
views
Free closed group action on varieties
Suppose we are given a reductive group $G$, its closed subgroup $H$ (not necessarily reductive), an affine $G$-variety $X$ and its closed subvariety $Y$ such that
(1) The $G$ action on $X$ is free and ...
- 55
1
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1
answer
222
views
Zariski density preserved under $p$-adic completion?
Let $G$ be an almost simple group defined over $\mathbb{Q}$. Assume that $\Gamma$ is a subgroup of $G(\mathbb{Q})$ which is Zariski dense. Consider now the $p$-adic completion $\mathbb{Q}_p$ of $\...
- 527