All Questions
2,543 questions
4
votes
0
answers
128
views
Tannakian reconstruction and the distribution algebra
$\DeclareMathOperator\Dist{Dist}\DeclareMathOperator\Lie{Lie}\DeclareMathOperator\Rep{Rep}\DeclareMathOperator\End{End}$Let $G$ be an affine group scheme over a commutative ring $k$ (I am mainly ...
- 3,446
16
votes
0
answers
429
views
Complete resource of Ngô's course notes on Algebraic Groups and Group Schemes
I'm looking for Ngô's M2 course notes on "Groupes algébriques et schémas en groupes".
The Wayback Machine has an incomplete capture here. However, it apparently lacks chapter 1, 3, and 5.
I ...
- 161
3
votes
1
answer
203
views
normalizer quotient is $\operatorname{GL}_2(p)$
Let $p$ be a prime and let $w$ be a primitive $p$-th root of unity in $\mathbb{C}$. There is an element $e$ of order $p$ in $G=\operatorname{PGL}_n(\mathbb{C})$ where $n=pk$ and
$$e=\left[\left(\begin{...
- 501
1
vote
0
answers
86
views
The local structure theorem for spherical varieties under quasi-split group action
I want to understand a simplified version of the general $k$-local structure theorem proved in the paper "Reductive group actions":
For $k$ a characteristic zero algebraically closed field, $...
- 121
1
vote
0
answers
116
views
List of automorphism groups of low-dimensional complex commutative algebras?
Let $\mathcal{A}$ be a finite-dimensional commutative associative unital $\mathbb{C}$-algebra. I am looking for a list (of further examples of) $\operatorname{Aut}_\mathbb{C}(\mathcal{A})$, the group ...
- 7,127
4
votes
1
answer
299
views
Can any pair of associate parabolics be related by opposite parabolics?
Let $G$ be a reductive group, say over an algebraically closed field of characteristic zero.
We have the following definitions for a pair of parabolic subgroups $P_1$ and $P_2$ with Levi quotients $...
1
vote
0
answers
150
views
Relative compactification without resolutions of singularities
Let $Y$ be a smooth proper variety over a field $k$, let $X$ be a smooth variety over $k$, $U\hookrightarrow X$ the complement of a strict normal crossing divisor and $\phi\colon U\to Y$ a map. By ...
- 309
3
votes
0
answers
112
views
What are the possibilities of the general fibres in an Iitaka fibration?
This question is motivated by complex algebraic geometry.
If $X$ is a complex algebraic variety with Kodaira dimension in $[1,\dim X-1]$, then the Iitaka fibration (the rational map induced by the ...
- 9,501
3
votes
0
answers
238
views
Centralizers and algebraic groups
Suppose $G$ is a linear algebraic group - I am also happy to assume $G$ is a simple algebraic group over an algebraically closed field of characteristic zero, but the question won't require this.
The ...
- 359
1
vote
0
answers
120
views
Geometric induction of modules for algebraic groups
Let $\Bbbk$ be an algebraically closed field (of any characteristic). Let $G$ be an algebraic group over $\Bbbk$, and $H$ a closed (hence algebraic) subgroup of $G$.
Let $V$ be a finite-dimensional $...
- 1,077
0
votes
1
answer
88
views
Isogeny of connected linear algebraic group stabilizes Borel subgroup
I am trying to understand a result on algebraic groups, namely that if $\sigma:G\to G$ is an isogeny of a connected linear algebraic group over an algebraically closed field, then $\sigma$ stabilizes ...
- 177
5
votes
1
answer
235
views
Has the determinant of a involution of the first kind ever been considered as an invariant?
$\DeclareMathOperator\Sym{Sym}\DeclareMathOperator\Aut{Aut}\newcommand\id{\mathrm{id}}$Let $k$ be a field of characteristic zero.
Let $A, B$ be central, simple algebras over $k$ of even degrees $n,m &...
- 1,479
2
votes
1
answer
429
views
Representation ring of the general linear group
The ring of representations of the symmetric group is isomorphic to the ring of symmetric functions. The Schur-Weyl duality relates the irreducible representations of the symmetric group and that of ...
- 673
3
votes
0
answers
120
views
Describing the outer automorphism of a special unitary group in terms of the Hermitian form
$\DeclareMathOperator\U{U}\DeclareMathOperator\SU{SU}\DeclareMathOperator\GL{GL}$Let $h$ be a non-degenerate Hermitian form on $\mathbb{C}^n$ with signature $(p,q)$. Let $\U_h$ denote the associated ...
- 7,547
1
vote
0
answers
104
views
Finite groups acting on algebraic groups and representations
Let $H$ be a connected algebraic group over an algebraically closed field $k$, and $I$ a finite group which acts on $H$ through group scheme morphisms. Denote by $Rep(H)$ the category of finite ...
- 11
4
votes
1
answer
435
views
Étale group schemes and specialization
If $A$ is an abelian variety over a finite field $\mathbf{F}_q$, then $A(\mathbf{F}_q)$ (resp. $A(\overline{\mathbf{F}}_q)$) is a finite (resp. infinite torsion) group, but $A(\mathbf{F}_q(t))$ is a ...
user501361
0
votes
0
answers
220
views
The largest value of $k$ for $\mathbb{Z}^k$ to be embedded in $GL(n,\mathbb{Z})$
This is just a question originated from This conversation (commented by Moishe Kohan). I tried to prove those two assertions but I don't know where to start:
If H is a free abelian subgroup of $SL(n, ...
4
votes
0
answers
103
views
Boundary divisor of an equivariant compactification of a non-trivial twist of $\mathbb{G}_a^n$
Let $K$ be a non-perfect field of positive characteristic. Then there exist non-trivial forms of $\mathbb{G}_a^n$ which split over finite purely inseparable extensions of $K$, i.e., algebraic groups $...
3
votes
0
answers
94
views
Dimension of a kernel of a cocycle map
Inspired by a previous question (Dimension of a kernel of a linear map) and some of the answers I was given I thought wheter I can generalize the question to the following:
Compute the kernel (or at ...
- 911
2
votes
2
answers
270
views
Zariski closure of the image of an induced representation
Let $G$ be a finitely generated discrete group, $H\le G$ a subgroup of finite index $d$, and let $\rho : H\rightarrow \operatorname{GL}(n,\mathbb{C})$ be a representation.
Let $\tilde{\rho} := \...
- 7,547
5
votes
1
answer
223
views
Commuting matrices and cyclic modules
Let $A, B\in M_n(\mathbb{C})$ be matrices that commute. We suppose that there exists a vector $v\in\mathbb{C}^{n}$ such that $(\mathbb{C}[A,B]).v$ generates $\mathbb{C}^{n}$. We call such a pair a ...
- 3,472
3
votes
0
answers
122
views
Torsion of Fermat hypersurfaces
An interesting invariant of a rationally chain-connected variety $X/k$ is the exponent of the group,
$$ \ker{(\mathrm{CH}_0(X_K) \xrightarrow{\deg} \mathbb{Z})} $$
where $K = k(X)$ is the function ...
- 3,730
0
votes
1
answer
122
views
Functions on products of tori
Let $T$ be an algebraic torus over an algebraically closed field $k$.
Let $d\in\mathbb{N}^{*}$. For every $d$-tuple of integers $\underline{n}=(n_1,\dotsc, n_d)$ and a function $f\in k[T]$, we can ...
- 3,472
4
votes
2
answers
297
views
Biquadratic extension of global function fields with cyclic decomposition groups
Let $F$ be a global function field, for example $F={\mathbb F}_q(t)$, the field of rational functions in one variable over a finite field ${\mathbb F}_q\,$.
Question. What would be an example of a ...
- 14.2k
2
votes
1
answer
107
views
Rationality of quasi-elementary group actions
I am learning the paper https://arxiv.org/pdf/1604.01005.pdf "Reductive group actions" by Knop and Krotz.
They defined that a linear k-algebraic group $H$ with unipotent radical $H_{u}$ is ...
- 121
2
votes
0
answers
239
views
Do we have $\underline{\operatorname{Ext}}^i_\text{fppf}(\widehat{\mathbb{G}}_a,\mathbb{G}_m)=0$ for $i>0$?
Let $k$ be a characteristic zero field and consider the category $(\mathsf{Sch}/k)_\text{fppf}$ of schemes over $k$ with the fppf topology. I know that $\underline{\operatorname{Hom}}(\mathbb{G}_a,\...
- 773
5
votes
1
answer
512
views
Do we have $\underline{\operatorname{Ext}}^i_\text{fppf}(\mathbb{G}_a,\mathbb{G}_m)=0$ for $i>0$?
Let $k$ be a characteristic zero field and consider the category $(\mathsf{Sch}/k)_\text{fppf}$ of schemes over $k$ with the fppf topology. I know that $\underline{\operatorname{Hom}}(\mathbb{G}_a,\...
- 773
1
vote
0
answers
119
views
Equivariant cohomology with compact support of the generalized Jacobian of a nodal elliptic curve
[Question forwarded from SE for lack of interaction]
Given an geometric genus $1$ curve with $1$ node $C$ (hence with arithmetic genus $2$), its normalization is given by the elliptic curve $\tilde{C}$...
- 33
3
votes
0
answers
87
views
Obstruction for points to be contained in smooth hypersurfaces in tterms of inseparability degree of residue field
Let $k$ be an imperfect field of char $p>0$ and
$x \in \mathbb{P}^n_k$ be closed point of projective space.
In this discussion Qing Liu wrote that
Over an imperfect field, a reduced point can not ...
- 619
1
vote
0
answers
252
views
Reference request: Weil's uniformization theorem
The Weil uniformization theorem says that if $k$ is an algebraically closed field, $G$ a reductive group, and $C$ a curve, we have an isomorphism of stacks $Bun_G(C)\cong G(F_C)\backslash G(\mathbb{A}...
- 1,969
2
votes
0
answers
68
views
On classifying space of normalizer of maximal torus
I am reading Marc Levine's paper 'Motivic Euler characteristics and Witt-valued characteristic classes'. In that paper he considers the $BN_T(SL_n)$, namely the classifying space of the group of ...
- 918
3
votes
0
answers
152
views
Disconnected reductive algebraic groups in Sage
All simply connected split simple groups have been implement on Sage and it is possible to find their highest roots, fundamental weights, Dynkin diagrams or compute the tensor of two of their ...
- 370
1
vote
0
answers
156
views
Software for computing invariant rings
I have an linearly reductive algebraic group $G$ acting regularly on an affine variety $X$(over an algebraically closed field of characteristic 0). I want to compute the invariant ring $\mathbb{K}[X]^{...
- 839
5
votes
1
answer
217
views
Fields of definition of conjugates
Let $k$ be a field, not necessarily algebraically closed, $G$ an affine group scheme over $k$, $H$ a subgroup of $G$, and $N$ a normal subgroup of $H$, none of them assumed to be smooth. Suppose that ...
- 12.9k
2
votes
0
answers
174
views
How are tangent spaces related via geometric quotient?
Let $G$ be a linearly reductive group acting regularly on an irreducible affine variety $X$ (over an algebraically closed field of characteristic zero). Suppose there's a $G$-stable open subvariety $U$...
- 839
3
votes
0
answers
105
views
Points with residue fields having big inseparability degree cannot be contained in smooth hypersurfaces
Let $X$ be a $k$-scheme over imperfect field $k$ and $x \in X $ some (reduced) point with residue field $\kappa(x) = \mathcal{O}_{X,x}/ \mathfrak{m}_x$. How to check that if $\kappa(x)$ has "big ...
- 619
6
votes
0
answers
306
views
Tits construction of algebraic groups of type D₆ and E₇ via C₃
As shown in the Freudenthal magic square, the Tits construction of $D_6$ takes as input
an quaternion algebra and the Jordan algebra of a quaternion algebra (see The Book of Involutions § 41). In ...
- 1,479
2
votes
0
answers
350
views
Galois cohomology of $\breve{\mathbb Q}_p \otimes_{\mathbb Q_p} \breve{\mathbb Q}_p$
Let $\breve{\mathbb Q}_p$ denote the completion of the maximal unramified extension of $\mathbb Q_p$. I‘d like to compute Galois cohomology groups and sets related to $\breve{\mathbb Q}_p \otimes_{\...
- 363
5
votes
2
answers
357
views
Integrating on orbits of algebraic groups
Suppose $G$ is a $\mathbb{Q}$-algebraic group (I am interested in the semisimple case) acting rationally on a vector space $V_\mathbb{Q}$. Let $x \in V_\mathbb{Q}$ be a non-zero rational vector. ...
- 885
11
votes
1
answer
580
views
Do people prefer working on $\mathrm{GSp}$ and $\mathrm{GU}$ rather than $\mathrm{Sp}$ and $\mathrm{U}$, and why?
I am a new learner of Iwasawa theory and currently reading the famous paper by Skinner-Urban in 2014, and the following-up works by many other people. When reading these papers, I found that some ...
- 639
2
votes
1
answer
344
views
Product decomposition into semisimple and unipotent parts of an algebraic group (in Borel’s LAG)
Let $G$ be an algebraic group, i.e., an affine reduced, separated $k$-scheme of finite type with structure of a group. In Borel’s Linear Algebraic Groups Theorem III.10.6(4) says
Theorem 10.6 (3): ...
- 5,986
1
vote
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
11
votes
4
answers
979
views
Is the set of rational points of an (almost) simple algebraic group simple?
Let $G$ be an almost simple algebraic group defined over a field $K$. Then we know that, for $H = G/Z(G)$, the set of rational points $H(\overline{K})$ is a simple group (when considered with the ...
- 20.2k
7
votes
3
answers
1k
views
Has anyone researched additive analogues of toric geometry in characteristic zero?
One definition of an $ n $-dimensional toric variety is that it is a variety $ Z $ for which there exists an equivariant embedding of
$ \mathbb{G}_{m}^{n} $ as a Zariski dense, open sub-variety of $ Z ...
- 912
1
vote
0
answers
98
views
Connected stabilisers for actions of reductive groups
Let $G$ be a connected split reductive group over a field $k$ acting on a variety $X$ over $k$. For each $x\in X$, let $G_x$ be the stabiliser. In general, $G_x$ may be disconnected.
Now suppose $G$ ...
- 2,751
1
vote
0
answers
80
views
When is $Y$ not an orbit closure?
Let $G$ be a linearly reductive algebraic group acting regularly on an affine space over $\mathbb{A}^n$ an algebraically closed field $\mathbb{K}$. Let $Y$ be a $G$-invariant (closed) affine ...
- 839
1
vote
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
2
votes
0
answers
119
views
Resolution of singularities of the resultant locus
We consider projective space of dimension $n$ as the parameter space of degree $n$ polynomials in one variable. Then, I am interested in resolving the singularities of the "resultant locus" $...
- 7,746
1
vote
0
answers
208
views
Is $\langle\chi,\lambda\rangle=0$, whenever the limit exists? Where is the mistake?
Suppose $G$ is a linearly reductive algebraic group acting linearly on a finite dimensional vector space $V$ over $\mathbb{C}$. This induces an action on the coordinate ring $\mathbb{C}[V]$ (see here)....
- 839
2
votes
0
answers
129
views
Getting an equivariant morphism
Let $X\subset\mathbb{A}^n$ be an irreducible affine variety over an algebraically closed field $\mathbb{K}$ of characteristic zero. Suppose we have two linearly reductive algebraic groups $G$, $G'$ ...
- 839