All Questions
Tagged with ag.algebraic-geometry algebraic-groups
914
questions
3
votes
1
answer
146
views
Extending Tannakian "dictionary" to gerbes
The following is Proposition 2.21 in Deligne and Milne's "Tannakian Categories".
Let $f: G \to G'$ be a homomorphism of affine group schemes over a field $k$ and let $\omega^f$ be the ...
- 323
5
votes
0
answers
126
views
Classification of visible actions for *reducible* representations?
Is there a classification of the pairs $(G,V)$ such that $G$ is reductive [and connected, if you like], and $G$ acts faithfully and visibly on $V$ - crucially, including all cases where $V$ is ...
- 2,968
6
votes
1
answer
235
views
Extensions of algebraic groups and extensions of fpqc sheaves
There are at least two ways to define $\operatorname{Ext}^p(F,G)$ when $F$ and $G$ are commutative algebraic groups over a field $k$:
Pass to the associated fppf sheaves and use an injective ...
- 7,894
4
votes
0
answers
142
views
Spaces of fixed points
I am reading the paper Space with $\mathbb{G}_{m}$-action, hyperbolic localization and nearby cycles by Timo Richarz and I am having some troubles in understanding the proof of Lemma 1.10.
The setting ...
- 823
8
votes
0
answers
278
views
Has the notion of a unipotent group scheme been studied?
The concept of a unipotent algebraic group over a field has been extensively studied and is fundamental in algebraic geometry. However, has the notion of a unipotent group scheme over a general base ...
- 1,139
2
votes
0
answers
66
views
Irreducibility of Białynicki-Birula cells
Let $X\subset \mathbb{P}^n$ be a smooth complex projective variety, and consider a non-trivial action of $\mathbb{C}^*$ on $X$. For any connected fixed component $Y$ of the fixed point locus, we may ...
2
votes
1
answer
262
views
Prodiscreteness of rational points of algebraic groups
Let $F$ be a field of characteristic 0 complete for a discrete non-archimedean valuation.
Let $G$ be a commutative smooth algebraic group over $F$.
Let us put on $G(F)$ the topology induced by the ...
- 95
3
votes
0
answers
49
views
Algebraicity of the group of equivariant automorphisms of an almost homogeneous variety
The base field is the field of complex numbers. Let $G$ be a connected linear algebraic group. Let $X$ be an almost homogeneous algebraic variety, i.e. $G$ acts on $X$ with a dense open orbit $U \...
- 543
3
votes
0
answers
117
views
Smooth unipotent algebraic groups over $\mathbb A^n$
Let $G\to \mathbb A^n_{\mathbb C}$ be a smooth morphism whose fibers at any point of $\mathbb A^n$ are unipotent groups. Can we conclude that $G\simeq \mathbb A^{n+N}_{\mathbb C}$ for some $N$, as a ...
- 435
0
votes
0
answers
113
views
Induced action on infinitesimal thickenings by an algebraic group
Let $X$ be an irreducible locally noetherian $k$-scheme (for $k$ any field), $G$ an algebraic group acting on $X$ via $a:G \times X \to X$ and $x \in X$ a closed point, which is by Zariski's lemma ...
- 5,780
3
votes
0
answers
209
views
Action of an algebraic group $G$ on a scheme $X$ with fixed rational point
Let $X$ be an irreducible locally noetherian $k$-scheme ($k$ any field) and $G$ an algebraic $k$-group acting on $X$.
Proposition 3.1.6 in these notes by M. Brion claims
Let $a : G \times X \to X$ be ...
- 5,780
3
votes
0
answers
150
views
Centers and conjugacy classes of groups relative to a pair of group homomorphisms
$\newcommand{\defeq}{\mathbin{\overset{\mathrm{def}}{=}}}$Given a group $G$, its center $\mathrm{Z}(G)$ and set of conjugacy classes $\mathrm{Cl}(G)$ are defined by
\begin{align*}
\mathrm{Z}(G) &\...
- 11.5k
4
votes
0
answers
189
views
Questions about the fixed point functor $X^G$ of a $G$-scheme
Let $X$ a (locally Noetherian; but not sure if that's really matter) $k$-scheme, $G$ a $k$-group scheme acting on $X$ via morphism $a:X \times G \to X$.
The fixed point functor of $X$ (where $X$ is ...
- 5,780
1
vote
1
answer
103
views
Torsor of finite presentation and surjectivity of map of $\overline{k}$-valued points
I have a question about the content of remark 2.6.6. (i) (p 18) from M. Brion's notes on structure of algebraic groups.
Let $G$ be a group scheme over certain fixed base field $k$ (as all other ...
- 5,780
2
votes
1
answer
239
views
Normalizer of Levi subgroup
Let $G$ be a reductive group (we can work on an algebraically closed field if needed) and let $L$ be a parabolic subgroup, i.e. the centralizer of a certain torus $T \subseteq G$.
Associated with this ...
- 53
2
votes
0
answers
152
views
GIT quotient and orbifolds
Let $G$ be a connected complex reductive group. Suppose $G$ acts on a smooth complex affine variety $X$. Assume the stabiliser $G_x$ of every point $x\in X$ is finite. Is it true that $X/\!/G$ is an ...
- 2,711
5
votes
1
answer
301
views
Parabolic subgroups of reductive group as stabilizers of flags
$\DeclareMathOperator\GL{GL}$Let $G$ be a linear algebraic group (probably reductive will be needed). Consider a faithful representation $G \to \GL(V)$. Given a parabolic subgroup $P < G$, we can ...
- 497
2
votes
0
answers
143
views
Classifying stack for finite flat group scheme
Let $G$ be a finite flat non-smooth group scheme over an algebraically closed field $k$, for example, $G$ can be $\operatorname{Spec}(\overline{\mathbb{F}}_p[t]/(t^p))$. Then the classifying stack $\...
- 435
4
votes
1
answer
259
views
Fppf or étale extension of group algebraic spaces
Let $S$ be a scheme and let
$$0 \to A \to B \to C \to 0$$
be an exact sequence of abelian sheaves on $(\mathrm{Sch}/S)_\text{fppf}$. Assume that $A$ and $C$ are representable by flat algebraic spaces. ...
- 41
10
votes
0
answers
263
views
Looking for counterexamples: Are maximal tori in the automorphism groups of smooth complex quasiprojective varieties conjugate?
Let $X$ be a smooth quasiprojective variety over $\mathbb{C}$. It has a group of (algebraic) automorphisms $
\DeclareMathOperator{\Aut}{Aut}
\Aut(X)$.
Define a torus in $\Aut(X)$ to be a faithful ...
- 1,111
6
votes
0
answers
269
views
Is every free additive action on the affine space conjugate to a translation?
Is every free action of the additive group $\mathbb{G}_a$ on the affine space $\mathbb{A}^3$ conjugate to a translation?
In characteristic zero, the answer is yes, and is due to Kaliman. [Kaliman, S. &...
- 7,682
4
votes
0
answers
270
views
GIT quotient of a reductive Lie algebra by the maximal torus
Let $G$ be a connected complex reductive group with Lie algebra $\mathfrak{g}$. One knows a lot about the GIT quotient $\mathfrak{g}/\!/G$: the invariant ring is a free polynomial algebra on $\mathrm{...
- 2,711
2
votes
0
answers
235
views
Action of algebraic group in cohomology of equivariant algebraic vector bundle
Let $X$ be a projective algebraic variety over an algebraically closed field. Let an algebraic group $G$ act algebraically on $X$. Let $\mathcal{F}$ be a $G$-equivariant vector bundle (or, more ...
- 21.3k
1
vote
0
answers
126
views
A Weierstrass product theorem for invertible formal Laurent series over local Artinian rings?
Let $(A,\mathfrak{m},\kappa)$ denote a commutative local Artinian ring. Somewhat by accident, I've stumbled across the following interesting decomposition:
$$
A(\!(t)\!)^\times = t^\mathbb{Z} \cdot (1 ...
- 6,996
4
votes
1
answer
254
views
Question regarding the definition of linearization of line bundles
I'm reading Dolgachev's book 'Lectures on invariant theory'. In Chapter 7, the linearization of a group action is discussed. Let $G$ be a linear algebraic group acting on a quasi-projective variety $X$...
- 737
3
votes
0
answers
61
views
Anisotropic kernel of groups of type A
I'm studying the results of classification of reductive groups using Tits index and anisotropic kernel.
It is known that simple groups with Tits index $^1 A_{n,r}^{(d)}$ are of the form $SL_{r+1}(D)$, ...
- 321
2
votes
0
answers
70
views
Arbitrary base change of a parahoric subgroup in split case
Assume $R\subset R'$ are henselien discretly valued rings with fraction field $K$ and $K'$, $G$ is a semisimple split group over $K$. Consider the parahoric group scheme $\mathcal{P}_F$ over $R$ ...
- 271
7
votes
0
answers
134
views
Quasisplit forms of wonderful varieties
I will assume that $k$ is a characteristic $0$ non-archimedean field. A classical result of Tits [T] states that a quasisplit connected reductive group $G$ over $k$ is classified up to strict isogeny ...
- 121
1
vote
0
answers
143
views
Is the functor $\mathrm{Hom}(\mathrm{spec}\,k[x^{1/{p^\infty}}]/(x), -)$ from the category of finite commutative group schemes exact?
Question. Let $B \twoheadrightarrow C$ be a fully faithful homomorphism of finite connected commutative group schemes over a perfect field $k$. Let $T = k[x^{1/p^\infty}]/(x) = \varinjlim k[t]/(t^p)$. ...
- 135
6
votes
1
answer
320
views
Exactness of the Weil restriction functor $\mathrm{Res}_{X/k}$
Question. Let $X$ be an Artinian scheme over a perfect field $k$. Consider the abelian category $\mathcal{C}$ of affine commutative group schemes of finite type. Is the Weil restriction $\mathrm{Res}_{...
- 135