# Tagged Questions

**4**

votes

**0**answers

159 views

### Tannaka categories and reductive groups

The group associated to a Tannaka category $T$ over a field is pro-reductive if and only if $T$ is semi-simple.
Pro-reductive groups make sense over any scheme.
Is there an extension of the theory ...

**0**

votes

**0**answers

61 views

### Group schemes decomposition

Given an abelian group scheme of finite type $(G,+)$ over $\mathbb{F}$ connected, and given two connected closed subgroup schemes of finite type $G$ over $\mathbb{F}$ connected $H$, $N$ of $G$. ...

**12**

votes

**3**answers

599 views

### Does every reductive group scheme admit a maximal torus?

A theorem of Grothendieck states that any smooth reductive algebraic group over a field $k$ admits a maximal torus over $k$. My question concerns what happens for schemes.
Let $S$ be a scheme and ...

**6**

votes

**1**answer

230 views

### Is $G_{\operatorname{red}}$ normal in $G$?

Let $G$ be an affine group scheme of finite type over a field $k$. It is well known that the associated reduced subscheme $G_{\operatorname{red}}$ of $G$ is a subgroup if $k$ is perfect. So let us ...

**3**

votes

**1**answer

355 views

### Orbits of group scheme action

I am interested in orbits of the action of a group scheme on a scheme and I'm particularly interested in the following special case: Let $k$ be an algebraically closed field, let $G$ be an affine ...

**1**

vote

**1**answer

150 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 ...

**4**

votes

**2**answers

191 views

### Splitting field of a Torus

Let $T$ be a torus over a non-necessary perfect field $k$. Let $\bar k$ an algebraic closure of $k$. Is there a smallest extension $k'$ of $k$ in $\bar k$ such that $T \times_{{\rm spec}\, k} {\rm ...

**5**

votes

**2**answers

250 views

### Quotient of a reductive group by a non-smooth subgroup

This is a continuation of my question Quotient of a reductive group by a non-smooth central finite subgroup.
Let $G$ be a smooth, connected, reductive $k$-group over a field $k$ of characteristic ...

**8**

votes

**1**answer

291 views

### Differential/difference algebraic groups as “group schemes”

While the common approach to algebraic groups is via representable functors, it seems that there is no such for differential algebraic groups (defined by differential polynomials). Neither the book by ...

**2**

votes

**0**answers

358 views

### On the structure of commutative group schemes

The structure of a commutative affine algebraic group $G$ over a field $k$ is understood (SGA 3): $G$ has a maximal subgroup of multiplicative type $M$ and the quotient $G/M$ is unipotent.
I am ...

**2**

votes

**1**answer

424 views

### finite non-commutative local group schemes

Can I have some examples of finite non-commutative connected group schemes over a field $k$?
I would like also to see some non-trivial torsors over a $k$-scheme $X$ under such group schemes. Thanks.
...

**1**

vote

**0**answers

254 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 ...

**6**

votes

**2**answers

574 views

### Character group of Frobenius kernels

Let $G$ be a semisimple algebraic group over an algebraically closed field $k$ of characteristic $p$ (e.g., $G=SL_n(k)$). Then $G$ is equal to its derived subgroup $[G,G]$. Consequently, the character ...

**12**

votes

**2**answers

745 views

### Can a reductive group act non-linearly on a vector group?

Let $k$ be a field; I'm going to discuss linear algebraic groups over $k$. The question I'll pose is only interesting when the characteristic is $p>0$.
1. Some motivation
A vector group is an ...

**10**

votes

**1**answer

775 views

### Are there “reasonable” criteria for existence/non-existence of Levi factors or their conjugacy in prime characteristic?

Classical theorems attributed to Levi, Mal'cev, Harish-Chandra for a finite
dimensional Lie algebra over a field of characteristic 0 state that it has a Levi decomposition (semisimple subalgebra plus ...