# Questions tagged [group-schemes]

The group-schemes tag has no usage guidance.

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### Are maximal tori conjugate Zariski-locally?

Let $S$ be a scheme and let $G\to S$ be a reductive group scheme. Then $G$ admits a maximal torus etale-locally, and any two maximal tori are conjugate etale-locally, by Theorem 3.2.6 and Corollary 3....

**3**

votes

**1**answer

175 views

### Extend Group Action of $\mathbb{A}^1 /G$ to Projective Line

My question refers to an argument used in Torsten Ekedahl's paper: https://arxiv.org/abs/0903.3148
in Example ii) (page 8):
We consider a finite subgroup of affine transformation of $\mathbb{A}^1$. ...

**1**

vote

**0**answers

149 views

### Computing hom sheaf

Let $\mu_{m}$ and $\mu_{p}$ be the group scheme representing the roots of unity over a field $k$ of characteristic $p>0$. Assume that $p$ divides $m$.
What does one compute the sheaf $\...

**1**

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**0**answers

95 views

### first infinitesimal sheaf and canonical sheaf

Please prove this problem for me.
Let $S$ be a scheme, $G$ be a group $S$-scheme and $\omega_{G} = e^{*}\Omega^{1}_{G/S} $ ($e: S \to G$ identity section).
If the first infinitesimal neighborhood of ...

**5**

votes

**1**answer

353 views

### What is $\mathrm{O}_q/\mathrm{SO}_q$ if $q$ is a quadratic $\mathbb{Z}$-form which is degenerate?

Any binary quadratic $\mathbb{Z}$-form $q$ induces a symmetric bilinear form
$$ B_q(u,v) = q(u+v) - q(u) -q(v) \ \ \forall u,v \ \in\mathbb{Z}^2 $$
and it is considered non-degenerate (over $\mathbb{...

**3**

votes

**2**answers

176 views

### Representability of Hom of two finite flat group schemes

I am reading a note at Page 63
ftp://ftp.math.ethz.ch/users/pink/FGS/CompleteNotes.pdf
It says whenever $G$ is finite and flat over $S$ the functor ${\rm Hom}(G,H)$ is representable. But it does not ...

**1**

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126 views

### Group scheme representation from action of a group scheme on a variety?

Let f be some homogenous polynomial of d.
Let $X = \operatorname{Proj} (k[x,y,z]/(f))$
where $k$ is algebraically closed field of characteristic $p>0$.
Now $G$ is a group scheme acting on $X$.
...

**26**

votes

**1**answer

776 views

### Serre's remark on group algebras and related questions

I've recently heard about an idea of Serre that for each finite group $G$ there exists a group scheme $X$ such that for each field $K$ the group $X(K)$ is naturally isomorphic to the unit group of $K[...

**2**

votes

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69 views

### Classical reductive group schemes vs. unitary groups of separable algebras with involution — reference request

Let $K$ be a field with $2\in K^\times$, let $A$ be a separable $K$-algebra (i.e. $A$ is finite-dimensional, semisimple and its center is an etale $K$-algebra), and let $\sigma:A\to A$ be a $K$-...

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193 views

### Ext group for commutative finite group schemes

EDIT: all group schemes are commutative. Thanks @R. van Dobben de Bruyn!
Could anyone provide a reference request about extensions of finite group schemes / Ext groups.
As far as I know the category ...

**9**

votes

**0**answers

107 views

### Can elements in the orthogonal group of a non-split Azumaya algebra with an orthogonal involution have reduced norm -1?

Let $R$ be a connected (commutative) ring with $2\in R^\times$.
Let $A$ be an Azumaya algebra over $R$ and let $\sigma:A\to A$ be an orthogonal involution. (This means that there is a faithfully flat ...

**7**

votes

**1**answer

325 views

### Exponential map of a Formal Group Scheme

Let $k$ be a field of characteristic $0$ and let $\mathfrak{g}$ be a finite dimensional Lie algebra over $k$. $\mathfrak{g}$ corresponds to a formal group scheme $\mathcal{G} = \text{Spf} (U(\mathfrak{...

**2**

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71 views

### Simply-connected covers of group schemes

Let $\mathbf G$ be a semisimple group scheme over a base $S$. I believe that one can always find a fiberwise simply-connected group scheme $\mathbf G^{sc}$ and a central isogeny $\mathbf G^{sc}\to\...

**2**

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110 views

### Representabillity of torsors and exact sequence of group schemes

I've seen the following property used at many places, I don't exactly why it is true.
Let $S$ be a site with an initial object $e$. There is an exact sequence of group objects
$$1 \to F \to G \to H \...

**2**

votes

**0**answers

116 views

### Quotient of a sheaf by group action and representabillity

Let $X$ be a scheme and $S$ be a sheaf of sets over the fppf topology of $X$. Let $G$ be a group scheme over $X$ and there is an action of $G$ on $S$. Now, I want to look at the quotient $G \setminus ...

**4**

votes

**1**answer

405 views

### What exactly is $\underset{n}{\varprojlim} \ \mu_{p^n}$?

Let $k$ be a field of characteristic $p> 0$.
Let $\mu_{p^n}$ denote the group scheme $\mu_{p^n}(R) =\{ x \in R: x^{p^n}=1 \}$. Then there are natural maps from $\mu_{p^n}$ to $\mu_{p^{n-1}}$. I'm ...

**7**

votes

**1**answer

340 views

### Example of a connected finite group scheme which is not solvable

What would be an example of a connected finite group scheme over a field $k$ that is not solvable? Here $k$ is algebraically closed.
Let $\operatorname{GL}_n$ be the general linear group scheme over ...

**2**

votes

**0**answers

72 views

### Isomorphism between Group Schemes over $\mathbb{Z}_2$

Consider the Matrix
$$M:= \left(\begin{matrix}
1 & 0 \\
0 & d
\end{matrix}\right)
$$
for an odd $d \in \mathbb{Z}$.
Define the group scheme $Sp(g)$ defined over $\mathbb{Z}$ with ...

**2**

votes

**1**answer

153 views

### How can we view the local Weil group as a group scheme over $\mathbf Q$?

Let $F$ be a $p$-adic local field with residue field $\kappa$, and $q = |\kappa|$. Then the residue field $\overline{\kappa}$ of $F^{\textrm{ur}}$ is an algebraic closure of $\kappa$. The local ...

**11**

votes

**0**answers

206 views

### Is the quotient of two linear group schemes linear?

Let $S$ be an affine scheme.
Call a group scheme $G\to S$ linear if there exists an $S$-group morphism $G\to \mathrm{GL}_{n,S}$ with trivial kernel.
Assuming this, suppose $H\to S$ is a central closed ...

**3**

votes

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234 views

### “Frobenius Descent”

The following proposition is there in Pink's lecture notes on finite group schemes.
Let $k$ be an algebraically closed field of characteristic $p$. The category of finite length $W(k)$-modules $N$ ...

**3**

votes

**1**answer

174 views

### Torsors of pushforward group schemes

I'm reading William Waterhouse's "Discriminants of etale algebras and related structures", and he makes a basic claim I'm struggling to justify.
Suppose $S/R$ is etale of rank $n$... and let $\pi$ ...

**3**

votes

**1**answer

394 views

### Representation of a group scheme

Let $\mathcal{G}$ be a affine algebraic group scheme(may not be reductive) over a scheme $S$. How to define a rational representaion of $\mathcal{G}$ (over $S$)? Is there always a faithful ...

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117 views

### Convolution of $\ell$-adic sheaves and group homomorphisms

This question follows this one , where I defined convolution of $\ell$-adic/perverse sheaves.
Here I am working with a perfect field $k$ ($char(k)\neq l$) and with a smooth separated groupscheme $G$ ...

**2**

votes

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191 views

### When is a quotient by a strictly free group action étale?

Let $S$ be a base scheme and $G$ an $S$-group scheme acting on an $S$-scheme $X$ and assume that the quotient scheme $Y := X/G$ exists. (This is for example the case if $S$ is finite locally free and ...

**2**

votes

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131 views

### understanding full set of sections as in katz-mazur

I was reading this question, specifically Brian's answer. In particular I am having a bit of trouble digesting the following sentence:
Being a "full set of sections" of $Z/S$ is something which is ...

**2**

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96 views

### G is p-divisible, about the affine rings of G[p]

Let $R$ be a ring and $G$ a $p$-divisible group over $R$. Since $G[p]$ is finite flat over $\text{Spec}(R)$, it is an affine (group) scheme, say $\text{Spec}(B)$. What can be said about $B$ beyond the ...

**3**

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185 views

### Schematic closure of maximal torus over a discrete valuation ring: smoothness of the special fibre

Let $R$ be a complete DVR (or just Henselian) with field of fractions
$F$ and algebraically closed residue field $k$ (of characteristic $p>0$). Let $G$ be a reductive group scheme over $R$ and let $...

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163 views

### When is the character group scheme of a group scheme representable? (Affine Case)

While reading Tate's article on Finite Flat Group Schemes in "Modular Forms and Fermat's Last Theorem" I was lead to this question. Let $S$ be a scheme, $G$ a group scheme over $S$, and $T$ an $S$-...

**2**

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40 views

### Special unitary group of an affine algebra is integral

Let $R$ be an affine $\mathbb C-$algebra with a linear involution $x\rightarrow \bar x=\iota(x)$, let $S=R/\iota$ and $\psi:R^n\times R^n\rightarrow R$ be an $R/S-$hermitian form. Finaly let $$SU_n(R)=...

**2**

votes

**1**answer

270 views

### Lie algebra of affine group scheme

Let $G$ be an affine group scheme over a characteristic zero field $k$ (the case I have in mind is $k=\mathbb{Q}$). Let $H$ be a subgroup of $G$ which satisfies $\mathrm{Lie}(H)=\mathrm{Lie}(G)$. Is ...

**9**

votes

**1**answer

358 views

### Is $G$ always the automorphism group of the trivial $G$-torsor?

If $G$ is just an ordinary set-theoretic group, then the answer to the question in the title is yes: the automorphisms of $G$ as a (left) $G$-set are all of the form "multiply (on the right) by an ...

**2**

votes

**1**answer

300 views

### Birational Group Law

Let $S$ be a scheme and $X$ a smooth separated faithfully flat over $S$.
An $S$-birational group law on $X$ is an $S$-rational map
$$m:X\times_S X\dashrightarrow X, (x,y)\mapsto xy$$
such that
...

**2**

votes

**1**answer

222 views

### Derived Group Scheme

If $G$ is a $k-$ group scheme (seeing as a functor) exist a good definition of what is the derived group scheme? (Or a good reference for a good definition). Where derived I'm talking in the sense of ...

**3**

votes

**2**answers

319 views

### Do all reductive group schemes over semilocal rings admit finite-dimensional free faithful representations?

The definition of a reductive group scheme is as in SGA III. Frankly, I only know that they exist for the adjoint group (the adjoint representation). In SGA III, I could only find a result for general ...

**4**

votes

**1**answer

185 views

### Flatness of Weil restriction

Let $X\rightarrow Y$ a ramified double cover of smooth projective curves, and let $$\mathcal G:=Res_{X/Y}(SL_n)$$
be the Weil restriction of the constant group scheme $SL_n$ over $X$.
Question: Is ...

**3**

votes

**1**answer

130 views

### Maximality of connected components of finite flat group schemes

Let $k$ be a perfect field of characteristic $p>0$. Let $K$ be a finite, totally ramified extension of $K_0:=\mathrm{Frac}\ W(k)$ and let $\mathcal{O}_K$ be the ring of integers of $K$. All group ...

**3**

votes

**1**answer

213 views

### Chevalley devissage

Let $G$ be an algebraic group over a perfect field $k$. Then it is know that it can be written as an extension of an affine algebraic group and a proper algebraic group.
Is there a similar result for ...

**5**

votes

**1**answer

458 views

### Lie algebra and base change

I am very confused by the following and would appreciate any help.
Let $\mu_p \subset \mathbb{G}_m$ be the $p$-torsion subgroup scheme of the multiplicative group over $\mathbb{Z}_p$. I would like to ...

**3**

votes

**2**answers

187 views

### Reference request: groups of multiplicative type are closed under extensions

I remember reading (quite a while ago, and I can't remember where!) that linear algebraic groups of multiplicative type over a field of characteristic zero are closed under extensions. This is ...

**9**

votes

**1**answer

577 views

### Is there a non-smooth algebraic group scheme in char $p$, all of whose defining relations have degree less than $p$?

Let $k$ be an algebraically closed field of characteristic $p>0$.
All the examples of non-smooth algebraic group schemes over $k$ that
I have seen (apart from "artificial" examples; see below) have ...

**3**

votes

**1**answer

134 views

### Approximation of Group Schemes over valuation rings

In his paper Approximation des schémas en groupes, quasi compacts sur un corps, Daniel Perrin shows that every quasi compact group scheme over a field then it is an inverse limit of group schemes of ...

**2**

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182 views

### From algebraic group actions to group scheme actions

I am trying to understand the basic results of geometric invariant theory. I want to pull off the band aid and use Mumford, but am a neophyte with respect to scheme theory. Thus, I have been trying to ...

**2**

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89 views

### split tori over local fields

Let $F$ be a non-archimedean local field, and $\mathscr O$ its ring of integers. Suppose $T$ is an $F$-split torus, i.e., $T = (\mathbb G_m)^r$ where $\mathbb G_m$ denotes the multiplicative group. ...

**4**

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**1**answer

246 views

### Normal Hopf ideals and Hopf subalgebras, quotient group schemes

There is a following theorem:
$H$ is a commutative Hopf algebra over a field $k$. Then there exists a bijective correspondence between
$$\{ \textrm{Hopf subalgebras }K\subset H \} \quad \...

**0**

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99 views

### Lifting points of étale group scheme

Consider the following setting: let $(R,\mathfrak{m})$ be a Noetherian complete discrete valuation ring (with maximal ideal $\mathfrak{m}$) and let $K$ be its field of fractions. Now take $L$ be the ...

**8**

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**1**answer

483 views

### Group schemes, adeles, double cosets, and étale cohomology

Let $K$ be a number field, $R$ the ring of integers of $K$,
${\mathbf{A}^f}$ the ring finite adeles of $K$, and ${\widehat{R}}\subset {\mathbf{A}^f}$ the ring of integral adeles.
Let $G$ be an affine ...

**13**

votes

**2**answers

478 views

### Infinitesimal deformations of the formal group of $\mathbb{G}_m$

For a commutative ring $R$, consider the formal group $\widehat{\mathbb{G}}_m$ over $R$ that is the completion of $\mathbb{G}_{m, R}$ along its identity section (naively, $\widehat{\mathbb{G}}_m$ is ...

**1**

vote

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116 views

### Condition for a finite group scheme to be étale [closed]

My question comes from the reading of Tate's paper $p$-divisible groups. In the last few pages there is an argument which gives as trivial the following fact. If we take a $p$-divisible group over a ...

**4**

votes

**0**answers

278 views

### A question about Weil restriction

Let $\pi:\tilde{C}\rightarrow C$ be a ramified cover between two smooth curves. And consider a group scheme $\mathcal G$ over $\tilde{C}$, I have found two definitions for Weil restriction:
$Res_{\...