# Questions tagged [group-schemes]

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148
questions

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votes

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149 views

### Some basic questions on quotient of group schemes

Let $S$ be a fixed base scheme and $G, H$ be group schemes over $S$. Since I am mainly interested in commutative group schemes over fields, we may assume that $G,H$ are commutative and $S$ is a field ...

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90 views

### Discriminant ideal in a member of Barsotti-Tate Group

Let $S = \operatorname{Spec} R$ an affine scheme (in our case latter a complete dvr) and $p$ a prime. Then Barsotti-Tate group or $p$-divisible group $G$ of height $h$ over $S$
is an inductive system
...

**1**

vote

**1**answer

232 views

### Viewing a finite group as a group scheme

I've read books which have this statement, without explanation : 'every finite group is an algebraic group'. I'm trying to understand what exactly they mean. The definition I have in my mind of a ...

**3**

votes

**1**answer

210 views

### Locally free group scheme étale

Let $R$ be a commutative ring, $p >0$ prime and $G$ a finite, locally free group scheme over $R$ of rank $p^n$; $n \in \mathbb{N}_{\ge 1}$. Assume $p \in R^*$ (i.e. is a unit in $R$).
Question: ...

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118 views

### Confusion about the classification of isotrivial group schemes

in SGA 3, exposé X, we find the following classification result (corollaire 1.2):
Let $S$ be a connected scheme and let $\xi:\mathrm{Spec}(\Omega) \to S$ be a geometric point. Let $\pi=\pi_1(S,\xi)$ ...

**4**

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184 views

### An application of Grothendieck's version of Hensel's Lemma

Assume $R$ is a Henselian local ring with $k=R/m_R$ ($m_R$ is the unique maximal ideal of $R$) and $G$ a finite flat group scheme over $R$. We denote by $G_k= G \otimes_R k$ the closed fiber.
There ...

**4**

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86 views

### Finite locally free group scheme killed by its order?

When I saw this paper of René Schoof, there are two questions on the first page and what confuses me is that how to reduce the first question to the second. This is expalined in the first paragraph on ...

**3**

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**2**answers

278 views

### Are open $\mathbb{G}_m$-invariant subschemes of an affine scheme precisely the homogeneous radical ideals of its coordinate ring?

This question is certainly not research level and in fact quite elementary which is why I asked it on math.stackexchange before: math.stackexchange. However it doesn't seem to get much attention there ...

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212 views

### Global functions on a product of schemes over artinian ring

For a morphism of schemes $f:X\to S$ with $S=\text{Spec}(R)$ affine, let's write $A(X)=H^0(X,\mathcal{O}_X)$. I'm interested in the morphism of $R$-algebras
$$
c:A(X)\otimes_R A(Y)\to A(X\times_SY)
$$
...

**1**

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145 views

### J. Tate's article on $p$-divisible groups

I have a question about a conclusion that J. Tate arrived at in his paper $p$-Divisible Groups (in: Springer T.A. (eds) Proceedings of a Conference on Local Fields. Springer, Berlin, Heidelberg (1967) ...

**1**

vote

**1**answer

191 views

### Property of representations of reductive group schemes over characteristic 0 field

I originally posted this on Maths SE, but then I thought it MO might be more fitting.
Let $k$ be a characteristic $0$ field and let $G$ be a linear algebraic group scheme over $k$. Then is it true ...

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159 views

### Finite Flat Group Scheme over a field $k$ of characteristic $0$ is always Etale

I have a question about a claim about classification of finite flat group schemes: https://en.wikipedia.org/wiki/Group_scheme#Finite_flat_group_schemes
A fin flat group scheme $G$ is of type $(a,b)$...

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134 views

### Is the Lie algebra of a flat group scheme still flat?

Let $\mathcal{O}$ be a discrete valuation ring, and $\mathcal{G}$ is a flat group scheme over $\mathcal{O}$, we may assume the generic fiber is reductive. Then can we define its Lie algebra, and ...

**5**

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236 views

### motivations of classifying $p$-divisible groups

Let $k$ be a perfect field of characteristic $p>0$ and $W:=W(k)$ is the witt ring. Let $K$ be a totally ramified extension of $K_0:=W(\frac{1}{p})$ and $\Lambda:=W[[u]]$ is the formal series ring ...

**2**

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201 views

### Étale group scheme exact sequence

Consider the exact sequence of finite flat group schemes over the $2$-adic integers ring $\mathbb{Z}_2$:
$$0\longrightarrow\mathbb{Z}/2\mathbb{Z}\longrightarrow A\longrightarrow\mathbb{Z}/2\mathbb{Z}\...

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142 views

### Standard application of Oort-Tate classification theorem

$\DeclareMathOperator{\Spec}{Spec}
\DeclareMathOperator{\tors}{tors}$In Mazur's paper “Modular curves and the Eisenstein ideals”, on the bottom of page 159, it says that if $T$ is a open subscheme of $...

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138 views

### Can one embed a group scheme into a locally constant one such that the quotient exists

Let $S$ be a good enough base scheme (say of finite type over an algebraic closed field) and $G\to S$ be a flat group scheme. I want to ask: can we always find a closed embedding $G\to H$ into another ...

**1**

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**1**answer

164 views

### P-torsion of elliptic curves

Suppose I have an ordinary elliptic curve $E$ over $\overline{\mathbb{F}}_p$. Then its $p$-torsion $E[p]$ is a finite flat group scheme of order $p^2$. My understanding is that it has $p+1$ subgroups ...

**2**

votes

**1**answer

167 views

### fppf-extension of algebraic groups is an algebraic group

The problem is the following:
Let $k$ be a field and let $N,G,Q: (\operatorname{Sch}/k)_{\operatorname{fppf}} \longrightarrow \operatorname{Grps}$
be fppf-sheaves from the big fppf-site of $\...

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vote

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60 views

### Derived subgroup of the connected component of an algebraic group

Let $G$ be an affine group variety (smooth) over a field $k$, let $G^0$ be the connected component of $G$.
Is it true that the derived subgroup $[G^0,G^0]$ (Or $[G,G^0]$) is the connected component ...

**1**

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**1**answer

142 views

### Maps to additive group scheme

Let $\underline{\mathbb{Q}_p/\mathbb{Z}_p}$ be constant p-divisible group over $\mathbb{F}_p$. And let $\mathbb{G}_a$ be the additive group over $\mathbb{F}_p$. Let me prove
$$
Hom(\underline{\mathbb{...

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224 views

### Tangent Space of Picard Scheme

Let $X$ be a scheme and it's Picard scheme $\underline{Pic}(X)$ exist. Denote by $\underline{Pic}^0(X)$ the connected component of the origin of
the Picard scheme. My question is what the geometric ...

**9**

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354 views

### Classification of finite flat group schemes over integers?

One can classify (commutative) finite flat group schemes (with order of $p$-powers) over $\mathbb Z_p$ using semi-linear algebraic datas such as Breuil-Kisin modules. And we can fix the special fiber ...

**7**

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**1**answer

337 views

### Lie Algebra of Automorphism Group of $\mathbb{P}_k^1$

Let $X$ be a scheme over an algebraically closed field $k$ and let $\operatorname{Aut}(X)$ denote the functor sending a $k$-scheme $T$ to the group $\operatorname{Aut}_T(X \times_k T)$ of ...

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87 views

### Representability of a subfunctor of the functor of points of a group scheme

Let $G$ be a group scheme over a scheme $S$ and $h_G:\rm{Sch}/S\longrightarrow \rm{Grp}$ the functor of points represented by $G$.
Let $k$ be a subfunctor of $h$. Is $k$ representable? If so, can we ...

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vote

**2**answers

318 views

### Motivating the Quotient of an Algebraic Variety

Let $X$ be a variety with a $G$-action by an algebraic group.
My question refers to a motivating example from:
https://web.maths.unsw.edu.au/~danielch/thesis/mbrassil.pdf
Here is the relevant ...

**2**

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77 views

### component group of Neron models

Let $A$ be an abelian variety over a discretely valued field $K$ and $\mathcal A$ its Neron model over $R$ (the ring of integers of $K$) and $\mathcal A^0$ the identity component of $\mathcal A$.
...

**3**

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209 views

### Given a semi-abelian scheme, is the set of points such that the fibres are abelian varities open?

Let $\pi:\mathcal{A}\rightarrow C$ be a semi-abelian scheme, i.e. $\mathcal{A}$ is a smooth separated commutative group scheme over $C$ via $\pi$ with geometrically connected fibres, such that each ...

**7**

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138 views

### 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**

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215 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 $\mathbb{A}^1$. ...

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174 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 $\...

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110 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**

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363 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

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275 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 ...

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140 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**

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**1**answer

870 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**

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80 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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285 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 ...

**11**

votes

**1**answer

147 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

446 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{...

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91 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\...

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147 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 \...

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173 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**

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**1**answer

429 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

376 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**

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76 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

170 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 ...

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261 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**

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297 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

232 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 $\pi$ ...