# Questions tagged [group-schemes]

The group-schemes tag has no usage guidance.

153
questions

**-1**

votes

**1**answer

326 views

### Are the prime number objects given by the prime numbers?

There is a way to define the notion of prime number in the framework of group theory, thanks to the following observation:
Observation: A non-trivial group $G$ is cyclic of prime order iff for any ...

**5**

votes

**1**answer

283 views

### Are equivariant perverse sheaves constructible with respect to the orbit stratification?

[Moved here from MSE]
Consider a variety $X$ over a field $k$ (complex numbers is fine) with the action of a group scheme $G$, and a $G$-equivariant perverse sheaf $F$ over $X$.
Question. Is it true ...

**1**

vote

**1**answer

414 views

### Fpqc-locally constant if and only if étale-locally constant?

Also in SE.
Let $\mathcal{F}$ be sheave over $S_\mathrm{fpqc}$. We say $\mathcal{F}$ is a fpqc-locally constant sheaf (of finitely generated abelian groups) if there exists a fpqc covering $(S_i\to S)...

**6**

votes

**1**answer

200 views

### Finite flat group schemes over $\mathbb{Z}$ that are extensions of $\mu_p$ by $\mathbb{Z}/p\mathbb{Z}$

Suppose $X$ is a finite flat group scheme over $\mathbb Z$, killed by a prime number $p$ and such that there exists an extension as finite flat group schemes defined over $\mathbb Z$:
$$0\to \mathbb{Z}...

**1**

vote

**0**answers

115 views

### pro-commutative group schemes

When $k$ is field, Demazure and Gabriel defined and worked with the category of commutative pro-algebraic groups over $k$. In their book, they proved that $Ext^n(\varprojlim G_i, H)= \varinjlim Ext^n(...

**1**

vote

**0**answers

185 views

### Any finite flat commutative group scheme of $p$-power order is etale if $p$ is invertible on the base

This question is immediately related to Discriminant ideal in a member of Barsotti-Tate Group
dealing with Barsotti–Tate groups and here I
would like to clarify a proof presented by
Anonymous in the ...

**6**

votes

**2**answers

353 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 ...

**1**

vote

**0**answers

134 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

317 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 ...

**2**

votes

**2**answers

350 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: ...

**2**

votes

**0**answers

143 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**

votes

**0**answers

222 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**

votes

**0**answers

104 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 ...

**4**

votes

**2**answers

330 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 ...

**8**

votes

**0**answers

225 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**

vote

**0**answers

156 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

201 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 ...

**2**

votes

**0**answers

192 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 $(a,b)$...

**2**

votes

**0**answers

143 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**

votes

**0**answers

255 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**

votes

**0**answers

220 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}\...

**2**

votes

**0**answers

173 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 $...

**3**

votes

**0**answers

142 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**

vote

**1**answer

238 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

203 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 $\...

**1**

vote

**0**answers

81 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**

vote

**1**answer

148 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{...

**2**

votes

**0**answers

292 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**

votes

**0**answers

381 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**

votes

**1**answer

405 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 ...

**2**

votes

**0**answers

105 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 ...

**0**

votes

**2**answers

332 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**

votes

**0**answers

106 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**

votes

**1**answer

393 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**

votes

**0**answers

140 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....

**2**

votes

**1**answer

236 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

181 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 $\...

**5**

votes

**1**answer

370 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{...

**5**

votes

**2**answers

362 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**

vote

**0**answers

147 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 $X$.
...

**27**

votes

**1**answer

951 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

**0**answers

88 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 $K$-...

**6**

votes

**0**answers

343 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

164 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

522 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{...

**3**

votes

**0**answers

117 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**

votes

**0**answers

160 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 \...

**3**

votes

**0**answers

199 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

457 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

416 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 ...