The group-schemes tag has no wiki summary.

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### 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 ...

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### 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. ...

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### 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 ...

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72 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 ...

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### 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 ...

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291 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 ...

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### 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 ...

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### 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:
...

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### Is there a unique commutative group structure on $\mathbb{G}_m$?

Let $S$ be a scheme and let $X := \mathrm{Spec}(\mathscr{O}_S[t, t^{-1}])$ be the underlying $S$-scheme of the $S$-group scheme $(\mathbb{G}_m)_S$. Is there only one structure of a commutative ...

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### How far is it to extend the results of SGA III Exp. VIB from group schemes to group spaces?

How far is it to extend the results of SGA III Exp. VIB from group schemes to group spaces?
In particular, does Corollary 4.4 from SGA III Exp. VIB hold for G/S being merely a group space? Here the ...

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### Galois cohomology out of the classifying stack

Suppose $G$ is a smooth and abelian $k$-group scheme, for $k$ a field.
Is it possible to get back galois cohomology groups $H^*(k,G)$ studying the cohomology of the classifying stack $BG=[*/G]$ ?

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443 views

### The difference between an étale finite group scheme and a finite group

I am trying to understand the statement that a Deligne-Mumford stack is locally a quotient $[U/G]$, where $G$ is a finite group. I don't understand why you can make $G$ a finite group, instead of a ...

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### Constant group scheme and torsors

Let $X$ be a scheme and $G$ a (commutative) constant group scheme. Consider a $G$-torsor $Y$ for $X$, by which I mean that there is a canonical isomorphism:
$$g_Y \colon Y \times_X Y \cong Y ...

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163 views

### Pathological behavior of Lie algebra under a map of abelian schemes

I am trying to understand Example 7.5/9 from the book "Neron models". There one has a discrete valuation ring $R'$ that is the localization of $\mathbb{Z}[\zeta_p]$ at $p$, so that the absolute ...

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### An extension of group schemes admitting Neron models

Let $R$ be a discrete valuation ring, $K$ its field of fractions, and
$$ 0 \rightarrow G_K' \rightarrow G_K \rightarrow G_K'' \rightarrow 0$$
a short exact sequence of smooth $K$-group schemes of ...

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### Decomposing quasi-finite separated group schemes

Let $U$ be a punctured disk, and let $G\to U$ be a quasi-finite separated group scheme. (Assume $K$ of char zero if it helps)
Why is $G = G_1\sqcup G_2$, where $G_1 \to U$ is finite and $G_2\to U$ ...

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186 views

### Relative identity component for group algebraic spaces

Let $S$ be a locally noetherian scheme and let $G$ be a separated and smooth $S$-group algebraic space of finite presentation. Does there exist an open sub-(group algebraic space) $G^0 \subset G$ ...

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### Is this $S$-birational map an open immersion on its domain of definition?

My question is about a claim on the bottom of p. 121 of the book "Neron models" by Bosch, Lutkebohmert, and Raynaud, so I will freely use the general terminology recalled in this book, but will ...

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### Is a “central” extension of $\mathbb{Z}/m\mathbb{Z}$ by $\mathrm{GL}_n$ necessarily split?

Let $m \ge 1$ be an integer, let $k$ be a field of characteristic $0$, and let
$$
1 \rightarrow \mathrm{GL}_n \rightarrow E \rightarrow \mathbb{Z}/m\mathbb{Z} \rightarrow 1
$$
be an extension of ...

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### Can we classify reductive group schemes over curves

Let $C$ be a smooth quasi-projective connected curve over the complex numbers.
Can one classify all reductive group schemes over $C$?
Certainly, you have the trivial ones (coming from pulling-back ...

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### 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 ...

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### Equivariant Sheaves, Local system

Let $(G,m)$ be a group scheme unipotent, and $L$ a local system of rank 1 on $G$ such that:
$m^*(L) \simeq L \boxtimes L $.
Then why is $L$ an equivariant sheaf on $G$ with the action the ...

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### 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 ...

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### 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 ...

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### A question on Kähler differentials and cotangent spaces on schemes

I have the following question (should be easy for those who know something about the field):
On page 92 (97 of the old edition) of Mumford's book "Abelian varieties", the author talks about an ...

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### Holomorphic convergence conditions on $\mathbb C((z))$-valued points of a group $G$

Let $G$ be a complex, connected, simply connected, semisimple group. I'm trying to compare the following two spaces: The free loop space $LG$ of $G$, and the $\mathbb C((z))$-valued points of $G$, ...

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### 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 ...

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### Representability of a certain group scheme quotient

Let $k$ be a field. Suppose we have an exact sequence of $k$-group schemes (not finite-type)
$$
1\to H\to G\to K\to 1
$$
In other words, the sheaf quotient $G/H$ is representable by a $k$-group ...

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### Flatness over Hopf subalgebra

Let $A\subset B$ be flat Hopf algebras over a Dedekind ring $R$.
J. Moore has proved in the article
Compléments sur les algèbres de Hopf, John C. Moore, Séminaire Henri Cartan (1959-1960), Volume: ...

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### 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 ...

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### Calculating components of finite group scheme

Suppose $X/k$ is a finite commutative group scheme over a perfect field. Then we know that the category $\mathcal{N}$ of finite commutative group schemes over $k$ is abelian and isomorphic to a direct ...

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### Group scheme counterexample

Could someone give me an example of a finite group scheme $G$ (over some base $S$) so that $G$ minus a point is still a group scheme over $S$, but not affine over $S$?
Oort mentions that there are ...

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### 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 ...

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### Semiabelian actions appearing in the toroidal campactification of a degenearting abelian varieties

Given a totally degenerated abelian variety $A_K$ (to make it easier) over a complete discrete valuation field $K$ with $R$, $\pi$ and $k$ the corresponding discrete valuation ring, uniformiser and ...

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### Degree of a finite locally free group scheme over a base scheme of characteristic p

Does a connected finite locally free group scheme G over a scheme S of characteristic p>0 has degree a power of p? I know that when S is the spectrum of a field k, it is true. Someone told me that ...

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### 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 ...

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### Submodule of a Kisin module

By M. Kisin, let $k$ be an algebraically closed field of characteristic $p$, and $K$ be a totally ramified extension of $B(k)$, the fraction field of the Witt vector ring $W(k)$, the category of ...

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### Kernel of powers of Frobenius on supersingular elliptic curves

I am trying to understand some things related to elliptic curves and finite flat group schemes but I am a little bit confused.
Let $A$ be a supersingular elliptic curve over an algebraically closed ...

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### Representability of sheaf of Ext^1 of a Néron model by $\mathbb{G}_m$

Let's work over a trait $S=\mathrm{Spec}R$, where $R$ is a dvr with fraction field $K$, residue field $k$. Given an abelian variety $A_K$ with semi-stable reduction, let $A$ over $S$ be its Néron ...

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### 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 ...

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### Is the n-torsion of an extension of an abelian variety by a torus, finite and flat?

I am looking for reference or hints how to prove the following result.
Let $G$ be a commutative $S$-group scheme which is the extension of an abelian scheme $A$ by a torus $T$. Then the n-torsion ...

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### 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 ...

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### Category of Hopf algebras.

Can you tell me, where I can find a proof of the following fact: Let $R$ be a commutative ring. Consider the category of commutative Hopf algebras over $R$. Then this category is equivalent to the ...

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### Degree of finite group schemes

Let $\pi: G \rightarrow S$ be a finite flat group scheme over a locally noetherian connected base scheme $S$.
Its degree is defined as the rank of the locally free $\mathcal O_S$-module $\pi_* ...

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### Proper morphisms: Lie groups vs. group schemes

A Lie group can (often) be recovered as the $\mathbb{R}$-points of a group scheme. I am wondering if this parallelism carries over to proper actions.
In particular, let $G$ be a Lie group acting on a ...

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### is a cartesian square of a group scheme with $\mathbb{G}_a^n$ fibres reduced?

Let $G$ be a group scheme over $S$ where $S$ is a reduced scheme of finite type over a field $k$ of characteristic 0, and let every fibre $G_s$ over a closed point of $S$ be isomorphic to ...

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### Etale group schemes over a local ring

Let $p$ be a prime number and $R$ be a Noetherian local ring of characteristic $p$ with residue field $k$. Let $G$ be a finite etale subgroup scheme over $R$ of order $p$. Suppose that the etale ...

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### 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.
...

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### Deformation of ordinary p-divisible groups via Grothendieck-Messing

I am hoping that someone can point out the error in the "proof" of the following "theorem":
Theorem: Let $k$ be a perfect field of characteristic $p>2$ and let $G$ be an ordinary $p$-divisible ...

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### isomorphism of fibre functors

If $\mathfrak{C}$ is a $k$-linear rigid abelian tensor category with End(1)=$k$(strictly speaking is isomorphic to $k$ as a $k$-algebra), and $k=\bar{k}$, and if $\omega_1$ and $\omega_2$ are two ...