Questions tagged [group-schemes]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
0 votes
0 answers
141 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 ...
rime's user avatar
  • 445
9 votes
1 answer
629 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 ...
Mikhail Borovoi's user avatar
13 votes
2 answers
664 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 ...
Lisa S.'s user avatar
  • 2,623
1 vote
0 answers
124 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 ...
rime's user avatar
  • 445
4 votes
0 answers
325 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_{\...
Z.A.Z.Z's user avatar
  • 1,871
13 votes
1 answer
757 views

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 $S$-...
Lisa S.'s user avatar
  • 2,623
2 votes
0 answers
205 views

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 ...
Heer's user avatar
  • 1,007
3 votes
1 answer
263 views

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 $BG=[*/G]$ ?
Bear's user avatar
  • 845
16 votes
1 answer
2k 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 ...
user38276's user avatar
  • 483
2 votes
0 answers
639 views

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 \...
Pierre MATSUMI's user avatar
3 votes
1 answer
190 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 ...
Question Mark's user avatar
4 votes
0 answers
217 views

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 ...
Question Mark's user avatar
1 vote
1 answer
212 views

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 $G_2\to U$ ...
Maksim Symirno's user avatar
2 votes
1 answer
360 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) $G^0 \subset G$ ...
Question Mark's user avatar
4 votes
0 answers
159 views

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 ...
Question Mark's user avatar
4 votes
2 answers
335 views

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 $k$-...
Kestutis Cesnavicius's user avatar
6 votes
0 answers
231 views

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 ...
User12345's user avatar
4 votes
0 answers
318 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 ...
User123456's user avatar
1 vote
0 answers
459 views

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 ...
João Dias's user avatar
17 votes
3 answers
2k 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 ...
Daniel Loughran's user avatar
6 votes
1 answer
304 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 ...
anonymous's user avatar
  • 352
3 votes
0 answers
221 views

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 ...
jadahue's user avatar
  • 39
1 vote
0 answers
89 views

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 $G$, $G(\...
Tyler Holden's user avatar
8 votes
1 answer
3k 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 ...
Jon's user avatar
  • 83
4 votes
1 answer
530 views

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 ...
KotelKanim's user avatar
  • 2,007
0 votes
0 answers
263 views

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: ...
Phung Ho Hai's user avatar
1 vote
1 answer
436 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 $P:=X\...
Jesko Hüttenhain's user avatar
1 vote
0 answers
134 views

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 ...
none's user avatar
  • 11
6 votes
1 answer
443 views

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 ...
LMN's user avatar
  • 3,525
4 votes
2 answers
1k 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 ...
Joël's user avatar
  • 25.7k
5 votes
0 answers
237 views

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 ...
Heer's user avatar
  • 1,007
2 votes
1 answer
190 views

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 ...
Qijun Yan's user avatar
5 votes
2 answers
562 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 $p&...
Mikhail Borovoi's user avatar
1 vote
1 answer
452 views

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 ...
Taisong Jing's user avatar
4 votes
1 answer
628 views

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 ...
user29513's user avatar
6 votes
1 answer
417 views

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 ...
Heer's user avatar
  • 1,007
11 votes
2 answers
643 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 ...
Andrei Smolensky's user avatar
4 votes
1 answer
557 views

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 $...
Tzanko Matev's user avatar
3 votes
0 answers
917 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 ...
Pooya's user avatar
  • 31
1 vote
1 answer
334 views

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 ...
philipp's user avatar
  • 147
2 votes
1 answer
460 views

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_* \...
Veen's user avatar
  • 649
1 vote
1 answer
340 views

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 ...
Earthliŋ's user avatar
  • 1,181
1 vote
1 answer
211 views

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 $\mathbb{G}...
Dima Sustretov's user avatar
2 votes
1 answer
493 views

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 $k$-...
A.E.'s user avatar
  • 163
2 votes
1 answer
714 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.
Lei's user avatar
  • 304
7 votes
0 answers
893 views

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 ...
B. Cais's user avatar
  • 1,589
2 votes
2 answers
1k views

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 ...
Lei's user avatar
  • 304
1 vote
0 answers
311 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 ...
user565739's user avatar
  • 1,099
2 votes
1 answer
540 views

Additive form of Hilbert 90 for schemes?

First, I am by no means well-versed on cohomology so I apologize if this is too elementary. I have been going through some basics of etale cohomology, with my ultimate goal being an understanding of ...
Randall's user avatar
  • 791
4 votes
0 answers
604 views

A 'standard patching argument' in Mazur's Eisenstein Ideal paper

On pp 46 of his Eisenstein Ideal paper, Mazur states Theorem I.4 and in the discussion that follows, he mentions 'a standard patching argument' that completes the proof. I was wondering whether this ...
Saikat Biswas's user avatar