Questions tagged [group-schemes]
The group-schemes tag has no usage guidance.
175
questions
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 ...
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 ...
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 ...
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 ...
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_{\...
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 $S$-...
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 ...
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 $BG=[*/G]$ ?
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 ...
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:
$$g_Y \colon Y \times_X Y \cong Y \...
3
votes
1
answer
190
views
Pathological behavior of Lie algebra under a map of abelian schemes
I am trying to understand Example 7.5/9 from the book "Neron models". There one has a discrete valuation ring $R'$ that is the localization of $\mathbb{Z}[\zeta_p]$ at $p$, so that the absolute ...
4
votes
0
answers
217
views
An extension of group schemes admitting Neron models
Let $R$ be a discrete valuation ring, $K$ its field of fractions, and
$$ 0 \rightarrow G_K' \rightarrow G_K \rightarrow G_K'' \rightarrow 0$$
a short exact sequence of smooth $K$-group schemes of ...
1
vote
1
answer
212
views
Decomposing quasi-finite separated group schemes
Let $U$ be a punctured disk, and let $G\to U$ be a quasi-finite separated group scheme. (Assume $K$ of char zero if it helps)
Why is $G = G_1\sqcup G_2$, where $G_1 \to U$ is finite and $G_2\to U$ ...
2
votes
1
answer
360
views
Relative identity component for group algebraic spaces
Let $S$ be a locally noetherian scheme and let $G$ be a separated and smooth $S$-group algebraic space of finite presentation. Does there exist an open sub-(group algebraic space) $G^0 \subset G$ ...
4
votes
0
answers
159
views
Is this $S$-birational map an open immersion on its domain of definition?
My question is about a claim on the bottom of p. 121 of the book "Neron models" by Bosch, Lutkebohmert, and Raynaud, so I will freely use the general terminology recalled in this book, but will ...
4
votes
2
answers
335
views
Is a "central" extension of $\mathbb{Z}/m\mathbb{Z}$ by $\mathrm{GL}_n$ necessarily split?
Let $m \ge 1$ be an integer, let $k$ be a field of characteristic $0$, and let
$$
1 \rightarrow \mathrm{GL}_n \rightarrow E \rightarrow \mathbb{Z}/m\mathbb{Z} \rightarrow 1
$$
be an extension of $k$-...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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(\...
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 ...
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 ...
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: ...
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\...
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 ...
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 ...
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 ...
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 ...
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 ...
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&...
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 ...
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 ...
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 ...
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 ...
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 $...
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 ...
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 ...
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_* \...
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 ...
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}...
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 $k$-...
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.
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 ...
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 ...
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 ...
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 ...
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 ...