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6
votes
1answer
212 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 ...
1
vote
0answers
70 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
0answers
66 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 $G$, ...
3
votes
1answer
303 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
1answer
184 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
0answers
101 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
1answer
142 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 ...
1
vote
0answers
90 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 ...
4
votes
0answers
180 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
2answers
174 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 ...
4
votes
0answers
124 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
1answer
106 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 ...
0
votes
0answers
113 views

exponential map for finite group schemes?

Hi, I am trying to define an exponential map for finite abelian group schemes. The following looks like it should work, but doesn't (see below). I am putting up this question hoping that someone will ...
5
votes
2answers
241 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 ...
0
votes
1answer
195 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 ...
3
votes
1answer
154 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
1answer
198 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 ...
7
votes
1answer
279 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 ...
2
votes
1answer
259 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 ...
2
votes
0answers
317 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
1answer
234 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 ...
1
vote
1answer
239 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_* ...
2
votes
1answer
225 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
1answer
162 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 ...
2
votes
1answer
333 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 ...
2
votes
1answer
409 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. ...
6
votes
0answers
680 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
1answer
451 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
0answers
243 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
1answer
294 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 ...
3
votes
0answers
444 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 ...
3
votes
1answer
422 views

Isomorphism on p-torsion of Neron models

Let $A$, $B$ be abelian varieties over $\mathbb{Q}$, with corresponding Neron models $\mathcal{A}$, $\mathcal{B}$ over $X=Spec{\mathbb{Z}}$. Let $p$ be an odd prime of good reduction for both $A$ and ...
1
vote
3answers
830 views

group scheme neither affine, nor an abelian variety

Does there exist a group scheme of finite type over a field $k$ , which is neither affine, nor an abelian variety?
5
votes
2answers
827 views

On a Theorem of Fontaine

Let $S=Spec(R)$ where $R$ is a Henselian local ring with fraction field $K$. Let $G$ and $G'$ be finite, flat group schemes of odd order over $S$ with isomorphic generic fibers (over $Spec(K)$). Does ...
4
votes
2answers
409 views

Tensor and Hom objects for finite flat group schemes

Is the category of finite flat group schemes equipped with "tensor products" and Hom-objects, encoding bilinear maps? I'm aware that the Cartier dual is $Hom(\mathbb{G}, \mathbb{G}_m)$, and want to ...
1
vote
1answer
166 views

Lifting of projective representations of a torus over a valuation ring

Let $R$ be a valuation ring. We don't assume it to be discrete or have a finite residue field. Let $T$ be a split torus over $R$, so $T\cong {\mathbb G}_m^r$ for some $r$. Left there be given a ...
6
votes
2answers
537 views

Character group of Frobenius kernels

Let $G$ be a semisimple algebraic group over an algebraically closed field $k$ of characteristic $p$ (e.g., $G=SL_n(k)$). Then $G$ is equal to its derived subgroup $[G,G]$. Consequently, the character ...
6
votes
2answers
596 views

finite flat commutative group schemes arising from Abelian varieties

How are the finite flat group schemes $\mathcal{A}[\ell^n]$ arising from an Abelian scheme $\mathcal{A}/S$ singled out among other finite flat commutative group schemes of exponent $\ell^n$?
1
vote
1answer
235 views

Group scheme of infinite dimensional linear groups ?

Hi there, I know there are fairly straightforward ways to write down the schemes of infinite dimensional projective spaces (not restricting myself to only countable dimensions), but what happens with ...
8
votes
5answers
1k views

Automorphism group of a scheme

Hi there, I have a probably stupid question on schemes ... Let S be a scheme, and let A be its automorphism group. Does A carry a scheme structure itself, that is, can one see A as a group scheme ? ...
7
votes
4answers
588 views

Twist of a group Hopf-algebra

Let $G$ be a finite group with identity element $e$, and $C[G]$ the ring of complex-valued functions on $G$, with pointwise addition and multiplication. Then $C[G]$ is naturally a Hopf algebra, with ...
5
votes
1answer
677 views

Generalization of Raynaud's (p, p, … p) result

Does Corollary 3.4.4 in Raynaud's paper ``Schemas en Groupes de Type (p, ..., p)'' apply also to the case where G is quasi-finite? If not, what is the more general statement? The corollary states: ...
20
votes
4answers
2k views

The Frobenius morphism

I found the following list on the "Frobenius Page" by David Ben-Zvi, described by the author as "an outdated collection of intuitive ways to think about raising to the p-th power". Generates a ...
2
votes
1answer
410 views

Quotient by p-th roots of unity in characteristic p

Let $X$ be a variety over $k$ of characteristic $p>0$ (you can assume $k$ algebraically closed and $X$ normal) with an action of the group scheme of $p$-th roots of unity $\mu_p = {\rm Spec}\ ...
14
votes
1answer
865 views

Universal homeomorphisms and the étale topology

Let $f:X\to S$ be a universal homeomorphism of schemes. Assume $X(S')\neq\emptyset$ for some étale surjective $S'\to S$. Does $f$ have a section? The answer is yes if $S$ is reduced, by descent. ...
2
votes
1answer
401 views

Cartier dual of \alpha_p

How can I prove that the Cartier dual of αp is again αp (using the Yoneda lemma)? It should be something like $\alpha_p(R) \to (\alpha_p(R) \to \mu_p(R)),x \mapsto (y \mapsto exp_{p−1}(x+y)$, where ...
14
votes
1answer
747 views

Is there a connected $k$-group scheme $G$ such that $G_{red}$ is not a subgroup?

I've been trying a learn a little more about group schemes by working through a set of exercises on Brian Conrad's website. Exercise 8.3 of http://math.stanford.edu/~conrad/papers/gpschemehw1.pdf ...
30
votes
3answers
2k views

Proper subgroup of GL(n,Z) isomorphic to GL(n,Z)?

This is just a question originated from some random thoughts. I hope it's nevertheless fit for mo. It's possible to find a proper subgroup of $GL(n,\mathbb{C})$ isomorphic to $GL(n,\mathbb{C})$ ...
12
votes
2answers
713 views

Can a reductive group act non-linearly on a vector group?

Let $k$ be a field; I'm going to discuss linear algebraic groups over $k$. The question I'll pose is only interesting when the characteristic is $p>0$. 1. Some motivation A vector group is an ...
10
votes
1answer
756 views

Are there “reasonable” criteria for existence/non-existence of Levi factors or their conjugacy in prime characteristic?

Classical theorems attributed to Levi, Mal'cev, Harish-Chandra for a finite dimensional Lie algebra over a field of characteristic 0 state that it has a Levi decomposition (semisimple subalgebra plus ...