Questions tagged [group-schemes]

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276 views

Connected-étale sequence $ 0 \to G^0 \to G \to G^{\text{ét}} \to 0$ for affine finite group scheme $G$

Let $G$ be an affine finite commutative group scheme over a complete (or at least Henselian; thanks to Jason Starr for calling attention to it) local ring $R$, and assume the residue field $\kappa=R/m$...
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3 votes
0 answers
62 views

Noncommutative group schemes corresponding to quantum groups

I'm not an expert on quantum groups by any stretch, so forgive me if this question seems overly naive. That said, I was wondering if there is a way (or if there has been any attempt in the literature) ...
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  • 1,241
3 votes
0 answers
46 views

Is the product of unipotent radicals of opposite Borels a closed immersion?

Let $G$ be a reductive group over a scheme $S$ and let $B \subset G$ and $B' \subset G$ be opposite Borel subgroups with their unipotent radicals $U \subset B \subset G$ and $U' \subset B' \subset G$. ...
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5 votes
1 answer
197 views

Group scheme with an isotrivial maximal torus

Let $G$ be a reductive group scheme over a normal ring $A$. Then, we know that Zariski locally it admits a maximal torus. Let us assume that it admits a maximal torus after a finite surjective (resp. ...
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3 votes
0 answers
161 views

A question regarding Corollary 4.12 in Mumford's "Analytic construction of deg. Ab. Var."

Let $S=Spec A$ be the spectrum of an integrally closed, excellent and Noetherian ring. In his paper, David Mumford constructs an $S$-group scheme $G$. He shows that the torsion of $G_s$ is $p$-...
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3 votes
0 answers
125 views

Finite commutative group schemes whose exponent coincides with its rank

In group theory, a finite commutative group $G$ contains an element whose order is the exponent of $G$. Thus, If the exponent of $G$ is the same as the order of $G$, it must be that $G$ is cyclic. ...
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10 votes
1 answer
312 views

Homomorphisms between Oort–Tate group schemes

Let $R$ be a complete local $\mathbf{Z}_p$-algebra, for some prime $p$. In the 1970 paper Group schemes of prime order by Oort and Tate, they write down an explicit finite flat group scheme $G_R(a, b)$...
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6 votes
0 answers
177 views

Semistable model of product of modular curves

Does the product $Y_1(Np) \times Y_1(Np)$ admit a semistable model over $\mathbf{Z}_p[\zeta_p]$ with a natural moduli-space interpretation? Less telegraphically: let $p$ be a prime, and $N \ge 4$ ...
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3 votes
0 answers
166 views

Reference request for Kummer-Artin-Schreier-Witt theory

I cannot find the following 4 papers by Sekiguchi–Suwa in their works on Kummer–Artin–Schreier–Witt theory: On the unified Kummer–Artin–Schreier–Witt theory, Prépublications du laboratoire de ...
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9 votes
1 answer
287 views

Cotangent spaces of finite flat group schemes in short exact sequences

Fix $(R,m)$ a complete DVR of mixed characteristic $(0,p)$ with perfect residue field, and consider finite flat commutative group schemes $G = Spec(A)$ over $R$. One can associate a differential ...
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1 answer
385 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 ...
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5 votes
1 answer
341 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 ...
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1 vote
1 answer
485 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)...
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  • 291
6 votes
1 answer
253 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}...
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1 vote
0 answers
118 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(...
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  • 21
1 vote
0 answers
225 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 ...
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  • 479
7 votes
2 answers
666 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 ...
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1 vote
0 answers
148 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 ...
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  • 479
1 vote
1 answer
521 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 ...
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2 votes
2 answers
454 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: ...
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4 votes
0 answers
256 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 ...
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4 votes
0 answers
115 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 ...
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5 votes
2 answers
394 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 ...
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8 votes
0 answers
230 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 $$ c:A(X)\otimes_R A(Y)\to A(X\times_SY) $$ ...
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1 vote
0 answers
191 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) ...
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1 vote
1 answer
220 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 ...
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2 votes
0 answers
269 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 $(a,b)$...
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2 votes
0 answers
159 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 ...
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5 votes
0 answers
269 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 ...
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2 votes
0 answers
235 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}\...
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2 votes
0 answers
208 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 $...
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3 votes
0 answers
146 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 ...
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2 votes
1 answer
337 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 ...
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  • 148
2 votes
1 answer
237 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 $\...
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  • 123
3 votes
0 answers
118 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 ...
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  • 123
1 vote
1 answer
154 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{...
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  • 367
2 votes
0 answers
375 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 ...
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  • 679
9 votes
0 answers
410 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 ...
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  • 6,040
7 votes
1 answer
460 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 ...
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  • 699
2 votes
0 answers
152 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 ...
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  • 511
0 votes
2 answers
356 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 ...
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  • 679
2 votes
0 answers
130 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 $\mathcal A$. ...
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  • 21
3 votes
1 answer
481 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 ...
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  • 291
7 votes
0 answers
145 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....
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2 votes
1 answer
250 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 $\mathbb{A}^1$. ...
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1 vote
0 answers
185 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 $\...
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  • 167
5 votes
1 answer
376 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{...
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5 votes
2 answers
478 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 ...
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  • 367
1 vote
0 answers
157 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 $X$. ...
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  • 167
27 votes
1 answer
1k 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[...
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