Questions tagged [group-schemes]
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175
questions
2
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Computing the Dieudonné module of $\mu_p$ from Fontaine's Witt Covector
In Groupes $p$-divisibles sur les corps locaux, Fontaine introduced a uniform construction of Dieudonné modules through the definition of the Witt covector. Consider a perfect field $k$ of ...
1
vote
0
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238
views
Unirational algebraic group scheme smooth
Let $G$ an unirational algebraic $k$-group over base field $k$ in sense of this book on Neron models, (ie separated $k$-group scheme of finite type).
On page 310 is claimed that unirationality implies ...
6
votes
2
answers
507
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 $\mathbb Z$:
$$0\to \mathbb{Z}...
3
votes
0
answers
250
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Does the orbit in geometric invariant theory have natural scheme structure
Let $X$ be a scheme locally of finite type over a sufficiently "nice" base scheme $S$ (nice in sense of reasonable "finiteness conditions", for sake of simplicity let's start as ...
16
votes
0
answers
412
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Complete resource of Ngô's course notes on Algebraic Groups and Group Schemes
I'm looking for Ngô's M2 course notes on "Groupes algébriques et schémas en groupes".
The Wayback Machine has an incomplete capture here. However, it apparently lacks chapter 1, 3, and 5.
I ...
2
votes
0
answers
115
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Endomorphisms of the multiplicative group over a non-reduced complex analytic space
Let $S_n$ be the (usually) non-reduced complex analytic space corresponding to the ring $\mathbb{C}[X]/(X^{n+1})$ (the underlying topological space of $S_n$ is a point, and $S_0$ is a reduced point).
...
4
votes
0
answers
207
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Do rational maps to abelian varieties extend across rational singularities?
Let $X$ be a normal proper variety with only rational singularities and $A$ an abelian variety. Does a rational map $X \supset U \to A$ extend to a morphism $X \to A$? If not, what is a ...
2
votes
0
answers
120
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"Vanishing locus" of forms in the $h$-topology
Let $\Omega_{h}^p$ be the sheaf of $p$-forms in the $h$-topology defined as the sheafification for the $h$-topology of the presheaf,
$$ Y \mapsto \Omega^p_Y(Y) $$
Kebekus and Schnell show that when $X$...
1
vote
1
answer
197
views
Lie algebras and pulled back group schemes
Suppose I have an extension of fields $L/K$, a group scheme $G_K$ over $\operatorname {Spec} K$. Let $G_L$ denote the pullback of $G_K$ to $\operatorname{Spec} L$. Then, under what conditions on the ...
8
votes
1
answer
605
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Example of a connected finite group scheme which is not solvable
What would be an example of a connected finite group scheme over a field $k$ that is not solvable? Here $k$ is algebraically closed.
Let $\operatorname{GL}_n$ be the general linear group scheme over ...
6
votes
1
answer
254
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Generating the coordinate ring of the Lubin-Tate formal group
Let $K$ be a $p$-adic local field with uniformizer $\pi \in \mathcal{O}_{K}$ and residue field $k = \mathcal{O}_{K}/\pi$. Let $G$ be a Lubin-Tate formal $\mathcal{O}_{K}$-module and $G_{0}$ its ...
4
votes
0
answers
225
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Lifting the connected-etale sequence of the $p$-torsion of an elliptic curve
Suppose that $R$ is a complete DVR with characteristic 0 fraction field $K$, maximal ideal generated by $p$ and characteristic $p>0$ residue field $k$ which is algebraically closed. Suppose that $\...
5
votes
0
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143
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Faltings' Cartier duality for A-modules in terms of Hopf algebras
$\newcommand\dual{^{\text{dual}}}\newcommand\GrpSch{\mathrm{GrpSch}}\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Vect{Vect}$If $G$ is a finite group scheme over a field $k$, we can define its ...
7
votes
1
answer
272
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Is there a "spherical building" for a reductive group over a Henselian local ring?
Let $A$ be a Henselian local ring and let $G$ be a split reductive $A$-group. I'm interested in some notion of a "building of parabolic subgroups" for the group scheme $G$.
In my specific ...
1
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0
answers
173
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Quotient stack is an algebraic space when $G$ is finite and acts freely
I have been following Jarod Alper's lecture series on YouTube on Stacks https://youtube.com/playlist?list=PLhFI5R_xInjdhtWuhgYlA8NZGXO-unnl4
From what I understand -
If a smooth affine group scheme $...
10
votes
1
answer
1k
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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 ...
2
votes
0
answers
385
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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$...
3
votes
0
answers
96
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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) ...
3
votes
0
answers
72
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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$. ...
5
votes
1
answer
251
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. ...
3
votes
0
answers
171
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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$-...
6
votes
0
answers
236
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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$ ...
3
votes
0
answers
147
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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. ...
10
votes
1
answer
425
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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)$...
9
votes
1
answer
367
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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 ...
4
votes
0
answers
293
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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 ...
3
votes
1
answer
585
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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 ...
0
votes
1
answer
464
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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 ...
4
votes
1
answer
308
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Understanding full set of sections as in Katz-Mazur
I was reading this question, specifically Brian's answer. In particular I am having a bit of trouble digesting the following sentence:
Being a "full set of sections" of $Z/S$ is something which is ...
17
votes
3
answers
2k
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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 ...
1
vote
1
answer
580
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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)...
5
votes
1
answer
479
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 ...
2
votes
2
answers
546
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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: ...
8
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2
answers
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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 ...
1
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0
answers
123
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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(...
1
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0
answers
309
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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 ...
2
votes
1
answer
252
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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 ...
11
votes
4
answers
2k
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The category of finite locally-free commutative group schemes
I'm trying to understand the properties of the category $\mathcal{FL}/S$ of finite locally-free commutative group schemes over an arbitrary base-scheme $S$. I know it is not in general an abelian ...
1
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0
answers
160
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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
...
20
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5
answers
5k
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Automorphism group of a scheme
I have a probably stupid question on schemes ...
Let $S$ be a scheme, and let $A = \mathsf{Aut}(S)$ be its automorphism group. Does $A$ carry
a scheme structure itself, that is, can one see $A$ as a ...
2
votes
2
answers
1k
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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 ...
2
votes
1
answer
971
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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 ...
9
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0
answers
466
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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 ...
4
votes
0
answers
280
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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 ...
5
votes
0
answers
141
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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 ...
6
votes
2
answers
531
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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 ...
8
votes
0
answers
253
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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)
$$
...
1
vote
0
answers
234
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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) ...
2
votes
0
answers
394
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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 $(a,b)$...
11
votes
1
answer
210
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Can elements in the orthogonal group of a non-split Azumaya algebra with an orthogonal involution have reduced norm -1?
Let $R$ be a connected (commutative) ring with $2\in R^\times$.
Let $A$ be an Azumaya algebra over $R$ and let $\sigma:A\to A$ be an orthogonal involution. (This means that there is a faithfully flat ...