# Questions tagged [group-schemes]

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172
questions

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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

112
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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).
...

2
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0
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111
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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 $X$...

4
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0
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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 ...

1
vote

1
answer

188
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### 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 ...

6
votes

1
answer

241
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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
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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
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0
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124
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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

245
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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 ...

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0
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148
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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 $...

2
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0
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348
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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
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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
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0
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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

230
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### 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
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170
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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 $p$-...

3
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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. ...

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1
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402
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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 $G_R(a, b)$...

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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 $N \ge 4$ ...

4
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0
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249
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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 ...

9
votes

1
answer

321
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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 ...

0
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1
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440
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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 ...

5
votes

1
answer

435
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### 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
answer

543
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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)...

6
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342
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### 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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0
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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(...

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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 ...

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 ...

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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
...

2
votes

1
answer

799
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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 ...

2
votes

2
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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: ...

4
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276
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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
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0
answers

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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 ...

5
votes

2
answers

481
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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
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0
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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
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0
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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) ...

1
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1
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245
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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 ...

2
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0
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347
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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 $(a,b)$...

2
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0
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### 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 ...

5
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### 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 ...

2
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0
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### 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}\...

2
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0
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### 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 $...

3
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0
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### 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 ...

2
votes

1
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493
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### 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 ...

2
votes

1
answer

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### 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 $\...

3
votes

0
answers

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### 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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1
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### 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{...

2
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0
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### 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 ...

9
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0
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459
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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 ...

7
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1
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### 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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### 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 ...