Questions tagged [group-schemes]

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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 ...
Modern_Hunter's user avatar
12 votes
0 answers
344 views

Is the quotient of two linear group schemes linear?

Let $S$ be an affine scheme. Call a group scheme $G\to S$ linear if there exists an $S$-group morphism $G\to \mathrm{GL}_{n,S}$ with trivial kernel. Assuming this, suppose $H\to S$ is a central closed ...
Uriya First's user avatar
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9 votes
0 answers
466 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 ...
sawdada's user avatar
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8 votes
0 answers
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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 $$ c:A(X)\otimes_R A(Y)\to A(X\times_SY) $$ ...
Matthieu Romagny's user avatar
7 votes
0 answers
153 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....
user avatar
7 votes
0 answers
359 views

When is the character group scheme of a group scheme representable? (Affine Case)

While reading Tate's article on Finite Flat Group Schemes in "Modular Forms and Fermat's Last Theorem" I was lead to this question. Let $S$ be a scheme, $G$ a group scheme over $S$, and $T$ an $S$-...
J. David Taylor's user avatar
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 ...
B. Cais's user avatar
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6 votes
0 answers
236 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 $N \ge 4$ ...
David Loeffler's user avatar
6 votes
0 answers
570 views

Ext group for commutative finite group schemes

EDIT: all group schemes are commutative. Thanks @R. van Dobben de Bruyn! Could anyone provide a reference request about extensions of finite group schemes / Ext groups. As far as I know the category ...
aytio's user avatar
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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 ...
User12345's user avatar
5 votes
0 answers
143 views

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 ...
Homotopy theorist 's user avatar
5 votes
0 answers
141 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 ...
user avatar
5 votes
0 answers
306 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 ...
user avatar
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 ...
Heer's user avatar
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4 votes
0 answers
206 views

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 ...
Ben C's user avatar
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4 votes
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225 views

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 $\...
David Hubbard's user avatar
4 votes
0 answers
293 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 ...
Ehsan Shahoseini's user avatar
4 votes
0 answers
280 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 ...
user267839's user avatar
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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_{\...
Z.A.Z.Z's user avatar
  • 1,871
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 ...
Question Mark's user avatar
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 ...
Question Mark's user avatar
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 ...
User123456's user avatar
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 ...
Saikat Biswas's user avatar
3 votes
0 answers
250 views

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 ...
user267839's user avatar
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3 votes
0 answers
96 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) ...
Dat Minh Ha's user avatar
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3 votes
0 answers
72 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 $U' \subset B' \subset G$. ...
cardiac.thrash87's user avatar
3 votes
0 answers
171 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 $p$-...
The Thin Whistler's user avatar
3 votes
0 answers
147 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. ...
FNH's user avatar
  • 329
3 votes
0 answers
152 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 ...
Lin Chen's user avatar
3 votes
0 answers
208 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 ...
Jrodri26's user avatar
  • 123
3 votes
0 answers
179 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 $\mathcal A$. ...
kkgz's user avatar
  • 31
3 votes
0 answers
165 views

Simply-connected covers of group schemes

Let $\mathbf G$ be a semisimple group scheme over a base $S$. I believe that one can always find a fiberwise simply-connected group scheme $\mathbf G^{sc}$ and a central isogeny $\mathbf G^{sc}\to\...
Roman Fedorov's user avatar
3 votes
0 answers
296 views

Quotient of a sheaf by group action and representabillity

Let $X$ be a scheme and $S$ be a sheaf of sets over the fppf topology of $X$. Let $G$ be a group scheme over $X$ and there is an action of $G$ on $S$. Now, I want to look at the quotient $G \setminus ...
john's user avatar
  • 1,257
3 votes
0 answers
435 views

"Frobenius Descent"

The following proposition is there in Pink's lecture notes on finite group schemes. Let $k$ be an algebraically closed field of characteristic $p$. The category of finite length $W(k)$-modules $N$ ...
Shubhodip Mondal's user avatar
3 votes
0 answers
733 views

When is a quotient by a strictly free group action étale?

Let $S$ be a base scheme and $G$ an $S$-group scheme acting on an $S$-scheme $X$ and assume that the quotient scheme $Y := X/G$ exists. (This is for example the case if $S$ is finite locally free and ...
Elvorfirilmathredia's user avatar
3 votes
0 answers
109 views

G is p-divisible, about the affine rings of G[p]

Let $R$ be a ring and $G$ a $p$-divisible group over $R$. Since $G[p]$ is finite flat over $\text{Spec}(R)$, it is an affine (group) scheme, say $\text{Spec}(B)$. What can be said about $B$ beyond the ...
aytio's user avatar
  • 371
3 votes
0 answers
251 views

Schematic closure of maximal torus over a discrete valuation ring: smoothness of the special fibre

Let $R$ be a complete DVR (or just Henselian) with field of fractions $F$ and algebraically closed residue field $k$ (of characteristic $p>0$). Let $G$ be a reductive group scheme over $R$ and let $...
A Stasinski's user avatar
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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 ...
jadahue's user avatar
  • 39
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 ...
Pooya's user avatar
  • 31
2 votes
0 answers
143 views

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 ...
HJK's user avatar
  • 135
2 votes
0 answers
115 views

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). ...
Martin Orr's user avatar
  • 1,500
2 votes
0 answers
120 views

"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 $X$...
Ben C's user avatar
  • 3,281
2 votes
0 answers
385 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 $\kappa=R/m$...
user267839's user avatar
  • 5,948
2 votes
0 answers
394 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 $(a,b)$...
user267839's user avatar
  • 5,948
2 votes
0 answers
175 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 ...
Bin Wang's user avatar
  • 193
2 votes
0 answers
266 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}\...
user avatar
2 votes
0 answers
233 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 $...
user avatar
2 votes
0 answers
553 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 ...
user267839's user avatar
  • 5,948
2 votes
0 answers
211 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 ...
Gaussian's user avatar
  • 519
2 votes
0 answers
98 views

Classical reductive group schemes vs. unitary groups of separable algebras with involution --- reference request

Let $K$ be a field with $2\in K^\times$, let $A$ be a separable $K$-algebra (i.e. $A$ is finite-dimensional, semisimple and its center is an etale $K$-algebra), and let $\sigma:A\to A$ be a $K$-...
Uriya First's user avatar
  • 2,846