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Questions tagged [group-schemes]

A group scheme over a scheme S is simply a group object in the category of schemes over S.

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36 views

closed embedding is open iff ideal sheaf idempotent

Let $f: \mathrm{Spec}(R) \to \mathrm{Spec}(A)$ be a closed embedding of affine Noetherian schemes, given by the ideal $I = \mathrm{ker}(A \twoheadrightarrow R)$. If $I = I^2$, then it's not too hard ...
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Is subgroup scheme of smooth group scheme flat over base scheme?

Let $\mathcal{A},\mathcal{B}$ be smooth separated commutative group scheme over $S$, where $S$ is Noetherian. Let $\iota:\mathcal{A}\rightarrow\mathcal{B}$ be a morphism of group scheme which is ...
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1answer
46 views

Different notions of torsors in algebraic geometry

In what follows $X$ will be a scheme and $G$ a group scheme. In the examples I will take $X=\mathbb{P}^1_k$ and $G=\mathbb{G}_{m}$. When reading about "the torsor..." I found many definitions, not ...
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19 views

Weil group and conjugacy classes of cocharacters

We have the following setup: $G$ a compact reductive group over $\mathbb{R}$, $T$ a maximal Torus, $U^1$ the real algebraic Torus and $W$ the Weil group of $T$ In Variétiés de Shimura Lemma 1.2.4 p....
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1answer
69 views

Base Change of Algebraic Group

I have some questions about the steps in the proof of COROLLARY 1.35 from Milne's "Algebraic Groups : The theory of group schemes of finite type over a field"(p. 17). Here the excerpt: Let $G$ be a ...
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33 views

Understanding the proof of Cartier duality

I'm trying to understand the above proof of Cartier Duality. The step I don't understand is the following It says $$ \phi\psi(a) = ((\phi \otimes \psi) \circ \Delta(a) = (\phi \otimes \psi)( a \...
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1answer
49 views

Composition of finite flat morphisms cancellation

Outline/some thoughts: Suppose we have morphisms of schemes $$ X \overset{f}{\rightarrow} Y \overset{g}{\rightarrow} S $$ where $f$ is finite, the composition $g \circ f$ is finite, and $S$ is locally ...
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65 views

$PGL_2(\Bbb R)$ as a scheme

How is $PGL_2(\Bbb R)$ a scheme? Here is my thought process $GL_2(\Bbb R)=Spec(\Bbb{R}[w,x,y,z,q]/((wz-xy)q-1))$ We want $PGL_2(\Bbb R)=GL_2(\Bbb R)/\Bbb{G}_m(\Bbb R)$ somehow. We can find $PGL_2(\...
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33 views

Sheaves of abealian groups and base change

Let $k$ be a field of characteristic $p > 0$ and $R_1$ and $R_2$ be two $k$-algebras. Let $X$ be a scheme over $k$. Let $f: R_1 \rightarrow R_2$ be a morphism of $k$-algebras induces the morphism ...
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33 views

Constructing a group scheme associated to a coherent module.

For a commutative ring $A$ and an $A$-module $M$, $\mathbb{V}(M)$ is defined to be $\text{Spec}(S^*M)$, where $S^*M = \bigoplus_k S^kM$ is the symmetric algebra. Thus $\mathbb{V}(M)$ is an affine ...
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52 views

A group scheme whose reduced underlying scheme is not normal

Let $G$ be a group scheme over a perfect field $k$. Let $G_{red} \hookrightarrow G$ be the reduced underlying scheme. It can be shown that $G_{red}$ is a closed subgroup scheme of $G$. Is there an ...
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111 views

How can an elliptic curve be regarded as a group scheme?

If I understand correctly: A scheme is a functor $\mathbf{CRing} \rightarrow \mathbf{Set}$ satisfying certain axioms. A morphism of schemes is a natural transformation. A group scheme is a group ...
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45 views

free action of group scheme

Let $f: G \to G'$ be a homomorphism of group schemes. Then we get a natural action of $G$ on $G'$, given on points by $(g,g') \mapsto f(g) \cdot g'$. Then it is claimed that this action is free if ...
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137 views

Subgroup scheme of a constant group scheme

I am studying the basics of group-schemes, and I found this statement in a book. Let $A$ be a finite abelian group, and let $S$ be a connected scheme. We denote by $(A)_S$ the constant group scheme ...
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36 views

Notion of inner automorphisms for group schemes

Let $G$ be a finite group scheme over the field $k$ ( you can assume $k$ is algebraically closed). I've seen the term "inner automorphisms" for $G$ at many places but I still don't understand what the ...
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Equivalence definition of affine (group) schemes

I am currently studying affine group schemes via Waterhouse. Since Waterhouse does not use schematic language in the first few chapters, I tried to "translate" the definitions in different languages. ...
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What is difference between constant group scheme associated with cyclic group and $\mu_n$

Let $k$ a field. Let $\operatorname{char}(k) \not\mid n$. Consider the two group schemes $\mu_n=spec(k[t]/(t^n-1))$ and $\underline{\mathbb{Z}/n\mathbb{Z}}$ the constant group scheme associated to ...
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38 views

Functorial relationship between torsors

Let $G$ be a finite group scheme. Let $T(G)$ denote the set of all $G$-torsors over some $k$-scheme $X$ (k is a field). Let there be a group scheme homomorphism from $H$ to $G$. Then should be some ...
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1answer
50 views

stabilizer of action of a group scheme on a scheme

Let $G$ be a group scheme over a basis $S$, and $X$ be a scheme over $S$. Let $\rho: G \times_SX \to X$ be an action of $G$ on $S$. If $T$ is an $S$-scheme and $x \in X(T)$, then the definition of the ...
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Are Abelian schemes of finite type?

I define an Abelian scheme $A$ over an arbitrary base scheme $S$ to be a smooth group scheme over $S$ such that the fibers are Abelian varieties. Is it true that A is of finite type? Smoothness ...
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On curves whose smooth compactification has genus $0$

Let $U$ be a smooth geometrically connected curve over a field $k$ that is elliptic or hyperbolic. Let's say $k$ has characteristic $0$. Let $X(U)$ be the smooth compactification of $U$ and suppose ...
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1answer
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Every profinite group is naturally an affine group scheme over $\mathbb Q$?

I saw a special case of this after reading about the Weil group and the Weil-Deligne group. Since I'm trying to become more technically proficient in algebraic geometry, I thought this was ...
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48 views

Examples of finite group schemes over a field which are not affine

Let $G$ be finite group scheme over a field $k$. What are some examples of $G$ such that it is not affine?
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Computing extension group of finite commutative group schemes over a field

Let $k$ be a perfect field with $\text{char} k=p>0$. Then the category of finite commutative group schemes over $k$ (denoted by $FC/k$) is abelian. It seems that $FC/k$ does not have enough ...
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98 views

Structure of p-torsion group scheme of elliptic curve

Let $k$ be an algebraically closed field with char=p>0, $E$ an elliptic curve over $k$, how to prove the fact that $E[p]$ is extension of $\alpha_p$ by $\alpha_p$ in the supersingular case as well as ...
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91 views

Torsors in fpqc and fppf topology

Let $X$ be scheme over a field $k$. I've seen two different definitions of torsors Let $G$ be a group scheme over $X$. Let $S$ be faithfully flat and locally of finite presentation over $X$ and ...
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1answer
41 views

Is the category of finite group schemes closed under fibred product?

Is the category of finite group schemes closed under fibred product? If not, what is the simplest counterexample? More precisely Let $G_1$ and $G_2$ be two finite group schemes over $k$. Let $f_1$ ...
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86 views

Convolution of l-adic sheaves is commutative

I am trying to figure out how to prove a very basic statement about convolution of $\ell$-adic/perverse sheaves in Katz's "Rigid local systems" (section 2.5.3, (1) ). The fact is fairly obvious and I ...
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1answer
78 views

Hom$(G,G_a)$Group scheme multiplicative type

I'd like to understand why Hom$(G,G_a)$ for $G$ of multiplicative type is trivial. Recall that $G$ is of multiplicative type if $G_{k_s}$ is diagonalizable, where $k_s$ is the splitting field of the ...
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Smooth morphism induced by the action of group schemes

Let $R$ be a discrete valuation ring of characteristic $0$ and furthermore let $G$ be a connected, simply connected,split semisimple, affine algebraic group scheme over $R$. Furthermore, let $X$ be a ...
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1answer
110 views

Antipode of Cocommutative Hopf Algebra

I’m reading about affine group schemes by Waterhouse and in the proof of showing the (Jordan) decomposition of Abelian affine group scheme (equivalently cocommutative Hopf algebra), I came across the ...
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2answers
90 views

Algebra representing the group scheme $\mathbb Z/ n\mathbb Z$

as title says, I can not figure out the hopf algebra structure to put on $k^n$ to make it represent the group scheme $\mathbb Z/n\mathbb Z$. With the usual example of group schemes like $G_a$ or $G_m$ ...
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117 views

surjective morphism of algebraic groups and fppf topology

I'm reading Milne's course notes on Affine Group Schemes: http://www.jmilne.org/math/CourseNotes/AGS.pdf In Definition 7.1 it says: " A homomorphism $G \to Q$ of affine groups is said to be ...
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1answer
49 views

Multiplication of two closed points of a group scheme

If $G$ is a group scheme, it's natural to study it's closed points $Cl(G)$. If $G$ is finite type over a field, then from the general fact that morphisms between algebraic varieties preserve closed ...
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1answer
204 views

Abelian group structure on roots of a polynomial

Assume $f \in \Bbb Z[x]$ is a monic polynomial, s.t for every commutative ring $R$, the solutions of $f(x)=0$ in $R$ can be endowed with an abelian group structure that is functorial respect to $R$. ...
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1answer
126 views

Generalization of the group scheme $\textrm{Spec}(A(T))$

Let $A$ be a ring, and let $S$ be the scheme $\textrm{Spec }A$. Then the $S$-scheme $G = \textrm{Spec }A[T]$ has the structure of a group scheme over $S$, via the morphisms of $S$-schemes $$m: G \...
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1answer
119 views

Group scheme endomorphisms of $G_a$ and $G_m$

I am working through some exercises on group schemes and had a few questions. $k$ is a ring (commutative with identity, always), $G_a = \textrm{Spec }k[T], G_m = \textrm{Spec } k[T,T^{-1}] = D(T)$, ...
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118 views

Smoothness of algebraic varieties: is there a finite type $k$-(group)scheme (variety?) $X$ without a regular point?

Recall that $x$ is regular point if the local ring $\mathcal{O}_{x}$ is a regular local ring. I ask because secretly I'm thinking $X_k=G$ is an algebraic group over $k$, so that the existence one ...
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61 views

Group Schemes and rational points

Suppose $0\to F \to G \to H \to 0$ is a short exact sequence of group schemes over a field $k$. Then $0 \to F(k) \to G(k) \to H(k) \to 0$ is not exact ($G(k) \to H(k)$ is not surjective). However, is ...
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1answer
146 views

Which ring represents the kernel of this homomorphism of group schemes?

Let $R = \mathbb{Z}[X,X^{-1}]$, and let $\mathbb{G}_m = \textrm{Spec } R$. If $A$ is a ring, the set of $A$-rational points $$\textrm{G}_m(A) = \textrm{Hom}_{\textrm{Sch}}(\textrm{Spec } A, \mathbb{G}...
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51 views

Reconstructing a subscheme from fibers

I'm not an expert in scheme theory, so I just ask this question to understand some peculiarity of this language. The situation I was thinking about is really concrete. Let me consider a group scheme $...
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119 views

Is a quotient of group schemes well defined?

I will first define the notions of exact,surjective for group schemes and then ask my question. Let $B$ and $C$ be fppf group schemes over S. The book I am reading defines a homomorphism $f:B\to C$ ...
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1answer
267 views

References for the representation theory of some algebraic groups

I'm interested in learning more about the representation theory of the group schemes $SL_2$, $GL_2$ and $\mathbb{G}_m$ (as group schemes over the rationals, say) - specifically, the structure of all ...
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1answer
160 views

What does $\mathbf G_m$ really mean?

My understanding is that $\mathbf G_m$ stands for $k^*$ (multiplicative group of the field $k$) as a group scheme. But I have also seen symobols like $H^1(X_{et},\mathbf G_m)$? Is this taking about ...
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69 views

Prove that a group scheme over a field is separated.

How to prove that a group scheme over a field is separated, i.e. to show that the unit $e:\mathrm{Spec}(K)\to G$ is a closed immersion.
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1answer
88 views

Why are representations of a finite constant group scheme determined by rational points?

I am trying to solve the following problem (3.3 in Waterhouse's Introduction to Affine Group Schemes): Let $\underline{\Gamma}$ be a finite constant group scheme over a field $k$ (corresponding to ...
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1answer
235 views

Representable functor from scheme to Groups

I want to do the following question (6.6k) in Vakli's Foundations of algebraic geometry Suppose we have a contravariant functor $F$ from $Sch$ to $Groups$. Suppose further that $F$ composed with ...
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1answer
85 views

Inverse limit of Hopfalgebras

My Question relates to Corollary 2.7 of http://www.jmilne.org/math/xnotes/tc.pdf So Let $k$ be a field and $\mathbb{G}_i$ be an projective system of affine $k$-groupschemes. I want to know if the ...
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0answers
105 views

When is Cartier dual of a finite group etale?

I am trying to solve the following exercise from Waterhouse: Introduction to affine group schemes (Chapter 6, Ex. 12 on page 53) without any success. Let $char(k)=p >0$ and let $G$ be an abelian ...
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1answer
144 views

Existence of categorical quotient $X/\mathbb{G}_{m,A}$.

Let $A$ be an $\bar{\mathbb{F}}_p$-Algebra of finite type (one might assume $A$ to be reduced). Let $X \subset \mathbb{A}_A^d\backslash \{0\}$ be a closed $A$-subscheme together with a group action of ...