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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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24 views

Given the multiplication morphism on the scheme functor, how to get it as scheme morphism?

I am studying group schemes. I would say I understand the definition of an $S$-group scheme as to give an $S$-scheme together with morphisms for multiplication law, identity, and inverse and also the ...
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2answers
99 views

Kernel of an algebra map and module of Kahler Differentials

Let $A$ be a $k$-algebra, $f:A\rightarrow k$ an algebra map with kernel $I$. I'd like to prove that $\Omega_A\otimes_f k $ is canonically isomorphic to $I/I^2$. This is from W.C.Waterhouse Intro to ...
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63 views

Kahler differentials of a Hopf Algebra

Let $A$ be a $k$-Hopf Algebra over some ring $k$, with augmentation ideal $J_A=$ ker $(\epsilon:A\rightarrow k)$ I would like to prove that the module of Khaler Differentials $\Omega_A$ of $A$ over $...
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30 views

Geometric invariant theory Affine Quotients

Let $V=Spec(R[V])$ an affine variey and $G$ a linear reductive group action on $V$. (I'm not sure if the linear reductive condition is neccessary for the question). According to GIT the "quotient" $V/...
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1answer
39 views

Affine Group Scheme Definition

From what I understand, a affine group scheme $G$ should be an affine scheme on which there exists a group structure in the sense that $$ \phi: G \times_k G \to G $$ is also a morphism of groups. ...
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65 views

Representations of $\mathbb{G}_m$

I know that the multiplicative affine group scheme $\mathbb{G}_m$ is diagonalizable, since the algebra that represents it is $k[X,X^{-1}]$, which is isomorphic to the group algebra $k[\mathbb{Z}]$. ...
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39 views

Characters of $\mathbb{G}_m$

Fix a field $k$. Let $\mathbb{G}_m$ be the multiplicative affine group scheme over $k$. A $k-$character $\chi$ of $\mathbb{G}_m$ is an endomorphism of affine group schemes $\mathbb{G}_m \to \mathbb{G}...
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1answer
73 views

Galois Action on underlying Topological space of a Group Scheme

I have a question arising from the answer of following thread: https://mathoverflow.net/questions/324887/why-is-for-a-group-scheme-of-finite-type-smooth-resp-irreducible-equivale Let $G/K$ be a group ...
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0answers
54 views

Action on Group Scheme

I have question which arises from the answer of my former thread: https://mathoverflow.net/questions/324887/why-is-for-a-group-scheme-of-finite-type-smooth-resp-irreducible-equivale Let $G/K$ be a ...
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1answer
60 views

Definition of Functor in Waterhouse

On page 21 of Waterhouse's Introduction to Affine Group Schemes he defines (on objects, which are $k$-algebras $R$) a functor $GL_V(R) = \mathrm{Aut}_R(V\otimes R)$ where $V$ is a fixed $k$-module ($k$...
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37 views

If $G$ is a smooth scheme over $S$ of characteristic $p$, is the relative Frobenius morphism $F_{G/S}$ faithfully flat?

Let $G$ be a smooth scheme over $S$ of characteristic $p$, do we have that the reltaive Frobenius morphism $F_{G/S}$ is faithfully flat? There is an excersice in Liu's book saying that this is true ...
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1answer
38 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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38 views

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
55 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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27 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
78 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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38 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
57 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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1answer
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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34 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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36 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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54 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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1answer
136 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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46 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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1answer
142 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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0answers
38 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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1answer
102 views

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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2answers
219 views

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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41 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
60 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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29 views

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
81 views

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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1answer
50 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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129 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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0answers
94 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
42 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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0answers
92 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
88 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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0answers
84 views

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
119 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
91 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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125 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
54 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
217 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
134 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
122 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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2answers
124 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 ...
3
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0answers
64 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
148 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}...